3.331 \(\int \frac{x^{3/2}}{\left (1+x^2\right )^3} \, dx\)

Optimal. Leaf size=129 \[ \frac{\sqrt{x}}{16 \left (x^2+1\right )}-\frac{\sqrt{x}}{4 \left (x^2+1\right )^2}-\frac{3 \log \left (x-\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}+\frac{3 \log \left (x+\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}-\frac{3 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}+\frac{3 \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{32 \sqrt{2}} \]

[Out]

-Sqrt[x]/(4*(1 + x^2)^2) + Sqrt[x]/(16*(1 + x^2)) - (3*ArcTan[1 - Sqrt[2]*Sqrt[x
]])/(32*Sqrt[2]) + (3*ArcTan[1 + Sqrt[2]*Sqrt[x]])/(32*Sqrt[2]) - (3*Log[1 - Sqr
t[2]*Sqrt[x] + x])/(64*Sqrt[2]) + (3*Log[1 + Sqrt[2]*Sqrt[x] + x])/(64*Sqrt[2])

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Rubi [A]  time = 0.159877, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.692 \[ \frac{\sqrt{x}}{16 \left (x^2+1\right )}-\frac{\sqrt{x}}{4 \left (x^2+1\right )^2}-\frac{3 \log \left (x-\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}+\frac{3 \log \left (x+\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}-\frac{3 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}+\frac{3 \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{32 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[x^(3/2)/(1 + x^2)^3,x]

[Out]

-Sqrt[x]/(4*(1 + x^2)^2) + Sqrt[x]/(16*(1 + x^2)) - (3*ArcTan[1 - Sqrt[2]*Sqrt[x
]])/(32*Sqrt[2]) + (3*ArcTan[1 + Sqrt[2]*Sqrt[x]])/(32*Sqrt[2]) - (3*Log[1 - Sqr
t[2]*Sqrt[x] + x])/(64*Sqrt[2]) + (3*Log[1 + Sqrt[2]*Sqrt[x] + x])/(64*Sqrt[2])

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Rubi in Sympy [A]  time = 20.5745, size = 116, normalized size = 0.9 \[ \frac{\sqrt{x}}{16 \left (x^{2} + 1\right )} - \frac{\sqrt{x}}{4 \left (x^{2} + 1\right )^{2}} - \frac{3 \sqrt{2} \log{\left (- \sqrt{2} \sqrt{x} + x + 1 \right )}}{128} + \frac{3 \sqrt{2} \log{\left (\sqrt{2} \sqrt{x} + x + 1 \right )}}{128} + \frac{3 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{64} + \frac{3 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{64} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(3/2)/(x**2+1)**3,x)

[Out]

sqrt(x)/(16*(x**2 + 1)) - sqrt(x)/(4*(x**2 + 1)**2) - 3*sqrt(2)*log(-sqrt(2)*sqr
t(x) + x + 1)/128 + 3*sqrt(2)*log(sqrt(2)*sqrt(x) + x + 1)/128 + 3*sqrt(2)*atan(
sqrt(2)*sqrt(x) - 1)/64 + 3*sqrt(2)*atan(sqrt(2)*sqrt(x) + 1)/64

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Mathematica [A]  time = 0.0800409, size = 121, normalized size = 0.94 \[ \frac{1}{128} \left (\frac{8 \sqrt{x}}{x^2+1}-\frac{32 \sqrt{x}}{\left (x^2+1\right )^2}-3 \sqrt{2} \log \left (x-\sqrt{2} \sqrt{x}+1\right )+3 \sqrt{2} \log \left (x+\sqrt{2} \sqrt{x}+1\right )-6 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )+6 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^(3/2)/(1 + x^2)^3,x]

[Out]

((-32*Sqrt[x])/(1 + x^2)^2 + (8*Sqrt[x])/(1 + x^2) - 6*Sqrt[2]*ArcTan[1 - Sqrt[2
]*Sqrt[x]] + 6*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[x]] - 3*Sqrt[2]*Log[1 - Sqrt[2]*S
qrt[x] + x] + 3*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[x] + x])/128

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Maple [A]  time = 0.015, size = 82, normalized size = 0.6 \[ 2\,{\frac{1}{ \left ({x}^{2}+1 \right ) ^{2}} \left ( 1/32\,{x}^{5/2}-{\frac{3\,\sqrt{x}}{32}} \right ) }+{\frac{3\,\sqrt{2}}{64}\arctan \left ( \sqrt{2}\sqrt{x}-1 \right ) }+{\frac{3\,\sqrt{2}}{128}\ln \left ({1 \left ( 1+x+\sqrt{2}\sqrt{x} \right ) \left ( 1+x-\sqrt{2}\sqrt{x} \right ) ^{-1}} \right ) }+{\frac{3\,\sqrt{2}}{64}\arctan \left ( 1+\sqrt{2}\sqrt{x} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)/(x^2+1)^3,x)

[Out]

2*(1/32*x^(5/2)-3/32*x^(1/2))/(x^2+1)^2+3/64*arctan(2^(1/2)*x^(1/2)-1)*2^(1/2)+3
/128*2^(1/2)*ln((1+x+2^(1/2)*x^(1/2))/(1+x-2^(1/2)*x^(1/2)))+3/64*arctan(1+2^(1/
2)*x^(1/2))*2^(1/2)

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Maxima [A]  time = 1.48985, size = 131, normalized size = 1.02 \[ \frac{3}{64} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) + \frac{3}{64} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) + \frac{3}{128} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{3}{128} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{x^{\frac{5}{2}} - 3 \, \sqrt{x}}{16 \,{\left (x^{4} + 2 \, x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3/2)/(x^2 + 1)^3,x, algorithm="maxima")

[Out]

