Optimal. Leaf size=68 \[ -\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{8 \sqrt{a}}-\frac{3 b \sqrt{a+b x^2}}{8 x^2}-\frac{\left (a+b x^2\right )^{3/2}}{4 x^4} \]
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Rubi [A] time = 0.0395221, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 47, 63, 208} \[ -\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{8 \sqrt{a}}-\frac{3 b \sqrt{a+b x^2}}{8 x^2}-\frac{\left (a+b x^2\right )^{3/2}}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 266
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^{3/2}}{x^5} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x^3} \, dx,x,x^2\right )\\ &=-\frac{\left (a+b x^2\right )^{3/2}}{4 x^4}+\frac{1}{8} (3 b) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac{3 b \sqrt{a+b x^2}}{8 x^2}-\frac{\left (a+b x^2\right )^{3/2}}{4 x^4}+\frac{1}{16} \left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^2\right )\\ &=-\frac{3 b \sqrt{a+b x^2}}{8 x^2}-\frac{\left (a+b x^2\right )^{3/2}}{4 x^4}+\frac{1}{8} (3 b) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^2}\right )\\ &=-\frac{3 b \sqrt{a+b x^2}}{8 x^2}-\frac{\left (a+b x^2\right )^{3/2}}{4 x^4}-\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{8 \sqrt{a}}\\ \end{align*}
Mathematica [A] time = 0.0354422, size = 76, normalized size = 1.12 \[ -\frac{2 a^2+3 b^2 x^4 \sqrt{\frac{b x^2}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{b x^2}{a}+1}\right )+7 a b x^2+5 b^2 x^4}{8 x^4 \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 102, normalized size = 1.5 \begin{align*} -{\frac{1}{4\,a{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{b}{8\,{a}^{2}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{{b}^{2}}{8\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{b}^{2}}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{3\,{b}^{2}}{8\,a}\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 = 1.54177, size = 317, normalized size = 4.66 \begin{align*} \left [\frac{3 \, \sqrt{a} b^{2} x^{4} \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) - 2 \,{\left (5 \, a b x^{2} + 2 \, a^{2}\right )} \sqrt{b x^{2} + a}}{16 \, a x^{4}}, \frac{3 \, \sqrt{-a} b^{2} x^{4} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) -{\left (5 \, a b x^{2} + 2 \, a^{2}\right )} \sqrt{b x^{2} + a}}{8 \, a x^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.00593, size = 71, normalized size = 1.04 \begin{align*} - \frac{a \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{4 x^{3}} - \frac{5 b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{8 x} - \frac{3 b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{8 \sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.6188, size = 82, normalized size = 1.21 \begin{align*} \frac{1}{8} \, b^{2}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} - 3 \, \sqrt{b x^{2} + a} a}{b^{2} x^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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