Optimal. Leaf size=86 \[ \frac{(2 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{2 c^{5/2}}-\frac{2 b c-3 a d}{2 c^2 \sqrt{c+\frac{d}{x^2}}}+\frac{a x^2}{2 c \sqrt{c+\frac{d}{x^2}}} \]
[Out]
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Rubi [A] time = 0.189811, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{(2 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{2 c^{5/2}}-\frac{2 b c-3 a d}{2 c^2 \sqrt{c+\frac{d}{x^2}}}+\frac{a x^2}{2 c \sqrt{c+\frac{d}{x^2}}} \]
Antiderivative was successfully verified.
[In] Int[((a + b/x^2)*x)/(c + d/x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 14.7331, size = 73, normalized size = 0.85 \[ \frac{a x^{2}}{2 c \sqrt{c + \frac{d}{x^{2}}}} + \frac{\frac{3 a d}{2} - b c}{c^{2} \sqrt{c + \frac{d}{x^{2}}}} - \frac{\left (\frac{3 a d}{2} - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + \frac{d}{x^{2}}}}{\sqrt{c}} \right )}}{c^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x**2)*x/(c+d/x**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.105913, size = 89, normalized size = 1.03 \[ \frac{\sqrt{c} x \left (a c x^2+3 a d-2 b c\right )+\sqrt{c x^2+d} (2 b c-3 a d) \log \left (\sqrt{c} \sqrt{c x^2+d}+c x\right )}{2 c^{5/2} x \sqrt{c+\frac{d}{x^2}}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b/x^2)*x)/(c + d/x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.016, size = 116, normalized size = 1.4 \[{\frac{c{x}^{2}+d}{2\,{x}^{3}} \left ({x}^{3}a{c}^{{\frac{7}{2}}}+3\,adx{c}^{5/2}-2\,xb{c}^{7/2}+2\,b\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ){c}^{3}\sqrt{c{x}^{2}+d}-3\,ad\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ){c}^{2}\sqrt{c{x}^{2}+d} \right ) \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}{c}^{-{\frac{9}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x^2)*x/(c+d/x^2)^(3/2),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((a + b/x^2)*x/(c + d/x^2)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.238511, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (2 \, b c d - 3 \, a d^{2} +{\left (2 \, b c^{2} - 3 \, a c d\right )} x^{2}\right )} \sqrt{c} \log \left (2 \, c x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} -{\left (2 \, c x^{2} + d\right )} \sqrt{c}\right ) - 2 \,{\left (a c^{2} x^{4} -{\left (2 \, b c^{2} - 3 \, a c d\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{4 \,{\left (c^{4} x^{2} + c^{3} d\right )}}, -\frac{{\left (2 \, b c d - 3 \, a d^{2} +{\left (2 \, b c^{2} - 3 \, a c d\right )} x^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c}}{\sqrt{\frac{c x^{2} + d}{x^{2}}}}\right ) -{\left (a c^{2} x^{4} -{\left (2 \, b c^{2} - 3 \, a c d\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{2 \,{\left (c^{4} x^{2} + c^{3} d\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)*x/(c + d/x^2)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 16.3772, size = 264, normalized size = 3.07 \[ a \left (\frac{x^{3}}{2 c \sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{3 \sqrt{d} x}{2 c^{2} \sqrt{\frac{c x^{2}}{d} + 1}} - \frac{3 d \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{2 c^{\frac{5}{2}}}\right ) + b \left (- \frac{2 c^{3} x^{2} \sqrt{1 + \frac{d}{c x^{2}}}}{2 c^{\frac{9}{2}} x^{2} + 2 c^{\frac{7}{2}} d} - \frac{c^{3} x^{2} \log{\left (\frac{d}{c x^{2}} \right )}}{2 c^{\frac{9}{2}} x^{2} + 2 c^{\frac{7}{2}} d} + \frac{2 c^{3} x^{2} \log{\left (\sqrt{1 + \frac{d}{c x^{2}}} + 1 \right )}}{2 c^{\frac{9}{2}} x^{2} + 2 c^{\frac{7}{2}} d} - \frac{c^{2} d \log{\left (\frac{d}{c x^{2}} \right )}}{2 c^{\frac{9}{2}} x^{2} + 2 c^{\frac{7}{2}} d} + \frac{2 c^{2} d \log{\left (\sqrt{1 + \frac{d}{c x^{2}}} + 1 \right )}}{2 c^{\frac{9}{2}} x^{2} + 2 c^{\frac{7}{2}} d}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x**2)*x/(c+d/x**2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.249726, size = 182, normalized size = 2.12 \[ -\frac{1}{2} \, d{\left (\frac{{\left (2 \, b c - 3 \, a d\right )} \arctan \left (\frac{\sqrt{\frac{c x^{2} + d}{x^{2}}}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{2} d} + \frac{2 \, b c^{2} - 2 \, a c d - \frac{2 \,{\left (c x^{2} + d\right )} b c}{x^{2}} + \frac{3 \,{\left (c x^{2} + d\right )} a d}{x^{2}}}{{\left (c \sqrt{\frac{c x^{2} + d}{x^{2}}} - \frac{{\left (c x^{2} + d\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{x^{2}}\right )} c^{2} d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)*x/(c + d/x^2)^(3/2),x, algorithm="giac")
[Out]