3.322 \(\int \frac{x^{5/2}}{\left (1+x^2\right )^2} \, dx\)

Optimal. Leaf size=113 \[ -\frac{x^{3/2}}{2 \left (x^2+1\right )}+\frac{3 \log \left (x-\sqrt{2} \sqrt{x}+1\right )}{8 \sqrt{2}}-\frac{3 \log \left (x+\sqrt{2} \sqrt{x}+1\right )}{8 \sqrt{2}}-\frac{3 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{4 \sqrt{2}}+\frac{3 \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{4 \sqrt{2}} \]

[Out]

-x^(3/2)/(2*(1 + x^2)) - (3*ArcTan[1 - Sqrt[2]*Sqrt[x]])/(4*Sqrt[2]) + (3*ArcTan
[1 + Sqrt[2]*Sqrt[x]])/(4*Sqrt[2]) + (3*Log[1 - Sqrt[2]*Sqrt[x] + x])/(8*Sqrt[2]
) - (3*Log[1 + Sqrt[2]*Sqrt[x] + x])/(8*Sqrt[2])

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Rubi [A]  time = 0.146961, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615 \[ -\frac{x^{3/2}}{2 \left (x^2+1\right )}+\frac{3 \log \left (x-\sqrt{2} \sqrt{x}+1\right )}{8 \sqrt{2}}-\frac{3 \log \left (x+\sqrt{2} \sqrt{x}+1\right )}{8 \sqrt{2}}-\frac{3 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{4 \sqrt{2}}+\frac{3 \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{4 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)/(1 + x^2)^2,x]

[Out]

-x^(3/2)/(2*(1 + x^2)) - (3*ArcTan[1 - Sqrt[2]*Sqrt[x]])/(4*Sqrt[2]) + (3*ArcTan
[1 + Sqrt[2]*Sqrt[x]])/(4*Sqrt[2]) + (3*Log[1 - Sqrt[2]*Sqrt[x] + x])/(8*Sqrt[2]
) - (3*Log[1 + Sqrt[2]*Sqrt[x] + x])/(8*Sqrt[2])

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Rubi in Sympy [A]  time = 19.5482, size = 102, normalized size = 0.9 \[ - \frac{x^{\frac{3}{2}}}{2 \left (x^{2} + 1\right )} + \frac{3 \sqrt{2} \log{\left (- \sqrt{2} \sqrt{x} + x + 1 \right )}}{16} - \frac{3 \sqrt{2} \log{\left (\sqrt{2} \sqrt{x} + x + 1 \right )}}{16} + \frac{3 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{8} + \frac{3 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)/(x**2+1)**2,x)

[Out]

-x**(3/2)/(2*(x**2 + 1)) + 3*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1)/16 - 3*sqrt(2
)*log(sqrt(2)*sqrt(x) + x + 1)/16 + 3*sqrt(2)*atan(sqrt(2)*sqrt(x) - 1)/8 + 3*sq
rt(2)*atan(sqrt(2)*sqrt(x) + 1)/8

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Mathematica [A]  time = 0.14101, size = 107, normalized size = 0.95 \[ \frac{1}{16} \left (-\frac{8 x^{3/2}}{x^2+1}+3 \sqrt{2} \log \left (x-\sqrt{2} \sqrt{x}+1\right )-3 \sqrt{2} \log \left (x+\sqrt{2} \sqrt{x}+1\right )-6 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )+6 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^(5/2)/(1 + x^2)^2,x]

[Out]

((-8*x^(3/2))/(1 + x^2) - 6*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[x]] + 6*Sqrt[2]*ArcT
an[1 + Sqrt[2]*Sqrt[x]] + 3*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[x] + x] - 3*Sqrt[2]*Log
[1 + Sqrt[2]*Sqrt[x] + x])/16

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Maple [A]  time = 0.014, size = 74, normalized size = 0.7 \[ -{\frac{1}{2\,{x}^{2}+2}{x}^{{\frac{3}{2}}}}+{\frac{3\,\sqrt{2}}{8}\arctan \left ( \sqrt{2}\sqrt{x}-1 \right ) }+{\frac{3\,\sqrt{2}}{16}\ln \left ({1 \left ( 1+x-\sqrt{2}\sqrt{x} \right ) \left ( 1+x+\sqrt{2}\sqrt{x} \right ) ^{-1}} \right ) }+{\frac{3\,\sqrt{2}}{8}\arctan \left ( 1+\sqrt{2}\sqrt{x} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)/(x^2+1)^2,x)

[Out]

-1/2*x^(3/2)/(x^2+1)+3/8*arctan(2^(1/2)*x^(1/2)-1)*2^(1/2)+3/16*2^(1/2)*ln((1+x-
2^(1/2)*x^(1/2))/(1+x+2^(1/2)*x^(1/2)))+3/8*arctan(1+2^(1/2)*x^(1/2))*2^(1/2)

