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

Optimal. Leaf size=129 \[ \frac{5 x^{3/2}}{16 \left (x^2+1\right )}+\frac{x^{3/2}}{4 \left (x^2+1\right )^2}+\frac{5 \log \left (x-\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}-\frac{5 \log \left (x+\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}-\frac{5 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}+\frac{5 \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{32 \sqrt{2}} \]

[Out]

x^(3/2)/(4*(1 + x^2)^2) + (5*x^(3/2))/(16*(1 + x^2)) - (5*ArcTan[1 - Sqrt[2]*Sqr
t[x]])/(32*Sqrt[2]) + (5*ArcTan[1 + Sqrt[2]*Sqrt[x]])/(32*Sqrt[2]) + (5*Log[1 -
Sqrt[2]*Sqrt[x] + x])/(64*Sqrt[2]) - (5*Log[1 + Sqrt[2]*Sqrt[x] + x])/(64*Sqrt[2
])

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Rubi [A]  time = 0.168566, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615 \[ \frac{5 x^{3/2}}{16 \left (x^2+1\right )}+\frac{x^{3/2}}{4 \left (x^2+1\right )^2}+\frac{5 \log \left (x-\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}-\frac{5 \log \left (x+\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}-\frac{5 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}+\frac{5 \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{32 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/(1 + x^2)^3,x]

[Out]

x^(3/2)/(4*(1 + x^2)^2) + (5*x^(3/2))/(16*(1 + x^2)) - (5*ArcTan[1 - Sqrt[2]*Sqr
t[x]])/(32*Sqrt[2]) + (5*ArcTan[1 + Sqrt[2]*Sqrt[x]])/(32*Sqrt[2]) + (5*Log[1 -
Sqrt[2]*Sqrt[x] + x])/(64*Sqrt[2]) - (5*Log[1 + Sqrt[2]*Sqrt[x] + x])/(64*Sqrt[2
])

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Rubi in Sympy [A]  time = 21.5784, size = 117, normalized size = 0.91 \[ \frac{5 x^{\frac{3}{2}}}{16 \left (x^{2} + 1\right )} + \frac{x^{\frac{3}{2}}}{4 \left (x^{2} + 1\right )^{2}} + \frac{5 \sqrt{2} \log{\left (- \sqrt{2} \sqrt{x} + x + 1 \right )}}{128} - \frac{5 \sqrt{2} \log{\left (\sqrt{2} \sqrt{x} + x + 1 \right )}}{128} + \frac{5 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{64} + \frac{5 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{64} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*x**(3/2)/(16*(x**2 + 1)) + x**(3/2)/(4*(x**2 + 1)**2) + 5*sqrt(2)*log(-sqrt(2)
*sqrt(x) + x + 1)/128 - 5*sqrt(2)*log(sqrt(2)*sqrt(x) + x + 1)/128 + 5*sqrt(2)*a
tan(sqrt(2)*sqrt(x) - 1)/64 + 5*sqrt(2)*atan(sqrt(2)*sqrt(x) + 1)/64

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Mathematica [A]  time = 0.0829428, size = 121, normalized size = 0.94 \[ \frac{1}{128} \left (\frac{40 x^{3/2}}{x^2+1}+\frac{32 x^{3/2}}{\left (x^2+1\right )^2}+5 \sqrt{2} \log \left (x-\sqrt{2} \sqrt{x}+1\right )-5 \sqrt{2} \log \left (x+\sqrt{2} \sqrt{x}+1\right )-10 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )+10 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((32*x^(3/2))/(1 + x^2)^2 + (40*x^(3/2))/(1 + x^2) - 10*Sqrt[2]*ArcTan[1 - Sqrt[
2]*Sqrt[x]] + 10*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[x]] + 5*Sqrt[2]*Log[1 - Sqrt[2]
*Sqrt[x] + x] - 5*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[x] + x])/128

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Maple [A]  time = 0.01, size = 86, normalized size = 0.7 \[{\frac{1}{4\, \left ({x}^{2}+1 \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{5}{16\,{x}^{2}+16}{x}^{{\frac{3}{2}}}}+{\frac{5\,\sqrt{2}}{64}\arctan \left ( 1+\sqrt{2}\sqrt{x} \right ) }+{\frac{5\,\sqrt{2}}{64}\arctan \left ( \sqrt{2}\sqrt{x}-1 \right ) }+{\frac{5\,\sqrt{2}}{128}\ln \left ({1 \left ( 1+x-\sqrt{2}\sqrt{x} \right ) \left ( 1+x+\sqrt{2}\sqrt{x} \right ) ^{-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(x^2+1)^3,x)

[Out]

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

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Maxima [A]  time = 1.49829, size = 134, normalized size = 1.04 \[ \frac{5}{64} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) + \frac{5}{64} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) - \frac{5}{128} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{5}{128} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{5 \, x^{\frac{7}{2}} + 9 \, x^{\frac{3}{2}}}{16 \,{\left (x^{4} + 2 \, x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/64*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) + 5/64*sqrt(2)*arctan(-1/
2*sqrt(2)*(sqrt(2) - 2*sqrt(x))) - 5/128*sqrt(2)*log(sqrt(2)*sqrt(x) + x + 1) +
5/128*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1) + 1/16*(5*x^(7/2) + 9*x^(3/2))/(x^4
+ 2*x^2 + 1)

