Optimal. Leaf size=71 \[ \frac{3 a \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{2 b^{5/2}}-\frac{3 \sqrt{a+\frac{b}{x^2}}}{2 b^2 x}+\frac{1}{b x^3 \sqrt{a+\frac{b}{x^2}}} \]
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Rubi [A] time = 0.110937, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{3 a \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{2 b^{5/2}}-\frac{3 \sqrt{a+\frac{b}{x^2}}}{2 b^2 x}+\frac{1}{b x^3 \sqrt{a+\frac{b}{x^2}}} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b/x^2)^(3/2)*x^6),x]
[Out]
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Rubi in Sympy [A] time = 10.8548, size = 63, normalized size = 0.89 \[ \frac{3 a \operatorname{atanh}{\left (\frac{\sqrt{b}}{x \sqrt{a + \frac{b}{x^{2}}}} \right )}}{2 b^{\frac{5}{2}}} + \frac{1}{b x^{3} \sqrt{a + \frac{b}{x^{2}}}} - \frac{3 \sqrt{a + \frac{b}{x^{2}}}}{2 b^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x**2)**(3/2)/x**6,x)
[Out]
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Mathematica [A] time = 0.0753506, size = 95, normalized size = 1.34 \[ \frac{-\sqrt{b} \left (3 a x^2+b\right )-3 a x^2 \log (x) \sqrt{a x^2+b}+3 a x^2 \sqrt{a x^2+b} \log \left (\sqrt{b} \sqrt{a x^2+b}+b\right )}{2 b^{5/2} x^3 \sqrt{a+\frac{b}{x^2}}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b/x^2)^(3/2)*x^6),x]
[Out]
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Maple [A] time = 0.012, size = 79, normalized size = 1.1 \[ -{\frac{a{x}^{2}+b}{2\,{x}^{5}} \left ( 3\,{b}^{3/2}{x}^{2}a-3\,\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{2}+b}+b}{x}} \right ) \sqrt{a{x}^{2}+b}{x}^{2}ab+{b}^{{\frac{5}{2}}} \right ) \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}{b}^{-{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x^2)^(3/2)/x^6,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(1/((a + b/x^2)^(3/2)*x^6),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.249167, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (a^{2} x^{3} + a b x\right )} \sqrt{b} \log \left (-\frac{2 \, b x \sqrt{\frac{a x^{2} + b}{x^{2}}} +{\left (a x^{2} + 2 \, b\right )} \sqrt{b}}{x^{2}}\right ) - 2 \,{\left (3 \, a b x^{2} + b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{4 \,{\left (a b^{3} x^{3} + b^{4} x\right )}}, -\frac{3 \,{\left (a^{2} x^{3} + a b x\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b}}{x \sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) +{\left (3 \, a b x^{2} + b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{2 \,{\left (a b^{3} x^{3} + b^{4} x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x^2)^(3/2)*x^6),x, algorithm="fricas")
[Out]
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Sympy [A] time = 16.6653, size = 73, normalized size = 1.03 \[ - \frac{3 \sqrt{a}}{2 b^{2} x \sqrt{1 + \frac{b}{a x^{2}}}} + \frac{3 a \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x} \right )}}{2 b^{\frac{5}{2}}} - \frac{1}{2 \sqrt{a} b x^{3} \sqrt{1 + \frac{b}{a x^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/x**2)**(3/2)/x**6,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x^2)^(3/2)*x^6),x, algorithm="giac")
[Out]