Optimal. Leaf size=69 \[ \frac{3 b}{2 a^2 \sqrt{a+\frac{b}{x^2}}}-\frac{3 b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{2 a^{5/2}}+\frac{x^2}{2 a \sqrt{a+\frac{b}{x^2}}} \]
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Rubi [A] time = 0.0330882, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {266, 51, 63, 208} \[ \frac{3 x^2 \sqrt{a+\frac{b}{x^2}}}{2 a^2}-\frac{3 b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{x^2}{a \sqrt{a+\frac{b}{x^2}}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{x}{\left (a+\frac{b}{x^2}\right )^{3/2}} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)^{3/2}} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\frac{x^2}{a \sqrt{a+\frac{b}{x^2}}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,\frac{1}{x^2}\right )}{2 a}\\ &=-\frac{x^2}{a \sqrt{a+\frac{b}{x^2}}}+\frac{3 \sqrt{a+\frac{b}{x^2}} x^2}{2 a^2}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x^2}\right )}{4 a^2}\\ &=-\frac{x^2}{a \sqrt{a+\frac{b}{x^2}}}+\frac{3 \sqrt{a+\frac{b}{x^2}} x^2}{2 a^2}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x^2}}\right )}{2 a^2}\\ &=-\frac{x^2}{a \sqrt{a+\frac{b}{x^2}}}+\frac{3 \sqrt{a+\frac{b}{x^2}} x^2}{2 a^2}-\frac{3 b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{2 a^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0371454, size = 74, normalized size = 1.07 \[ \frac{\sqrt{a} x \left (a x^2+3 b\right )-3 b^{3/2} \sqrt{\frac{a x^2}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 a^{5/2} x \sqrt{a+\frac{b}{x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 73, normalized size = 1.1 \begin{align*}{\frac{a{x}^{2}+b}{2\,{x}^{3}} \left ({x}^{3}{a}^{{\frac{5}{2}}}+3\,{a}^{3/2}xb-3\,\ln \left ( x\sqrt{a}+\sqrt{a{x}^{2}+b} \right ) \sqrt{a{x}^{2}+b}ab \right ) \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}{a}^{-{\frac{7}{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 = 1.53647, size = 421, normalized size = 6.1 \begin{align*} \left [\frac{3 \,{\left (a b x^{2} + b^{2}\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 (a^{2} x^{4} + 3 \, a b x^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{4 \,{\left (a^{4} x^{2} + a^{3} b\right )}}, \frac{3 \,{\left (a b x^{2} + b^{2}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a} x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) +{\left (a^{2} x^{4} + 3 \, a b x^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{2 \,{\left (a^{4} x^{2} + a^{3} b\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.1326, size = 71, normalized size = 1.03 \begin{align*} \frac{x^{3}}{2 a \sqrt{b} \sqrt{\frac{a x^{2}}{b} + 1}} + \frac{3 \sqrt{b} x}{2 a^{2} \sqrt{\frac{a x^{2}}{b} + 1}} - \frac{3 b \operatorname{asinh}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{2 a^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30483, size = 131, normalized size = 1.9 \begin{align*} \frac{1}{2} \, b{\left (\frac{3 \, \arctan \left (\frac{\sqrt{\frac{a x^{2} + b}{x^{2}}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{2 \, a - \frac{3 \,{\left (a x^{2} + b\right )}}{x^{2}}}{{\left (a \sqrt{\frac{a x^{2} + b}{x^{2}}} - \frac{{\left (a x^{2} + b\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{x^{2}}\right )} a^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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