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Fricas [A]  time = 0.249519, size = 231, normalized size = 1.79 \[ -\frac{20 \, \sqrt{2}{\left (x^{4} + 2 \, x^{2} + 1\right )} \arctan \left (\frac{1}{\sqrt{2} \sqrt{x} + \sqrt{2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2} + 1}\right ) + 20 \, \sqrt{2}{\left (x^{4} + 2 \, x^{2} + 1\right )} \arctan \left (\frac{1}{\sqrt{2} \sqrt{x} + \sqrt{-2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2} - 1}\right ) + 5 \, \sqrt{2}{\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2\right ) - 5 \, \sqrt{2}{\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (-2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2\right ) - 8 \,{\left (5 \, x^{3} + 9 \, x\right )} \sqrt{x}}{128 \,{\left (x^{4} + 2 \, x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/128*(20*sqrt(2)*(x^4 + 2*x^2 + 1)*arctan(1/(sqrt(2)*sqrt(x) + sqrt(2*sqrt(2)*
sqrt(x) + 2*x + 2) + 1)) + 20*sqrt(2)*(x^4 + 2*x^2 + 1)*arctan(1/(sqrt(2)*sqrt(x
) + sqrt(-2*sqrt(2)*sqrt(x) + 2*x + 2) - 1)) + 5*sqrt(2)*(x^4 + 2*x^2 + 1)*log(2
*sqrt(2)*sqrt(x) + 2*x + 2) - 5*sqrt(2)*(x^4 + 2*x^2 + 1)*log(-2*sqrt(2)*sqrt(x)
+ 2*x + 2) - 8*(5*x^3 + 9*x)*sqrt(x))/(x^4 + 2*x^2 + 1)

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Sympy [A]  time = 46.9091, size = 481, normalized size = 3.73 \[ \frac{40 x^{\frac{7}{2}}}{128 x^{4} + 256 x^{2} + 128} + \frac{72 x^{\frac{3}{2}}}{128 x^{4} + 256 x^{2} + 128} + \frac{5 \sqrt{2} x^{4} \log{\left (- 4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} - \frac{5 \sqrt{2} x^{4} \log{\left (4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{10 \sqrt{2} x^{4} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{10 \sqrt{2} x^{4} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{10 \sqrt{2} x^{2} \log{\left (- 4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} - \frac{10 \sqrt{2} x^{2} \log{\left (4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{20 \sqrt{2} x^{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{20 \sqrt{2} x^{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{5 \sqrt{2} \log{\left (- 4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} - \frac{5 \sqrt{2} \log{\left (4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{10 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{10 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{128 x^{4} + 256 x^{2} + 128} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

40*x**(7/2)/(128*x**4 + 256*x**2 + 128) + 72*x**(3/2)/(128*x**4 + 256*x**2 + 128
) + 5*sqrt(2)*x**4*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**4 + 256*x**2 + 128)
- 5*sqrt(2)*x**4*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**4 + 256*x**2 + 128) +
10*sqrt(2)*x**4*atan(sqrt(2)*sqrt(x) - 1)/(128*x**4 + 256*x**2 + 128) + 10*sqrt
(2)*x**4*atan(sqrt(2)*sqrt(x) + 1)/(128*x**4 + 256*x**2 + 128) + 10*sqrt(2)*x**2
*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**4 + 256*x**2 + 128) - 10*sqrt(2)*x**2
*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**4 + 256*x**2 + 128) + 20*sqrt(2)*x**2*
atan(sqrt(2)*sqrt(x) - 1)/(128*x**4 + 256*x**2 + 128) + 20*sqrt(2)*x**2*atan(sqr
t(2)*sqrt(x) + 1)/(128*x**4 + 256*x**2 + 128) + 5*sqrt(2)*log(-4*sqrt(2)*sqrt(x)
+ 4*x + 4)/(128*x**4 + 256*x**2 + 128) - 5*sqrt(2)*log(4*sqrt(2)*sqrt(x) + 4*x
+ 4)/(128*x**4 + 256*x**2 + 128) + 10*sqrt(2)*atan(sqrt(2)*sqrt(x) - 1)/(128*x**
4 + 256*x**2 + 128) + 10*sqrt(2)*atan(sqrt(2)*sqrt(x) + 1)/(128*x**4 + 256*x**2
+ 128)

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GIAC/XCAS [A]  time = 0.211035, size = 127, normalized size = 0.98 \[ \frac{5}{64} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) + \frac{5}{64} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) - \frac{5}{128} \, \sqrt{2}{\rm ln}\left (\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{5}{128} \, \sqrt{2}{\rm ln}\left (-\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{5 \, x^{\frac{7}{2}} + 9 \, x^{\frac{3}{2}}}{16 \,{\left (x^{2} + 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/64*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) + 5/64*sqrt(2)*arctan(-1/
2*sqrt(2)*(sqrt(2) - 2*sqrt(x))) - 5/128*sqrt(2)*ln(sqrt(2)*sqrt(x) + x + 1) + 5
/128*sqrt(2)*ln(-sqrt(2)*sqrt(x) + x + 1) + 1/16*(5*x^(7/2) + 9*x^(3/2))/(x^2 +
1)^2