Optimal. Leaf size=41 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{1}{a \sqrt{a+\frac{b}{x^2}}} \]
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Rubi [A] time = 0.0236517, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 51, 63, 208} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{1}{a \sqrt{a+\frac{b}{x^2}}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^2}\right )^{3/2} x} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{3/2}} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\frac{1}{a \sqrt{a+\frac{b}{x^2}}}-\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x^2}\right )}{2 a}\\ &=-\frac{1}{a \sqrt{a+\frac{b}{x^2}}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x^2}}\right )}{a b}\\ &=-\frac{1}{a \sqrt{a+\frac{b}{x^2}}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{a^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0554261, size = 62, normalized size = 1.51 \[ \frac{\sqrt{b} \sqrt{\frac{a x^2}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )-\sqrt{a} x}{a^{3/2} x \sqrt{a+\frac{b}{x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 63, normalized size = 1.5 \begin{align*} -{\frac{a{x}^{2}+b}{{x}^{3}} \left ( x{a}^{{\frac{3}{2}}}-\ln \left ( x\sqrt{a}+\sqrt{a{x}^{2}+b} \right ) a\sqrt{a{x}^{2}+b} \right ) \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}{a}^{-{\frac{5}{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 [B] time = 1.50511, size = 359, normalized size = 8.76 \begin{align*} \left [-\frac{2 \, a x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} -{\left (a x^{2} + b\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^{3} x^{2} + a^{2} b\right )}}, -\frac{a x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} +{\left (a x^{2} + b\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a} x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right )}{a^{3} x^{2} + a^{2} b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.8982, size = 187, normalized size = 4.56 \begin{align*} - \frac{2 a^{3} x^{2} \sqrt{1 + \frac{b}{a x^{2}}}}{2 a^{\frac{9}{2}} x^{2} + 2 a^{\frac{7}{2}} b} - \frac{a^{3} x^{2} \log{\left (\frac{b}{a x^{2}} \right )}}{2 a^{\frac{9}{2}} x^{2} + 2 a^{\frac{7}{2}} b} + \frac{2 a^{3} x^{2} \log{\left (\sqrt{1 + \frac{b}{a x^{2}}} + 1 \right )}}{2 a^{\frac{9}{2}} x^{2} + 2 a^{\frac{7}{2}} b} - \frac{a^{2} b \log{\left (\frac{b}{a x^{2}} \right )}}{2 a^{\frac{9}{2}} x^{2} + 2 a^{\frac{7}{2}} b} + \frac{2 a^{2} b \log{\left (\sqrt{1 + \frac{b}{a x^{2}}} + 1 \right )}}{2 a^{\frac{9}{2}} x^{2} + 2 a^{\frac{7}{2}} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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