3.313 \(\int \frac{x^{7/2}}{1+x^2} \, dx\)

Optimal. Leaf size=108 \[ \frac{2 x^{5/2}}{5}-2 \sqrt{x}-\frac{\log \left (x-\sqrt{2} \sqrt{x}+1\right )}{2 \sqrt{2}}+\frac{\log \left (x+\sqrt{2} \sqrt{x}+1\right )}{2 \sqrt{2}}-\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{\sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{\sqrt{2}} \]

[Out]

-2*Sqrt[x] + (2*x^(5/2))/5 - ArcTan[1 - Sqrt[2]*Sqrt[x]]/Sqrt[2] + ArcTan[1 + Sq
rt[2]*Sqrt[x]]/Sqrt[2] - Log[1 - Sqrt[2]*Sqrt[x] + x]/(2*Sqrt[2]) + Log[1 + Sqrt
[2]*Sqrt[x] + x]/(2*Sqrt[2])

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Rubi [A]  time = 0.145155, antiderivative size = 108, 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 \[ \frac{2 x^{5/2}}{5}-2 \sqrt{x}-\frac{\log \left (x-\sqrt{2} \sqrt{x}+1\right )}{2 \sqrt{2}}+\frac{\log \left (x+\sqrt{2} \sqrt{x}+1\right )}{2 \sqrt{2}}-\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{\sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[x^(7/2)/(1 + x^2),x]

[Out]

-2*Sqrt[x] + (2*x^(5/2))/5 - ArcTan[1 - Sqrt[2]*Sqrt[x]]/Sqrt[2] + ArcTan[1 + Sq
rt[2]*Sqrt[x]]/Sqrt[2] - Log[1 - Sqrt[2]*Sqrt[x] + x]/(2*Sqrt[2]) + Log[1 + Sqrt
[2]*Sqrt[x] + x]/(2*Sqrt[2])

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Rubi in Sympy [A]  time = 19.6065, size = 99, normalized size = 0.92 \[ \frac{2 x^{\frac{5}{2}}}{5} - 2 \sqrt{x} - \frac{\sqrt{2} \log{\left (- \sqrt{2} \sqrt{x} + x + 1 \right )}}{4} + \frac{\sqrt{2} \log{\left (\sqrt{2} \sqrt{x} + x + 1 \right )}}{4} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{2} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(7/2)/(x**2+1),x)

[Out]

2*x**(5/2)/5 - 2*sqrt(x) - sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1)/4 + sqrt(2)*log
(sqrt(2)*sqrt(x) + x + 1)/4 + sqrt(2)*atan(sqrt(2)*sqrt(x) - 1)/2 + sqrt(2)*atan
(sqrt(2)*sqrt(x) + 1)/2

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Mathematica [A]  time = 0.0558188, size = 107, normalized size = 0.99 \[ \frac{1}{20} \left (8 x^{5/2}-40 \sqrt{x}-5 \sqrt{2} \log \left (x-\sqrt{2} \sqrt{x}+1\right )+5 \sqrt{2} \log \left (x+\sqrt{2} \sqrt{x}+1\right )-10 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )+10 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^(7/2)/(1 + x^2),x]

[Out]

(-40*Sqrt[x] + 8*x^(5/2) - 10*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[x]] + 10*Sqrt[2]*A
rcTan[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])/20

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Maple [A]  time = 0.011, size = 72, normalized size = 0.7 \[{\frac{2}{5}{x}^{{\frac{5}{2}}}}-2\,\sqrt{x}+{\frac{\sqrt{2}}{2}\arctan \left ( 1+\sqrt{2}\sqrt{x} \right ) }+{\frac{\sqrt{2}}{2}\arctan \left ( \sqrt{2}\sqrt{x}-1 \right ) }+{\frac{\sqrt{2}}{4}\ln \left ({1 \left ( 1+x+\sqrt{2}\sqrt{x} \right ) \left ( 1+x-\sqrt{2}\sqrt{x} \right ) ^{-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(7/2)/(x^2+1),x)

[Out]

2/5*x^(5/2)-2*x^(1/2)+1/2*arctan(1+2^(1/2)*x^(1/2))*2^(1/2)+1/2*arctan(2^(1/2)*x
^(1/2)-1)*2^(1/2)+1/4*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 = 1.4949, size = 113, normalized size = 1.05 \[ \frac{2}{5} \, x^{\frac{5}{2}} + \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) + \frac{1}{2} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) + \frac{1}{4} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{1}{4} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{x} + x + 1\right ) - 2 \, \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(7/2)/(x^2 + 1),x, algorithm="maxima")

[Out]

2/5*x^(5/2) + 1/2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) + 1/2*sqrt(2
)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(x))) + 1/4*sqrt(2)*log(sqrt(2)*sqrt(x) +
x + 1) - 1/4*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1) - 2*sqrt(x)

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Fricas [A]  time = 0.254332, size = 153, normalized size = 1.42 \[ \frac{2}{5} \,{\left (x^{2} - 5\right )} \sqrt{x} - \sqrt{2} \arctan \left (\frac{1}{\sqrt{2} \sqrt{x} + \sqrt{2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2} + 1}\right ) - \sqrt{2} \arctan \left (\frac{1}{\sqrt{2} \sqrt{x} + \sqrt{-2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2} - 1}\right ) + \frac{1}{4} \, \sqrt{2} \log \left (2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2\right ) - \frac{1}{4} \, \sqrt{2} \log \left (-2 \, \sqrt{2} \sqrt{x} + 2 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(7/2)/(x^2 + 1),x, algorithm="fricas")

[Out]

2/5*(x^2 - 5)*sqrt(x) - sqrt(2)*arctan(1/(sqrt(2)*sqrt(x) + sqrt(2*sqrt(2)*sqrt(
x) + 2*x + 2) + 1)) - sqrt(2)*arctan(1/(sqrt(2)*sqrt(x) + sqrt(-2*sqrt(2)*sqrt(x
) + 2*x + 2) - 1)) + 1/4*sqrt(2)*log(2*sqrt(2)*sqrt(x) + 2*x + 2) - 1/4*sqrt(2)*
log(-2*sqrt(2)*sqrt(x) + 2*x + 2)

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Sympy [A]  time = 22.1255, size = 105, normalized size = 0.97 \[ \frac{2 x^{\frac{5}{2}}}{5} - 2 \sqrt{x} - \frac{\sqrt{2} \log{\left (- 4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{4} + \frac{\sqrt{2} \log{\left (4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{4} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{2} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(7/2)/(x**2+1),x)

[Out]

2*x**(5/2)/5 - 2*sqrt(x) - sqrt(2)*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/4 + sqrt(2)
*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/4 + sqrt(2)*atan(sqrt(2)*sqrt(x) - 1)/2 + sqrt
(2)*atan(sqrt(2)*sqrt(x) + 1)/2

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GIAC/XCAS [A]  time = 0.209826, size = 113, normalized size = 1.05 \[ \frac{2}{5} \, x^{\frac{5}{2}} + \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) + \frac{1}{2} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) + \frac{1}{4} \, \sqrt{2}{\rm ln}\left (\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{1}{4} \, \sqrt{2}{\rm ln}\left (-\sqrt{2} \sqrt{x} + x + 1\right ) - 2 \, \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(7/2)/(x^2 + 1),x, algorithm="giac")

[Out]

2/5*x^(5/2) + 1/2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) + 1/2*sqrt(2
)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(x))) + 1/4*sqrt(2)*ln(sqrt(2)*sqrt(x) +
x + 1) - 1/4*sqrt(2)*ln(-sqrt(2)*sqrt(x) + x + 1) - 2*sqrt(x)