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Maxima [A]  time = 1.50652, size = 116, normalized size = 1.03 \[ \frac{3}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) + \frac{3}{8} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) - \frac{3}{16} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{3}{16} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{x^{\frac{3}{2}}}{2 \,{\left (x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/(x^2 + 1)^2,x, algorithm="maxima")

[Out]

3/8*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) + 3/8*sqrt(2)*arctan(-1/2*
sqrt(2)*(sqrt(2) - 2*sqrt(x))) - 3/16*sqrt(2)*log(sqrt(2)*sqrt(x) + x + 1) + 3/1
6*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1) - 1/2*x^(3/2)/(x^2 + 1)

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Fricas [A]  time = 0.251427, size = 185, normalized size = 1.64 \[ -\frac{12 \, \sqrt{2}{\left (x^{2} + 1\right )} \arctan \left (\frac{1}{\sqrt{2} \sqrt{x} + \sqrt{2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2} + 1}\right ) + 12 \, \sqrt{2}{\left (x^{2} + 1\right )} \arctan \left (\frac{1}{\sqrt{2} \sqrt{x} + \sqrt{-2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2} - 1}\right ) + 3 \, \sqrt{2}{\left (x^{2} + 1\right )} \log \left (2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2\right ) - 3 \, \sqrt{2}{\left (x^{2} + 1\right )} \log \left (-2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2\right ) + 8 \, x^{\frac{3}{2}}}{16 \,{\left (x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/(x^2 + 1)^2,x, algorithm="fricas")

[Out]

-1/16*(12*sqrt(2)*(x^2 + 1)*arctan(1/(sqrt(2)*sqrt(x) + sqrt(2*sqrt(2)*sqrt(x) +
2*x + 2) + 1)) + 12*sqrt(2)*(x^2 + 1)*arctan(1/(sqrt(2)*sqrt(x) + sqrt(-2*sqrt(
2)*sqrt(x) + 2*x + 2) - 1)) + 3*sqrt(2)*(x^2 + 1)*log(2*sqrt(2)*sqrt(x) + 2*x +
2) - 3*sqrt(2)*(x^2 + 1)*log(-2*sqrt(2)*sqrt(x) + 2*x + 2) + 8*x^(3/2))/(x^2 + 1
)

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Sympy [A]  time = 35.1342, size = 264, normalized size = 2.34 \[ - \frac{8 x^{\frac{3}{2}}}{16 x^{2} + 16} + \frac{3 \sqrt{2} x^{2} \log{\left (- 4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{16 x^{2} + 16} - \frac{3 \sqrt{2} x^{2} \log{\left (4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{16 x^{2} + 16} + \frac{6 \sqrt{2} x^{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{16 x^{2} + 16} + \frac{6 \sqrt{2} x^{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{16 x^{2} + 16} + \frac{3 \sqrt{2} \log{\left (- 4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{16 x^{2} + 16} - \frac{3 \sqrt{2} \log{\left (4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{16 x^{2} + 16} + \frac{6 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{16 x^{2} + 16} + \frac{6 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{16 x^{2} + 16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(5/2)/(x**2+1)**2,x)

[Out]

-8*x**(3/2)/(16*x**2 + 16) + 3*sqrt(2)*x**2*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/(1
6*x**2 + 16) - 3*sqrt(2)*x**2*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(16*x**2 + 16) +
6*sqrt(2)*x**2*atan(sqrt(2)*sqrt(x) - 1)/(16*x**2 + 16) + 6*sqrt(2)*x**2*atan(sq
rt(2)*sqrt(x) + 1)/(16*x**2 + 16) + 3*sqrt(2)*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/
(16*x**2 + 16) - 3*sqrt(2)*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(16*x**2 + 16) + 6*s
qrt(2)*atan(sqrt(2)*sqrt(x) - 1)/(16*x**2 + 16) + 6*sqrt(2)*atan(sqrt(2)*sqrt(x)
+ 1)/(16*x**2 + 16)

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GIAC/XCAS [A]  time = 0.215082, size = 116, normalized size = 1.03 \[ \frac{3}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) + \frac{3}{8} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) - \frac{3}{16} \, \sqrt{2}{\rm ln}\left (\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{3}{16} \, \sqrt{2}{\rm ln}\left (-\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{x^{\frac{3}{2}}}{2 \,{\left (x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/(x^2 + 1)^2,x, algorithm="giac")

[Out]

3/8*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) + 3/8*sqrt(2)*arctan(-1/2*
sqrt(2)*(sqrt(2) - 2*sqrt(x))) - 3/16*sqrt(2)*ln(sqrt(2)*sqrt(x) + x + 1) + 3/16
*sqrt(2)*ln(-sqrt(2)*sqrt(x) + x + 1) - 1/2*x^(3/2)/(x^2 + 1)