No crash with versions 12.0 and 11.3 on Windows 7 x64. I've tried evaluating the code both in a fresh and non-fresh Kernel - the result is the same. I can only note that version 12.0 takes 77 seconds, while version 11.3 only 18 seconds for evaluating all the integrals. The result returned by version 12 for the last integral is mush shorter than of version 11.3.

`LeafCount`s for the expressions returned by version 12:

    $Version
    
>     "12.0.0 for Microsoft Windows (64-bit) (April 6, 2019)"

    ClearAll[x, a, b, c, e, d, f];
    LeafCount /@ {Integrate[(1 + x^2)^3/(1 + x^2 + x^4)^(3/2), x],
      Integrate[(1 + x^2)^2/(1 + x^2 + x^4)^(3/2), x],
      Integrate[(1 + x^2)/(1 + x^2 + x^4)^(3/2), x],
      Integrate[(7 + 5*x^2)^3/(2 + 3*x^2 + x^4)^(3/2), x],
      Integrate[(7 + 5*x^2)^2/(2 + 3*x^2 + x^4)^(3/2), x],
      Integrate[(7 + 5*x^2)/(2 + 3*x^2 + x^4)^(3/2), x],
      Integrate[(2 + 3*x^2 + x^4)^(3/2)*(7 + 5*x^2)^3, x],
      Integrate[(2 + 3*x^2 + x^4)^(3/2)*(7 + 5*x^2)^2, x],
      Integrate[(2 + 3*x^2 + x^4)^(3/2)*(7 + 5*x^2), x],
      Integrate[(7 + 5*x^2)^4/(2 + 3*x^2 + x^4)^(3/2), x],
      Integrate[(7 + 5*x^2)^2/(2 + 3*x^2 + x^4)^(3/2), x],
      Integrate[(4 + 3*x^2 + x^4)^(3/2)*(7 + 5*x^2), x],
      Integrate[(d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2)/(f*x)^(1/2), x]}
    
>     {136, 158, 160, 99, 99, 97, 129, 124, 119, 104, 99, 349, 487}

`LeafCount`s for the expressions returned by version 11.3:

    $Version
    
>     "11.3.0 for Microsoft Windows (64-bit) (March 7, 2018)"

    ClearAll[x, a, b, c, e, d, f];
    LeafCount /@ {Integrate[(1 + x^2)^3/(1 + x^2 + x^4)^(3/2), x],
      Integrate[(1 + x^2)^2/(1 + x^2 + x^4)^(3/2), x],
      Integrate[(1 + x^2)/(1 + x^2 + x^4)^(3/2), x],
      Integrate[(7 + 5*x^2)^3/(2 + 3*x^2 + x^4)^(3/2), x],
      Integrate[(7 + 5*x^2)^2/(2 + 3*x^2 + x^4)^(3/2), x],
      Integrate[(7 + 5*x^2)/(2 + 3*x^2 + x^4)^(3/2), x],
      Integrate[(2 + 3*x^2 + x^4)^(3/2)*(7 + 5*x^2)^3, x],
      Integrate[(2 + 3*x^2 + x^4)^(3/2)*(7 + 5*x^2)^2, x],
      Integrate[(2 + 3*x^2 + x^4)^(3/2)*(7 + 5*x^2), x],
      Integrate[(7 + 5*x^2)^4/(2 + 3*x^2 + x^4)^(3/2), x],
      Integrate[(7 + 5*x^2)^2/(2 + 3*x^2 + x^4)^(3/2), x],
      Integrate[(4 + 3*x^2 + x^4)^(3/2)*(7 + 5*x^2), x],
      Integrate[(d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2)/(f*x)^(1/2), x]}
    
>     {136, 158, 160, 99, 99, 97, 129, 124, 119, 104, 99, 349, 3656}