Optimal. Leaf size=59 \[ -a \sqrt{c+\frac{d}{x^2}}+a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )-\frac{b \left (c+\frac{d}{x^2}\right )^{3/2}}{3 d} \]
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Rubi [A] time = 0.0431102, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 5, 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} \[ -a \sqrt{c+\frac{d}{x^2}}+a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )-\frac{b \left (c+\frac{d}{x^2}\right )^{3/2}}{3 d} \]
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 ) \sqrt{c+\frac{d}{x^2}}}{x} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x) \sqrt{c+d x}}{x} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\frac{b \left (c+\frac{d}{x^2}\right )^{3/2}}{3 d}-\frac{1}{2} a \operatorname{Subst}\left (\int \frac{\sqrt{c+d x}}{x} \, dx,x,\frac{1}{x^2}\right )\\ &=-a \sqrt{c+\frac{d}{x^2}}-\frac{b \left (c+\frac{d}{x^2}\right )^{3/2}}{3 d}-\frac{1}{2} (a c) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+d x}} \, dx,x,\frac{1}{x^2}\right )\\ &=-a \sqrt{c+\frac{d}{x^2}}-\frac{b \left (c+\frac{d}{x^2}\right )^{3/2}}{3 d}-\frac{(a c) \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+\frac{d}{x^2}}\right )}{d}\\ &=-a \sqrt{c+\frac{d}{x^2}}-\frac{b \left (c+\frac{d}{x^2}\right )^{3/2}}{3 d}+a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )\\ \end{align*}
Mathematica [A] time = 0.116312, size = 82, normalized size = 1.39 \[ \frac{\sqrt{c+\frac{d}{x^2}} \left (\frac{3 a \sqrt{c} \sqrt{d} x^3 \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )}{\sqrt{\frac{c x^2}{d}+1}}-3 a d x^2-b \left (c x^2+d\right )\right )}{3 d x^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.011, size = 110, normalized size = 1.9 \begin{align*}{\frac{1}{3\,{x}^{2}d}\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}} \left ( 3\,{c}^{3/2}\sqrt{c{x}^{2}+d}{x}^{4}a-3\,\sqrt{c} \left ( c{x}^{2}+d \right ) ^{3/2}{x}^{2}a+3\,\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ){x}^{3}acd-\sqrt{c} \left ( c{x}^{2}+d \right ) ^{{\frac{3}{2}}}b \right ){\frac{1}{\sqrt{c{x}^{2}+d}}}{\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 = 1.3588, size = 385, normalized size = 6.53 \begin{align*} \left [\frac{3 \, a \sqrt{c} d x^{2} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c} x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} - d\right ) - 2 \,{\left ({\left (b c + 3 \, a d\right )} x^{2} + b d\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{6 \, d x^{2}}, -\frac{3 \, a \sqrt{-c} d x^{2} \arctan \left (\frac{\sqrt{-c} x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) +{\left ({\left (b c + 3 \, a d\right )} x^{2} + b d\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{3 \, d x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 17.0405, size = 75, normalized size = 1.27 \begin{align*} \frac{a \left (- \frac{2 c \operatorname{atan}{\left (\frac{\sqrt{c + \frac{d}{x^{2}}}}{\sqrt{- c}} \right )}}{\sqrt{- c}} - 2 \sqrt{c + \frac{d}{x^{2}}}\right )}{2} + \frac{b \left (\begin{cases} - \frac{\sqrt{c}}{x^{2}} & \text{for}\: d = 0 \\- \frac{2 \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3 d} & \text{otherwise} \end{cases}\right )}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.39762, size = 220, normalized size = 3.73 \begin{align*} -\frac{1}{2} \, a \sqrt{c} \log \left ({\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2}\right ) \mathrm{sgn}\left (x\right ) + \frac{2 \,{\left (3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} b c^{\frac{3}{2}} \mathrm{sgn}\left (x\right ) + 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} a \sqrt{c} d \mathrm{sgn}\left (x\right ) - 6 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} a \sqrt{c} d^{2} \mathrm{sgn}\left (x\right ) + b c^{\frac{3}{2}} d^{2} \mathrm{sgn}\left (x\right ) + 3 \, a \sqrt{c} d^{3} \mathrm{sgn}\left (x\right )\right )}}{3 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} - d\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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