Optimal. Leaf size=123 \[ \frac{d^2 (6 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{16 c^{3/2}}+\frac{d x^2 \sqrt{c+\frac{d}{x^2}} (6 b c-a d)}{16 c}+\frac{x^4 \left (c+\frac{d}{x^2}\right )^{3/2} (6 b c-a d)}{24 c}+\frac{a x^6 \left (c+\frac{d}{x^2}\right )^{5/2}}{6 c} \]
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Rubi [A] time = 0.264568, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{d^2 (6 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{16 c^{3/2}}+\frac{d x^2 \sqrt{c+\frac{d}{x^2}} (6 b c-a d)}{16 c}+\frac{x^4 \left (c+\frac{d}{x^2}\right )^{3/2} (6 b c-a d)}{24 c}+\frac{a x^6 \left (c+\frac{d}{x^2}\right )^{5/2}}{6 c} \]
Antiderivative was successfully verified.
[In] Int[(a + b/x^2)*(c + d/x^2)^(3/2)*x^5,x]
[Out]
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Rubi in Sympy [A] time = 19.1018, size = 105, normalized size = 0.85 \[ \frac{a x^{6} \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{6 c} - \frac{d x^{2} \sqrt{c + \frac{d}{x^{2}}} \left (a d - 6 b c\right )}{16 c} - \frac{x^{4} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}} \left (a d - 6 b c\right )}{24 c} - \frac{d^{2} \left (a d - 6 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + \frac{d}{x^{2}}}}{\sqrt{c}} \right )}}{16 c^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x**2)*(c+d/x**2)**(3/2)*x**5,x)
[Out]
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Mathematica [A] time = 0.158235, size = 124, normalized size = 1.01 \[ \frac{x \sqrt{c+\frac{d}{x^2}} \left (\sqrt{c} x \sqrt{c x^2+d} \left (a \left (8 c^2 x^4+14 c d x^2+3 d^2\right )+6 b c \left (2 c x^2+5 d\right )\right )-3 d^2 (a d-6 b c) \log \left (\sqrt{c} \sqrt{c x^2+d}+c x\right )\right )}{48 c^{3/2} \sqrt{c x^2+d}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b/x^2)*(c + d/x^2)^(3/2)*x^5,x]
[Out]
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Maple [A] time = 0.014, size = 165, normalized size = 1.3 \[{\frac{{x}^{3}}{48} \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{{\frac{3}{2}}} \left ( 8\,ax \left ( c{x}^{2}+d \right ) ^{5/2}{c}^{3/2}-2\,adx \left ( c{x}^{2}+d \right ) ^{3/2}{c}^{3/2}+12\,bx \left ( c{x}^{2}+d \right ) ^{3/2}{c}^{5/2}-3\,a{d}^{2}x\sqrt{c{x}^{2}+d}{c}^{3/2}+18\,bdx\sqrt{c{x}^{2}+d}{c}^{5/2}+18\,b{d}^{2}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ){c}^{2}-3\,a{d}^{3}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ) c \right ) \left ( c{x}^{2}+d \right ) ^{-{\frac{3}{2}}}{c}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x^2)*(c+d/x^2)^(3/2)*x^5,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)*(c + d/x^2)^(3/2)*x^5,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.253723, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (6 \, b c d^{2} - a d^{3}\right )} \sqrt{c} \log \left (2 \, c x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} -{\left (2 \, c x^{2} + d\right )} \sqrt{c}\right ) - 2 \,{\left (8 \, a c^{3} x^{6} + 2 \,{\left (6 \, b c^{3} + 7 \, a c^{2} d\right )} x^{4} + 3 \,{\left (10 \, b c^{2} d + a c d^{2}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{96 \, c^{2}}, -\frac{3 \,{\left (6 \, b c d^{2} - a d^{3}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c}}{\sqrt{\frac{c x^{2} + d}{x^{2}}}}\right ) -{\left (8 \, a c^{3} x^{6} + 2 \,{\left (6 \, b c^{3} + 7 \, a c^{2} d\right )} x^{4} + 3 \,{\left (10 \, b c^{2} d + a c d^{2}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{48 \, c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)*(c + d/x^2)^(3/2)*x^5,x, algorithm="fricas")
[Out]
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Sympy [A] time = 48.9777, size = 253, normalized size = 2.06 \[ \frac{a c^{2} x^{7}}{6 \sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{11 a c \sqrt{d} x^{5}}{24 \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{17 a d^{\frac{3}{2}} x^{3}}{48 \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{a d^{\frac{5}{2}} x}{16 c \sqrt{\frac{c x^{2}}{d} + 1}} - \frac{a d^{3} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{16 c^{\frac{3}{2}}} + \frac{b c^{2} x^{5}}{4 \sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{3 b c \sqrt{d} x^{3}}{8 \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{b d^{\frac{3}{2}} x \sqrt{\frac{c x^{2}}{d} + 1}}{2} + \frac{b d^{\frac{3}{2}} x}{8 \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{3 b d^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{8 \sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x**2)*(c+d/x**2)**(3/2)*x**5,x)
[Out]
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GIAC/XCAS [A] time = 0.235508, size = 197, normalized size = 1.6 \[ \frac{1}{48} \,{\left (2 \,{\left (4 \, a c x^{2}{\rm sign}\left (x\right ) + \frac{6 \, b c^{5}{\rm sign}\left (x\right ) + 7 \, a c^{4} d{\rm sign}\left (x\right )}{c^{4}}\right )} x^{2} + \frac{3 \,{\left (10 \, b c^{4} d{\rm sign}\left (x\right ) + a c^{3} d^{2}{\rm sign}\left (x\right )\right )}}{c^{4}}\right )} \sqrt{c x^{2} + d} x - \frac{{\left (6 \, b c d^{2}{\rm sign}\left (x\right ) - a d^{3}{\rm sign}\left (x\right )\right )}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + d} \right |}\right )}{16 \, c^{\frac{3}{2}}} + \frac{{\left (6 \, b c d^{2}{\rm ln}\left (\sqrt{d}\right ) - a d^{3}{\rm ln}\left (\sqrt{d}\right )\right )}{\rm sign}\left (x\right )}{16 \, c^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)*(c + d/x^2)^(3/2)*x^5,x, algorithm="giac")
[Out]