Optimal. Leaf size=84 \[ -\frac{\sqrt{c+\frac{d}{x^2}} (a d+2 b c)}{2 c}+\frac{(a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{2 \sqrt{c}}+\frac{a x^2 \left (c+\frac{d}{x^2}\right )^{3/2}}{2 c} \]
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Rubi [A] time = 0.0579218, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {446, 78, 50, 63, 208} \[ -\frac{\sqrt{c+\frac{d}{x^2}} (a d+2 b c)}{2 c}+\frac{(a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{2 \sqrt{c}}+\frac{a x^2 \left (c+\frac{d}{x^2}\right )^{3/2}}{2 c} \]
Antiderivative was successfully verified.
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Rule 446
Rule 78
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \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^2} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=\frac{a \left (c+\frac{d}{x^2}\right )^{3/2} x^2}{2 c}-\frac{(2 b c+a d) \operatorname{Subst}\left (\int \frac{\sqrt{c+d x}}{x} \, dx,x,\frac{1}{x^2}\right )}{4 c}\\ &=-\frac{(2 b c+a d) \sqrt{c+\frac{d}{x^2}}}{2 c}+\frac{a \left (c+\frac{d}{x^2}\right )^{3/2} x^2}{2 c}-\frac{1}{4} (2 b c+a d) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+d x}} \, dx,x,\frac{1}{x^2}\right )\\ &=-\frac{(2 b c+a d) \sqrt{c+\frac{d}{x^2}}}{2 c}+\frac{a \left (c+\frac{d}{x^2}\right )^{3/2} x^2}{2 c}-\frac{(2 b c+a d) \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+\frac{d}{x^2}}\right )}{2 d}\\ &=-\frac{(2 b c+a d) \sqrt{c+\frac{d}{x^2}}}{2 c}+\frac{a \left (c+\frac{d}{x^2}\right )^{3/2} x^2}{2 c}+\frac{(2 b c+a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{2 \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.160214, size = 71, normalized size = 0.85 \[ \frac{1}{2} \sqrt{c+\frac{d}{x^2}} \left (\frac{x (a d+2 b c) \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )}{\sqrt{c} \sqrt{d} \sqrt{\frac{c x^2}{d}+1}}+a x^2-2 b\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 127, normalized size = 1.5 \begin{align*}{\frac{1}{2\,d}\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}} \left ( 2\,{c}^{3/2}\sqrt{c{x}^{2}+d}{x}^{2}b+\sqrt{c}\sqrt{c{x}^{2}+d}{x}^{2}ad-2\,\sqrt{c} \left ( c{x}^{2}+d \right ) ^{3/2}b+\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ) xa{d}^{2}+2\,\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ) xbcd \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 = 1.32003, size = 358, normalized size = 4.26 \begin{align*} \left [\frac{{\left (2 \, b c + a d\right )} \sqrt{c} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c} x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} - d\right ) + 2 \,{\left (a c x^{2} - 2 \, b c\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{4 \, c}, -\frac{{\left (2 \, b c + a d\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) -{\left (a c x^{2} - 2 \, b c\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{2 \, c}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 32.2153, size = 107, normalized size = 1.27 \begin{align*} \frac{a \sqrt{d} x \sqrt{\frac{c x^{2}}{d} + 1}}{2} + \frac{a d \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{2 \sqrt{c}} + b \sqrt{c} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )} - \frac{b c x}{\sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}} - \frac{b \sqrt{d}}{x \sqrt{\frac{c x^{2}}{d} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25489, size = 124, normalized size = 1.48 \begin{align*} \frac{1}{2} \, \sqrt{c x^{2} + d} a x \mathrm{sgn}\left (x\right ) + \frac{2 \, b \sqrt{c} d \mathrm{sgn}\left (x\right )}{{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} - d} - \frac{{\left (2 \, b c^{\frac{3}{2}} \mathrm{sgn}\left (x\right ) + a \sqrt{c} d \mathrm{sgn}\left (x\right )\right )} \log \left ({\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2}\right )}{4 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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