Optimal. Leaf size=64 \[ x \left (a+\frac{b}{x^2}\right )^{3/2}-\frac{3 b \sqrt{a+\frac{b}{x^2}}}{2 x}-\frac{3}{2} a \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right ) \]
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Rubi [A] time = 0.0263427, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {242, 277, 195, 217, 206} \[ x \left (a+\frac{b}{x^2}\right )^{3/2}-\frac{3 b \sqrt{a+\frac{b}{x^2}}}{2 x}-\frac{3}{2} a \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right ) \]
Antiderivative was successfully verified.
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Rule 242
Rule 277
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \left (a+\frac{b}{x^2}\right )^{3/2} \, dx &=-\operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^{3/2}}{x^2} \, dx,x,\frac{1}{x}\right )\\ &=\left (a+\frac{b}{x^2}\right )^{3/2} x-(3 b) \operatorname{Subst}\left (\int \sqrt{a+b x^2} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{3 b \sqrt{a+\frac{b}{x^2}}}{2 x}+\left (a+\frac{b}{x^2}\right )^{3/2} x-\frac{1}{2} (3 a b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{3 b \sqrt{a+\frac{b}{x^2}}}{2 x}+\left (a+\frac{b}{x^2}\right )^{3/2} x-\frac{1}{2} (3 a b) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{1}{\sqrt{a+\frac{b}{x^2}} x}\right )\\ &=-\frac{3 b \sqrt{a+\frac{b}{x^2}}}{2 x}+\left (a+\frac{b}{x^2}\right )^{3/2} x-\frac{3}{2} a \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{a+\frac{b}{x^2}} x}\right )\\ \end{align*}
Mathematica [C] time = 0.0107218, size = 47, normalized size = 0.73 \[ \frac{a x^3 \left (a+\frac{b}{x^2}\right )^{3/2} \left (a x^2+b\right ) \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};\frac{a x^2}{b}+1\right )}{5 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 100, normalized size = 1.6 \begin{align*} -{\frac{x}{2\,b} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{{\frac{3}{2}}} \left ( - \left ( a{x}^{2}+b \right ) ^{{\frac{3}{2}}}{x}^{2}a+3\,{b}^{3/2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{2}+b}+b}{x}} \right ){x}^{2}a+ \left ( a{x}^{2}+b \right ) ^{{\frac{5}{2}}}-3\,\sqrt{a{x}^{2}+b}{x}^{2}ab \right ) \left ( a{x}^{2}+b \right ) ^{-{\frac{3}{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 = 1.56836, size = 327, normalized size = 5.11 \begin{align*} \left [\frac{3 \, a \sqrt{b} x \log \left (-\frac{a x^{2} - 2 \, \sqrt{b} x \sqrt{\frac{a x^{2} + b}{x^{2}}} + 2 \, b}{x^{2}}\right ) + 2 \,{\left (2 \, a x^{2} - b\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{4 \, x}, \frac{3 \, a \sqrt{-b} x \arctan \left (\frac{\sqrt{-b} x \sqrt{\frac{a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) +{\left (2 \, a x^{2} - b\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{2 \, x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.53202, size = 88, normalized size = 1.38 \begin{align*} \frac{a^{\frac{3}{2}} x}{\sqrt{1 + \frac{b}{a x^{2}}}} + \frac{\sqrt{a} b}{2 x \sqrt{1 + \frac{b}{a x^{2}}}} - \frac{3 a \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x} \right )}}{2} - \frac{b^{2}}{2 \sqrt{a} x^{3} \sqrt{1 + \frac{b}{a x^{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20065, size = 80, normalized size = 1.25 \begin{align*} \frac{1}{2} \,{\left (\frac{3 \, b \arctan \left (\frac{\sqrt{a x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} + 2 \, \sqrt{a x^{2} + b} - \frac{\sqrt{a x^{2} + b} b}{a x^{2}}\right )} a \mathrm{sgn}\left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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