Optimal. Leaf size=118 \[ \frac{x^2 (4 b c-5 a d)}{8 c^2 \sqrt{c+\frac{d}{x^2}}}+\frac{3 d (4 b c-5 a d)}{8 c^3 \sqrt{c+\frac{d}{x^2}}}-\frac{3 d (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{8 c^{7/2}}+\frac{a x^4}{4 c \sqrt{c+\frac{d}{x^2}}} \]
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Rubi [A] time = 0.0848099, antiderivative size = 120, normalized size of antiderivative = 1.02, 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, 78, 51, 63, 208} \[ \frac{3 x^2 \sqrt{c+\frac{d}{x^2}} (4 b c-5 a d)}{8 c^3}-\frac{x^2 (4 b c-5 a d)}{4 c^2 \sqrt{c+\frac{d}{x^2}}}-\frac{3 d (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{8 c^{7/2}}+\frac{a x^4}{4 c \sqrt{c+\frac{d}{x^2}}} \]
Antiderivative was successfully verified.
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Rule 446
Rule 78
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x^2}\right ) x^3}{\left (c+\frac{d}{x^2}\right )^{3/2}} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{a+b x}{x^3 (c+d x)^{3/2}} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=\frac{a x^4}{4 c \sqrt{c+\frac{d}{x^2}}}-\frac{\left (2 b c-\frac{5 a d}{2}\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 (c+d x)^{3/2}} \, dx,x,\frac{1}{x^2}\right )}{4 c}\\ &=-\frac{(4 b c-5 a d) x^2}{4 c^2 \sqrt{c+\frac{d}{x^2}}}+\frac{a x^4}{4 c \sqrt{c+\frac{d}{x^2}}}-\frac{(3 (4 b c-5 a d)) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{c+d x}} \, dx,x,\frac{1}{x^2}\right )}{8 c^2}\\ &=-\frac{(4 b c-5 a d) x^2}{4 c^2 \sqrt{c+\frac{d}{x^2}}}+\frac{3 (4 b c-5 a d) \sqrt{c+\frac{d}{x^2}} x^2}{8 c^3}+\frac{a x^4}{4 c \sqrt{c+\frac{d}{x^2}}}+\frac{(3 d (4 b c-5 a d)) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+d x}} \, dx,x,\frac{1}{x^2}\right )}{16 c^3}\\ &=-\frac{(4 b c-5 a d) x^2}{4 c^2 \sqrt{c+\frac{d}{x^2}}}+\frac{3 (4 b c-5 a d) \sqrt{c+\frac{d}{x^2}} x^2}{8 c^3}+\frac{a x^4}{4 c \sqrt{c+\frac{d}{x^2}}}+\frac{(3 (4 b c-5 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^3}\\ &=-\frac{(4 b c-5 a d) x^2}{4 c^2 \sqrt{c+\frac{d}{x^2}}}+\frac{3 (4 b c-5 a d) \sqrt{c+\frac{d}{x^2}} x^2}{8 c^3}+\frac{a x^4}{4 c \sqrt{c+\frac{d}{x^2}}}-\frac{3 d (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{8 c^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.14451, size = 111, normalized size = 0.94 \[ \frac{\sqrt{c} x \left (a \left (2 c^2 x^4-5 c d x^2-15 d^2\right )+4 b c \left (c x^2+3 d\right )\right )+3 d^{3/2} \sqrt{\frac{c x^2}{d}+1} (5 a d-4 b c) \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )}{8 c^{7/2} x \sqrt{c+\frac{d}{x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 140, normalized size = 1.2 \begin{align*}{\frac{c{x}^{2}+d}{8\,{x}^{3}} \left ( 2\,{c}^{7/2}{x}^{5}a-5\,{c}^{5/2}{x}^{3}ad+4\,{c}^{7/2}{x}^{3}b-15\,{c}^{3/2}xa{d}^{2}+12\,{c}^{5/2}xbd+15\,\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ) \sqrt{c{x}^{2}+d}ac{d}^{2}-12\,\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ) \sqrt{c{x}^{2}+d}b{c}^{2}d \right ) \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}{c}^{-{\frac{9}{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 = 1.88141, size = 662, normalized size = 5.61 \begin{align*} \left [-\frac{3 \,{\left (4 \, b c d^{2} - 5 \, a d^{3} +{\left (4 \, b c^{2} d - 5 \, a c d^{2}\right )} x^{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^{3} x^{6} +{\left (4 \, b c^{3} - 5 \, a c^{2} d\right )} x^{4} + 3 \,{\left (4 \, b c^{2} d - 5 \, a c d^{2}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{16 \,{\left (c^{5} x^{2} + c^{4} d\right )}}, \frac{3 \,{\left (4 \, b c d^{2} - 5 \, a d^{3} +{\left (4 \, b c^{2} d - 5 \, a c d^{2}\right )} x^{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^{3} x^{6} +{\left (4 \, b c^{3} - 5 \, a c^{2} d\right )} x^{4} + 3 \,{\left (4 \, b c^{2} d - 5 \, a c d^{2}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{8 \,{\left (c^{5} x^{2} + c^{4} d\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 52.4764, size = 177, normalized size = 1.5 \begin{align*} a \left (\frac{x^{5}}{4 c \sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}} - \frac{5 \sqrt{d} x^{3}}{8 c^{2} \sqrt{\frac{c x^{2}}{d} + 1}} - \frac{15 d^{\frac{3}{2}} x}{8 c^{3} \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{15 d^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{8 c^{\frac{7}{2}}}\right ) + b \left (\frac{x^{3}}{2 c \sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{3 \sqrt{d} x}{2 c^{2} \sqrt{\frac{c x^{2}}{d} + 1}} - \frac{3 d \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{2 c^{\frac{5}{2}}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1696, size = 261, normalized size = 2.21 \begin{align*} \frac{1}{8} \, d^{2}{\left (\frac{3 \,{\left (4 \, b c - 5 \, a d\right )} \arctan \left (\frac{\sqrt{\frac{c x^{2} + d}{x^{2}}}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{3} d} + \frac{8 \,{\left (b c - a d\right )}}{c^{3} d \sqrt{\frac{c x^{2} + d}{x^{2}}}} - \frac{4 \, b c^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} - 9 \, a c d \sqrt{\frac{c x^{2} + d}{x^{2}}} - \frac{4 \,{\left (c x^{2} + d\right )} b c \sqrt{\frac{c x^{2} + d}{x^{2}}}}{x^{2}} + \frac{7 \,{\left (c x^{2} + d\right )} a d \sqrt{\frac{c x^{2} + d}{x^{2}}}}{x^{2}}}{{\left (c - \frac{c x^{2} + d}{x^{2}}\right )}^{2} c^{3} d}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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