Optimal. Leaf size=95 \[ -\frac{15 b^2}{8 a^3 \sqrt{a+\frac{b}{x^2}}}+\frac{15 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{8 a^{7/2}}-\frac{5 b x^2}{8 a^2 \sqrt{a+\frac{b}{x^2}}}+\frac{x^4}{4 a \sqrt{a+\frac{b}{x^2}}} \]
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Rubi [A] time = 0.0454288, antiderivative size = 93, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 51, 63, 208} \[ \frac{15 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{8 a^{7/2}}+\frac{5 x^4 \sqrt{a+\frac{b}{x^2}}}{4 a^2}-\frac{15 b x^2 \sqrt{a+\frac{b}{x^2}}}{8 a^3}-\frac{x^4}{a \sqrt{a+\frac{b}{x^2}}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{x^3}{\left (a+\frac{b}{x^2}\right )^{3/2}} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^3 (a+b x)^{3/2}} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\frac{x^4}{a \sqrt{a+\frac{b}{x^2}}}-\frac{5 \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{a+b x}} \, dx,x,\frac{1}{x^2}\right )}{2 a}\\ &=-\frac{x^4}{a \sqrt{a+\frac{b}{x^2}}}+\frac{5 \sqrt{a+\frac{b}{x^2}} x^4}{4 a^2}+\frac{(15 b) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,\frac{1}{x^2}\right )}{8 a^2}\\ &=-\frac{15 b \sqrt{a+\frac{b}{x^2}} x^2}{8 a^3}-\frac{x^4}{a \sqrt{a+\frac{b}{x^2}}}+\frac{5 \sqrt{a+\frac{b}{x^2}} x^4}{4 a^2}-\frac{\left (15 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x^2}\right )}{16 a^3}\\ &=-\frac{15 b \sqrt{a+\frac{b}{x^2}} x^2}{8 a^3}-\frac{x^4}{a \sqrt{a+\frac{b}{x^2}}}+\frac{5 \sqrt{a+\frac{b}{x^2}} x^4}{4 a^2}-\frac{(15 b) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x^2}}\right )}{8 a^3}\\ &=-\frac{15 b \sqrt{a+\frac{b}{x^2}} x^2}{8 a^3}-\frac{x^4}{a \sqrt{a+\frac{b}{x^2}}}+\frac{5 \sqrt{a+\frac{b}{x^2}} x^4}{4 a^2}+\frac{15 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{8 a^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0555257, size = 86, normalized size = 0.91 \[ \frac{\sqrt{a} x \left (2 a^2 x^4-5 a b x^2-15 b^2\right )+15 b^{5/2} \sqrt{\frac{a x^2}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{7/2} x \sqrt{a+\frac{b}{x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 87, normalized size = 0.9 \begin{align*}{\frac{a{x}^{2}+b}{8\,{x}^{3}} \left ( 2\,{x}^{5}{a}^{7/2}-5\,{a}^{5/2}{x}^{3}b-15\,{a}^{3/2}x{b}^{2}+15\,\ln \left ( x\sqrt{a}+\sqrt{a{x}^{2}+b} \right ) \sqrt{a{x}^{2}+b}a{b}^{2} \right ) \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}{a}^{-{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59922, size = 483, normalized size = 5.08 \begin{align*} \left [\frac{15 \,{\left (a b^{2} x^{2} + b^{3}\right )} \sqrt{a} \log \left (-2 \, a x^{2} - 2 \, \sqrt{a} x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} - b\right ) + 2 \,{\left (2 \, a^{3} x^{6} - 5 \, a^{2} b x^{4} - 15 \, a b^{2} x^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{16 \,{\left (a^{5} x^{2} + a^{4} b\right )}}, -\frac{15 \,{\left (a b^{2} x^{2} + b^{3}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a} x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) -{\left (2 \, a^{3} x^{6} - 5 \, a^{2} b x^{4} - 15 \, a b^{2} x^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{8 \,{\left (a^{5} x^{2} + a^{4} b\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.20888, size = 100, normalized size = 1.05 \begin{align*} \frac{x^{5}}{4 a \sqrt{b} \sqrt{\frac{a x^{2}}{b} + 1}} - \frac{5 \sqrt{b} x^{3}}{8 a^{2} \sqrt{\frac{a x^{2}}{b} + 1}} - \frac{15 b^{\frac{3}{2}} x}{8 a^{3} \sqrt{\frac{a x^{2}}{b} + 1}} + \frac{15 b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{8 a^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25262, size = 158, normalized size = 1.66 \begin{align*} -\frac{1}{8} \, b^{2}{\left (\frac{15 \, \arctan \left (\frac{\sqrt{\frac{a x^{2} + b}{x^{2}}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} + \frac{8}{a^{3} \sqrt{\frac{a x^{2} + b}{x^{2}}}} - \frac{9 \, a \sqrt{\frac{a x^{2} + b}{x^{2}}} - \frac{7 \,{\left (a x^{2} + b\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{x^{2}}}{{\left (a - \frac{a x^{2} + b}{x^{2}}\right )}^{2} a^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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