Optimal. Leaf size=76 \[ a c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )-\frac{1}{3} a \left (c+\frac{d}{x^2}\right )^{3/2}-a c \sqrt{c+\frac{d}{x^2}}-\frac{b \left (c+\frac{d}{x^2}\right )^{5/2}}{5 d} \]
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Rubi [A] time = 0.0541351, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {446, 80, 50, 63, 208} \[ a c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )-\frac{1}{3} a \left (c+\frac{d}{x^2}\right )^{3/2}-a c \sqrt{c+\frac{d}{x^2}}-\frac{b \left (c+\frac{d}{x^2}\right )^{5/2}}{5 d} \]
Antiderivative was successfully verified.
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Rule 446
Rule 80
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x^2}\right ) \left (c+\frac{d}{x^2}\right )^{3/2}}{x} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x) (c+d x)^{3/2}}{x} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\frac{b \left (c+\frac{d}{x^2}\right )^{5/2}}{5 d}-\frac{1}{2} a \operatorname{Subst}\left (\int \frac{(c+d x)^{3/2}}{x} \, dx,x,\frac{1}{x^2}\right )\\ &=-\frac{1}{3} a \left (c+\frac{d}{x^2}\right )^{3/2}-\frac{b \left (c+\frac{d}{x^2}\right )^{5/2}}{5 d}-\frac{1}{2} (a c) \operatorname{Subst}\left (\int \frac{\sqrt{c+d x}}{x} \, dx,x,\frac{1}{x^2}\right )\\ &=-a c \sqrt{c+\frac{d}{x^2}}-\frac{1}{3} a \left (c+\frac{d}{x^2}\right )^{3/2}-\frac{b \left (c+\frac{d}{x^2}\right )^{5/2}}{5 d}-\frac{1}{2} \left (a c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+d x}} \, dx,x,\frac{1}{x^2}\right )\\ &=-a c \sqrt{c+\frac{d}{x^2}}-\frac{1}{3} a \left (c+\frac{d}{x^2}\right )^{3/2}-\frac{b \left (c+\frac{d}{x^2}\right )^{5/2}}{5 d}-\frac{\left (a c^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+\frac{d}{x^2}}\right )}{d}\\ &=-a c \sqrt{c+\frac{d}{x^2}}-\frac{1}{3} a \left (c+\frac{d}{x^2}\right )^{3/2}-\frac{b \left (c+\frac{d}{x^2}\right )^{5/2}}{5 d}+a c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )\\ \end{align*}
Mathematica [C] time = 0.046791, size = 90, normalized size = 1.18 \[ -\frac{\sqrt{c+\frac{d}{x^2}} \left (5 a d^2 x^2 \, _2F_1\left (-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};-\frac{c x^2}{d}\right )+3 b \left (c x^2+d\right )^2 \sqrt{\frac{c x^2}{d}+1}\right )}{15 d x^4 \sqrt{\frac{c x^2}{d}+1}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.014, size = 153, normalized size = 2. \begin{align*}{\frac{1}{15\,{x}^{2}{d}^{2}} \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{{\frac{3}{2}}} \left ( 10\,{c}^{5/2} \left ( c{x}^{2}+d \right ) ^{3/2}{x}^{6}a+15\,{c}^{5/2}\sqrt{c{x}^{2}+d}{x}^{6}ad-10\,{c}^{3/2} \left ( c{x}^{2}+d \right ) ^{5/2}{x}^{4}a+15\,\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ){x}^{5}a{c}^{2}{d}^{2}-5\,\sqrt{c} \left ( c{x}^{2}+d \right ) ^{5/2}{x}^{2}ad-3\,\sqrt{c} \left ( c{x}^{2}+d \right ) ^{5/2}bd \right ) \left ( c{x}^{2}+d \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{c}}}} \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 = 1.34711, size = 493, normalized size = 6.49 \begin{align*} \left [\frac{15 \, a c^{\frac{3}{2}} d x^{4} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c} x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} - d\right ) - 2 \,{\left ({\left (3 \, b c^{2} + 20 \, a c d\right )} x^{4} + 3 \, b d^{2} +{\left (6 \, b c d + 5 \, a d^{2}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{30 \, d x^{4}}, -\frac{15 \, a \sqrt{-c} c d x^{4} \arctan \left (\frac{\sqrt{-c} x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) +{\left ({\left (3 \, b c^{2} + 20 \, a c d\right )} x^{4} + 3 \, b d^{2} +{\left (6 \, b c d + 5 \, a d^{2}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{15 \, d x^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 37.2231, size = 73, normalized size = 0.96 \begin{align*} - \frac{a c^{2} \operatorname{atan}{\left (\frac{\sqrt{c + \frac{d}{x^{2}}}}{\sqrt{- c}} \right )}}{\sqrt{- c}} - a c \sqrt{c + \frac{d}{x^{2}}} - \frac{a \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3} - \frac{b \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5 d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.80624, size = 343, normalized size = 4.51 \begin{align*} -\frac{1}{2} \, a c^{\frac{3}{2}} \log \left ({\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2}\right ) \mathrm{sgn}\left (x\right ) + \frac{2 \,{\left (15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{8} b c^{\frac{5}{2}} \mathrm{sgn}\left (x\right ) + 30 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{8} a c^{\frac{3}{2}} d \mathrm{sgn}\left (x\right ) - 90 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{6} a c^{\frac{3}{2}} d^{2} \mathrm{sgn}\left (x\right ) + 30 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} b c^{\frac{5}{2}} d^{2} \mathrm{sgn}\left (x\right ) + 110 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} a c^{\frac{3}{2}} d^{3} \mathrm{sgn}\left (x\right ) - 70 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} a c^{\frac{3}{2}} d^{4} \mathrm{sgn}\left (x\right ) + 3 \, b c^{\frac{5}{2}} d^{4} \mathrm{sgn}\left (x\right ) + 20 \, a c^{\frac{3}{2}} d^{5} \mathrm{sgn}\left (x\right )\right )}}{15 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} - d\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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