Optimal. Leaf size=92 \[ -\frac{5 b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{2 a^{7/2}}+\frac{5 x^2 \sqrt{a+\frac{b}{x^2}}}{2 a^3}-\frac{5 x^2}{3 a^2 \sqrt{a+\frac{b}{x^2}}}-\frac{x^2}{3 a \left (a+\frac{b}{x^2}\right )^{3/2}} \]
[Out]
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Rubi [A] time = 0.136101, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{5 b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{2 a^{7/2}}+\frac{5 x^2 \sqrt{a+\frac{b}{x^2}}}{2 a^3}-\frac{5 x^2}{3 a^2 \sqrt{a+\frac{b}{x^2}}}-\frac{x^2}{3 a \left (a+\frac{b}{x^2}\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[x/(a + b/x^2)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 12.8146, size = 83, normalized size = 0.9 \[ - \frac{x^{2}}{3 a \left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}} - \frac{5 x^{2}}{3 a^{2} \sqrt{a + \frac{b}{x^{2}}}} + \frac{5 x^{2} \sqrt{a + \frac{b}{x^{2}}}}{2 a^{3}} - \frac{5 b \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{2}}}}{\sqrt{a}} \right )}}{2 a^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(a+b/x**2)**(5/2),x)
[Out]
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Mathematica [A] time = 0.0807643, size = 97, normalized size = 1.05 \[ \frac{\sqrt{a} x \left (3 a^2 x^4+20 a b x^2+15 b^2\right )-15 b \left (a x^2+b\right )^{3/2} \log \left (\sqrt{a} \sqrt{a x^2+b}+a x\right )}{6 a^{7/2} x \sqrt{a+\frac{b}{x^2}} \left (a x^2+b\right )} \]
Antiderivative was successfully verified.
[In] Integrate[x/(a + b/x^2)^(5/2),x]
[Out]
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Maple [A] time = 0.015, size = 85, normalized size = 0.9 \[{\frac{a{x}^{2}+b}{6\,{x}^{5}} \left ( 3\,{x}^{5}{a}^{7/2}+20\,{a}^{5/2}{x}^{3}b+15\,{a}^{3/2}x{b}^{2}-15\,\ln \left ( \sqrt{a}x+\sqrt{a{x}^{2}+b} \right ) \left ( a{x}^{2}+b \right ) ^{3/2}ab \right ) \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{5}{2}}}{a}^{-{\frac{9}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(a+b/x^2)^(5/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(a + b/x^2)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.271512, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (a^{2} b x^{4} + 2 \, a b^{2} x^{2} + b^{3}\right )} \sqrt{a} \log \left (2 \, a x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} -{\left (2 \, a x^{2} + b\right )} \sqrt{a}\right ) + 2 \,{\left (3 \, a^{3} x^{6} + 20 \, a^{2} b x^{4} + 15 \, a b^{2} x^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{12 \,{\left (a^{6} x^{4} + 2 \, a^{5} b x^{2} + a^{4} b^{2}\right )}}, \frac{15 \,{\left (a^{2} b x^{4} + 2 \, a b^{2} x^{2} + b^{3}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a}}{\sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) +{\left (3 \, a^{3} x^{6} + 20 \, a^{2} b x^{4} + 15 \, a b^{2} x^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{6 \,{\left (a^{6} x^{4} + 2 \, a^{5} b x^{2} + a^{4} b^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(a + b/x^2)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 18.7551, size = 819, normalized size = 8.9 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(a+b/x**2)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.272632, size = 151, normalized size = 1.64 \[ \frac{1}{6} \, b{\left (\frac{2 \,{\left (a + \frac{6 \,{\left (a x^{2} + b\right )}}{x^{2}}\right )} x^{2}}{{\left (a x^{2} + b\right )} a^{3} \sqrt{\frac{a x^{2} + b}{x^{2}}}} + \frac{15 \, \arctan \left (\frac{\sqrt{\frac{a x^{2} + b}{x^{2}}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} - \frac{3 \, \sqrt{\frac{a x^{2} + b}{x^{2}}}}{{\left (a - \frac{a x^{2} + b}{x^{2}}\right )} a^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(a + b/x^2)^(5/2),x, algorithm="giac")
[Out]