Optimal. Leaf size=63 \[ -\frac{\left (a+b x^2\right )^{3/2}}{2 x^2}+\frac{3}{2} b \sqrt{a+b x^2}-\frac{3}{2} \sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right ) \]
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Rubi [A] time = 0.0367053, antiderivative size = 63, 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, Rules used = {266, 47, 50, 63, 208} \[ -\frac{\left (a+b x^2\right )^{3/2}}{2 x^2}+\frac{3}{2} b \sqrt{a+b x^2}-\frac{3}{2} \sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
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Rule 266
Rule 47
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^{3/2}}{x^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x^2} \, dx,x,x^2\right )\\ &=-\frac{\left (a+b x^2\right )^{3/2}}{2 x^2}+\frac{1}{4} (3 b) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,x^2\right )\\ &=\frac{3}{2} b \sqrt{a+b x^2}-\frac{\left (a+b x^2\right )^{3/2}}{2 x^2}+\frac{1}{4} (3 a b) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^2\right )\\ &=\frac{3}{2} b \sqrt{a+b x^2}-\frac{\left (a+b x^2\right )^{3/2}}{2 x^2}+\frac{1}{2} (3 a) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^2}\right )\\ &=\frac{3}{2} b \sqrt{a+b x^2}-\frac{\left (a+b x^2\right )^{3/2}}{2 x^2}-\frac{3}{2} \sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [C] time = 0.0089778, size = 37, normalized size = 0.59 \[ \frac{b \left (a+b x^2\right )^{5/2} \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};\frac{b x^2}{a}+1\right )}{5 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 75, normalized size = 1.2 \begin{align*} -{\frac{1}{2\,a{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{b}{2\,a} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,b}{2}\sqrt{a}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ) }+{\frac{3\,b}{2}\sqrt{b{x}^{2}+a}} \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 = 1.61507, size = 284, normalized size = 4.51 \begin{align*} \left [\frac{3 \, \sqrt{a} b x^{2} \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (2 \, b x^{2} - a\right )} \sqrt{b x^{2} + a}}{4 \, x^{2}}, \frac{3 \, \sqrt{-a} b x^{2} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) +{\left (2 \, b x^{2} - a\right )} \sqrt{b x^{2} + a}}{2 \, x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.34298, size = 88, normalized size = 1.4 \begin{align*} - \frac{3 \sqrt{a} b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2} - \frac{a^{2}}{2 \sqrt{b} x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{a \sqrt{b}}{2 x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{b^{\frac{3}{2}} x}{\sqrt{\frac{a}{b x^{2}} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.59864, size = 77, normalized size = 1.22 \begin{align*} \frac{1}{2} \,{\left (\frac{3 \, a \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + 2 \, \sqrt{b x^{2} + a} - \frac{\sqrt{b x^{2} + a} a}{b x^{2}}\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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