Optimal. Leaf size=90 \[ \frac{d (4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{8 c^{3/2}}+\frac{x^2 \sqrt{c+\frac{d}{x^2}} (4 b c-a d)}{8 c}+\frac{a x^4 \left (c+\frac{d}{x^2}\right )^{3/2}}{4 c} \]
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Rubi [A] time = 0.0678375, antiderivative size = 90, 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, 78, 47, 63, 208} \[ \frac{d (4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{8 c^{3/2}}+\frac{x^2 \sqrt{c+\frac{d}{x^2}} (4 b c-a d)}{8 c}+\frac{a x^4 \left (c+\frac{d}{x^2}\right )^{3/2}}{4 c} \]
Antiderivative was successfully verified.
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Rule 446
Rule 78
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \left (a+\frac{b}{x^2}\right ) \sqrt{c+\frac{d}{x^2}} x^3 \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x) \sqrt{c+d x}}{x^3} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=\frac{a \left (c+\frac{d}{x^2}\right )^{3/2} x^4}{4 c}-\frac{\left (2 b c-\frac{a d}{2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c+d x}}{x^2} \, dx,x,\frac{1}{x^2}\right )}{4 c}\\ &=\frac{(4 b c-a d) \sqrt{c+\frac{d}{x^2}} x^2}{8 c}+\frac{a \left (c+\frac{d}{x^2}\right )^{3/2} x^4}{4 c}-\frac{(d (4 b c-a d)) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+d x}} \, dx,x,\frac{1}{x^2}\right )}{16 c}\\ &=\frac{(4 b c-a d) \sqrt{c+\frac{d}{x^2}} x^2}{8 c}+\frac{a \left (c+\frac{d}{x^2}\right )^{3/2} x^4}{4 c}-\frac{(4 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 )}{8 c}\\ &=\frac{(4 b c-a d) \sqrt{c+\frac{d}{x^2}} x^2}{8 c}+\frac{a \left (c+\frac{d}{x^2}\right )^{3/2} x^4}{4 c}+\frac{d (4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{8 c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.104588, size = 100, normalized size = 1.11 \[ \frac{x \sqrt{c+\frac{d}{x^2}} \left (\sqrt{c} x \sqrt{\frac{c x^2}{d}+1} \left (a \left (2 c x^2+d\right )+4 b c\right )+\sqrt{d} (4 b c-a d) \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )\right )}{8 c^{3/2} \sqrt{\frac{c x^2}{d}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 122, normalized size = 1.4 \begin{align*}{\frac{x}{8}\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}} \left ( 2\,\sqrt{c} \left ( c{x}^{2}+d \right ) ^{3/2}xa-\sqrt{c}\sqrt{c{x}^{2}+d}xad+4\,{c}^{3/2}\sqrt{c{x}^{2}+d}xb-\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ) a{d}^{2}+4\,\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ) bcd \right ){\frac{1}{\sqrt{c{x}^{2}+d}}}{c}^{-{\frac{3}{2}}}} \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 = 1.37069, size = 431, normalized size = 4.79 \begin{align*} \left [-\frac{{\left (4 \, b c d - a d^{2}\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 (2 \, a c^{2} x^{4} +{\left (4 \, b c^{2} + a c d\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{16 \, c^{2}}, -\frac{{\left (4 \, b c d - a d^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) -{\left (2 \, a c^{2} x^{4} +{\left (4 \, b c^{2} + a c d\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{8 \, c^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 28.8253, size = 144, normalized size = 1.6 \begin{align*} \frac{a c x^{5}}{4 \sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{3 a \sqrt{d} x^{3}}{8 \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{a d^{\frac{3}{2}} x}{8 c \sqrt{\frac{c x^{2}}{d} + 1}} - \frac{a d^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{8 c^{\frac{3}{2}}} + \frac{b \sqrt{d} x \sqrt{\frac{c x^{2}}{d} + 1}}{2} + \frac{b d \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{2 \sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14636, size = 142, normalized size = 1.58 \begin{align*} \frac{1}{8} \,{\left (2 \, a x^{2} \mathrm{sgn}\left (x\right ) + \frac{4 \, b c^{2} \mathrm{sgn}\left (x\right ) + a c d \mathrm{sgn}\left (x\right )}{c^{2}}\right )} \sqrt{c x^{2} + d} x - \frac{{\left (4 \, b c d \mathrm{sgn}\left (x\right ) - a d^{2} \mathrm{sgn}\left (x\right )\right )} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + d} \right |}\right )}{8 \, c^{\frac{3}{2}}} + \frac{{\left (4 \, b c d \log \left ({\left | d \right |}\right ) - a d^{2} \log \left ({\left | d \right |}\right )\right )} \mathrm{sgn}\left (x\right )}{16 \, c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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