Optimal. Leaf size=52 \[ \frac{2}{27 d^2 \sqrt{c+d x^3}}+\frac{16 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{81 \sqrt{c} d^2} \]
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Rubi [A] time = 0.0457229, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {446, 78, 63, 206} \[ \frac{2}{27 d^2 \sqrt{c+d x^3}}+\frac{16 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{81 \sqrt{c} d^2} \]
Antiderivative was successfully verified.
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Rule 446
Rule 78
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{x^5}{\left (8 c-d x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x}{(8 c-d x) (c+d x)^{3/2}} \, dx,x,x^3\right )\\ &=\frac{2}{27 d^2 \sqrt{c+d x^3}}+\frac{8 \operatorname{Subst}\left (\int \frac{1}{(8 c-d x) \sqrt{c+d x}} \, dx,x,x^3\right )}{27 d}\\ &=\frac{2}{27 d^2 \sqrt{c+d x^3}}+\frac{16 \operatorname{Subst}\left (\int \frac{1}{9 c-x^2} \, dx,x,\sqrt{c+d x^3}\right )}{27 d^2}\\ &=\frac{2}{27 d^2 \sqrt{c+d x^3}}+\frac{16 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{81 \sqrt{c} d^2}\\ \end{align*}
Mathematica [A] time = 0.030077, size = 49, normalized size = 0.94 \[ \frac{2 \left (\frac{3}{\sqrt{c+d x^3}}+\frac{8 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{\sqrt{c}}\right )}{81 d^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.01, size = 456, normalized size = 8.8 \begin{align*}{\frac{2}{3\,{d}^{2}}{\frac{1}{\sqrt{d{x}^{3}+c}}}}-8\,{\frac{c}{d} \left ({\frac{2}{27\,cd}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{c}{d}} \right ) d}}}}+{\frac{{\frac{i}{243}}\sqrt{2}}{{d}^{3}{c}^{2}}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{3}d-8\,c \right ) }{\frac{\sqrt [3]{-{d}^{2}c} \left ( i\sqrt [3]{-{d}^{2}c}{\it \_alpha}\,\sqrt{3}d-i \left ( -{d}^{2}c \right ) ^{2/3}\sqrt{3}+2\,{{\it \_alpha}}^{2}{d}^{2}-\sqrt [3]{-{d}^{2}c}{\it \_alpha}\,d- \left ( -{d}^{2}c \right ) ^{2/3} \right ) }{\sqrt{d{x}^{3}+c}}\sqrt{{\frac{i/2d}{\sqrt [3]{-{d}^{2}c}} \left ( 2\,x+{\frac{-i\sqrt{3}\sqrt [3]{-{d}^{2}c}+\sqrt [3]{-{d}^{2}c}}{d}} \right ) }}\sqrt{{\frac{d}{-3\,\sqrt [3]{-{d}^{2}c}+i\sqrt{3}\sqrt [3]{-{d}^{2}c}} \left ( x-{\frac{\sqrt [3]{-{d}^{2}c}}{d}} \right ) }}\sqrt{{\frac{-i/2d}{\sqrt [3]{-{d}^{2}c}} \left ( 2\,x+{\frac{i\sqrt{3}\sqrt [3]{-{d}^{2}c}+\sqrt [3]{-{d}^{2}c}}{d}} \right ) }}{\it EllipticPi} \left ( 1/3\,\sqrt{3}\sqrt{{\frac{i\sqrt{3}d}{\sqrt [3]{-{d}^{2}c}} \left ( x+1/2\,{\frac{\sqrt [3]{-{d}^{2}c}}{d}}-{\frac{i/2\sqrt{3}\sqrt [3]{-{d}^{2}c}}{d}} \right ) }},-1/18\,{\frac{2\,i\sqrt [3]{-{d}^{2}c}\sqrt{3}{{\it \_alpha}}^{2}d-i \left ( -{d}^{2}c \right ) ^{2/3}\sqrt{3}{\it \_alpha}+i\sqrt{3}cd-3\, \left ( -{d}^{2}c \right ) ^{2/3}{\it \_alpha}-3\,cd}{cd}},\sqrt{{\frac{i\sqrt{3}\sqrt [3]{-{d}^{2}c}}{d} \left ( -3/2\,{\frac{\sqrt [3]{-{d}^{2}c}}{d}}+{\frac{i/2\sqrt{3}\sqrt [3]{-{d}^{2}c}}{d}} \right ) ^{-1}}} \right ) }} \right ) } \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 = 1.54801, size = 344, normalized size = 6.62 \begin{align*} \left [\frac{2 \,{\left (4 \,{\left (d x^{3} + c\right )} \sqrt{c} \log \left (\frac{d x^{3} + 6 \, \sqrt{d x^{3} + c} \sqrt{c} + 10 \, c}{d x^{3} - 8 \, c}\right ) + 3 \, \sqrt{d x^{3} + c} c\right )}}{81 \,{\left (c d^{3} x^{3} + c^{2} d^{2}\right )}}, -\frac{2 \,{\left (8 \,{\left (d x^{3} + c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{d x^{3} + c} \sqrt{-c}}{3 \, c}\right ) - 3 \, \sqrt{d x^{3} + c} c\right )}}{81 \,{\left (c d^{3} x^{3} + c^{2} d^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 19.0056, size = 58, normalized size = 1.12 \begin{align*} \begin{cases} \frac{2 \left (\frac{1}{27 d \sqrt{c + d x^{3}}} - \frac{8 \operatorname{atan}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{- c}} \right )}}{81 d \sqrt{- c}}\right )}{d} & \text{for}\: d \neq 0 \\\frac{x^{6}}{48 c^{\frac{5}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12979, size = 63, normalized size = 1.21 \begin{align*} -\frac{2 \,{\left (\frac{8 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} d} - \frac{3}{\sqrt{d x^{3} + c} d}\right )}}{81 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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