3/64*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) + 3/64*sqrt(2)*arctan(-1/
2*sqrt(2)*(sqrt(2) - 2*sqrt(x))) + 3/128*sqrt(2)*log(sqrt(2)*sqrt(x) + x + 1) -
3/128*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1) + 1/16*(x^(5/2) - 3*sqrt(x))/(x^4 +
2*x^2 + 1)

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Fricas [A]  time = 0.245325, size = 225, normalized size = 1.74 \[ -\frac{12 \, \sqrt{2}{\left (x^{4} + 2 \, x^{2} + 1\right )} \arctan \left (\frac{1}{\sqrt{2} \sqrt{x} + \sqrt{2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2} + 1}\right ) + 12 \, \sqrt{2}{\left (x^{4} + 2 \, x^{2} + 1\right )} \arctan \left (\frac{1}{\sqrt{2} \sqrt{x} + \sqrt{-2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2} - 1}\right ) - 3 \, \sqrt{2}{\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2\right ) + 3 \, \sqrt{2}{\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (-2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2\right ) - 8 \,{\left (x^{2} - 3\right )} \sqrt{x}}{128 \,{\left (x^{4} + 2 \, x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3/2)/(x^2 + 1)^3,x, algorithm="fricas")

[Out]

-1/128*(12*sqrt(2)*(x^4 + 2*x^2 + 1)*arctan(1/(sqrt(2)*sqrt(x) + sqrt(2*sqrt(2)*
sqrt(x) + 2*x + 2) + 1)) + 12*sqrt(2)*(x^4 + 2*x^2 + 1)*arctan(1/(sqrt(2)*sqrt(x
) + sqrt(-2*sqrt(2)*sqrt(x) + 2*x + 2) - 1)) - 3*sqrt(2)*(x^4 + 2*x^2 + 1)*log(2
*sqrt(2)*sqrt(x) + 2*x + 2) + 3*sqrt(2)*(x^4 + 2*x^2 + 1)*log(-2*sqrt(2)*sqrt(x)
+ 2*x + 2) - 8*(x^2 - 3)*sqrt(x))/(x^4 + 2*x^2 + 1)

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Sympy [A]  time = 83.8601, size = 481, normalized size = 3.73 \[ \frac{8 x^{\frac{5}{2}}}{128 x^{4} + 256 x^{2} + 128} - \frac{24 \sqrt{x}}{128 x^{4} + 256 x^{2} + 128} - \frac{3 \sqrt{2} x^{4} \log{\left (- 4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{3 \sqrt{2} x^{4} \log{\left (4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{6 \sqrt{2} x^{4} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{6 \sqrt{2} x^{4} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{128 x^{4} + 256 x^{2} + 128} - \frac{6 \sqrt{2} x^{2} \log{\left (- 4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{6 \sqrt{2} x^{2} \log{\left (4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{12 \sqrt{2} x^{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{12 \sqrt{2} x^{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{128 x^{4} + 256 x^{2} + 128} - \frac{3 \sqrt{2} \log{\left (- 4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{3 \sqrt{2} \log{\left (4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{6 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{6 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{128 x^{4} + 256 x^{2} + 128} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(3/2)/(x**2+1)**3,x)

[Out]

8*x**(5/2)/(128*x**4 + 256*x**2 + 128) - 24*sqrt(x)/(128*x**4 + 256*x**2 + 128)
- 3*sqrt(2)*x**4*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**4 + 256*x**2 + 128) +
3*sqrt(2)*x**4*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**4 + 256*x**2 + 128) + 6
*sqrt(2)*x**4*atan(sqrt(2)*sqrt(x) - 1)/(128*x**4 + 256*x**2 + 128) + 6*sqrt(2)*
x**4*atan(sqrt(2)*sqrt(x) + 1)/(128*x**4 + 256*x**2 + 128) - 6*sqrt(2)*x**2*log(
-4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**4 + 256*x**2 + 128) + 6*sqrt(2)*x**2*log(4
*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**4 + 256*x**2 + 128) + 12*sqrt(2)*x**2*atan(s
qrt(2)*sqrt(x) - 1)/(128*x**4 + 256*x**2 + 128) + 12*sqrt(2)*x**2*atan(sqrt(2)*s
qrt(x) + 1)/(128*x**4 + 256*x**2 + 128) - 3*sqrt(2)*log(-4*sqrt(2)*sqrt(x) + 4*x
+ 4)/(128*x**4 + 256*x**2 + 128) + 3*sqrt(2)*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(
128*x**4 + 256*x**2 + 128) + 6*sqrt(2)*atan(sqrt(2)*sqrt(x) - 1)/(128*x**4 + 256
*x**2 + 128) + 6*sqrt(2)*atan(sqrt(2)*sqrt(x) + 1)/(128*x**4 + 256*x**2 + 128)

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GIAC/XCAS [A]  time = 0.211146, size = 124, normalized size = 0.96 \[ \frac{3}{64} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) + \frac{3}{64} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) + \frac{3}{128} \, \sqrt{2}{\rm ln}\left (\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{3}{128} \, \sqrt{2}{\rm ln}\left (-\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{x^{\frac{5}{2}} - 3 \, \sqrt{x}}{16 \,{\left (x^{2} + 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3/2)/(x^2 + 1)^3,x, algorithm="giac")

[Out]

3/64*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) + 3/64*sqrt(2)*arctan(-1/
2*sqrt(2)*(sqrt(2) - 2*sqrt(x))) + 3/128*sqrt(2)*ln(sqrt(2)*sqrt(x) + x + 1) - 3
/128*sqrt(2)*ln(-sqrt(2)*sqrt(x) + x + 1) + 1/16*(x^(5/2) - 3*sqrt(x))/(x^2 + 1)
^2