Optimal. Leaf size=104 \[ \frac{9 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{256 c^{3/2}}-\frac{37 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{768 c^{3/2}}-\frac{11 d \sqrt{c+d x^3}}{192 c x^3}-\frac{\sqrt{c+d x^3}}{48 x^6} \]
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Rubi [A] time = 0.0946632, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {446, 98, 151, 156, 63, 208, 206} \[ \frac{9 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{256 c^{3/2}}-\frac{37 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{768 c^{3/2}}-\frac{11 d \sqrt{c+d x^3}}{192 c x^3}-\frac{\sqrt{c+d x^3}}{48 x^6} \]
Antiderivative was successfully verified.
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Rule 446
Rule 98
Rule 151
Rule 156
Rule 63
Rule 208
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (c+d x^3\right )^{3/2}}{x^7 \left (8 c-d x^3\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(c+d x)^{3/2}}{x^3 (8 c-d x)} \, dx,x,x^3\right )\\ &=-\frac{\sqrt{c+d x^3}}{48 x^6}-\frac{\operatorname{Subst}\left (\int \frac{-22 c^2 d-\frac{35}{2} c d^2 x}{x^2 (8 c-d x) \sqrt{c+d x}} \, dx,x,x^3\right )}{48 c}\\ &=-\frac{\sqrt{c+d x^3}}{48 x^6}-\frac{11 d \sqrt{c+d x^3}}{192 c x^3}+\frac{\operatorname{Subst}\left (\int \frac{74 c^3 d^2+11 c^2 d^3 x}{x (8 c-d x) \sqrt{c+d x}} \, dx,x,x^3\right )}{384 c^3}\\ &=-\frac{\sqrt{c+d x^3}}{48 x^6}-\frac{11 d \sqrt{c+d x^3}}{192 c x^3}+\frac{\left (37 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+d x}} \, dx,x,x^3\right )}{1536 c}+\frac{\left (27 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{(8 c-d x) \sqrt{c+d x}} \, dx,x,x^3\right )}{512 c}\\ &=-\frac{\sqrt{c+d x^3}}{48 x^6}-\frac{11 d \sqrt{c+d x^3}}{192 c x^3}+\frac{(37 d) \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+d x^3}\right )}{768 c}+\frac{\left (27 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{9 c-x^2} \, dx,x,\sqrt{c+d x^3}\right )}{256 c}\\ &=-\frac{\sqrt{c+d x^3}}{48 x^6}-\frac{11 d \sqrt{c+d x^3}}{192 c x^3}+\frac{9 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{256 c^{3/2}}-\frac{37 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{768 c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0516253, size = 96, normalized size = 0.92 \[ \frac{27 d^2 x^6 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )-37 d^2 x^6 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )-4 \sqrt{c} \sqrt{c+d x^3} \left (4 c+11 d x^3\right )}{768 c^{3/2} x^6} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.026, size = 617, normalized size = 5.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\left (d x^{3} + c\right )}^{\frac{3}{2}}}{{\left (d x^{3} - 8 \, c\right )} x^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43754, size = 532, normalized size = 5.12 \begin{align*} \left [\frac{27 \, \sqrt{c} d^{2} x^{6} \log \left (\frac{d x^{3} + 6 \, \sqrt{d x^{3} + c} \sqrt{c} + 10 \, c}{d x^{3} - 8 \, c}\right ) + 37 \, \sqrt{c} d^{2} x^{6} \log \left (\frac{d x^{3} - 2 \, \sqrt{d x^{3} + c} \sqrt{c} + 2 \, c}{x^{3}}\right ) - 8 \,{\left (11 \, c d x^{3} + 4 \, c^{2}\right )} \sqrt{d x^{3} + c}}{1536 \, c^{2} x^{6}}, \frac{37 \, \sqrt{-c} d^{2} x^{6} \arctan \left (\frac{\sqrt{d x^{3} + c} \sqrt{-c}}{c}\right ) - 27 \, \sqrt{-c} d^{2} x^{6} \arctan \left (\frac{\sqrt{d x^{3} + c} \sqrt{-c}}{3 \, c}\right ) - 4 \,{\left (11 \, c d x^{3} + 4 \, c^{2}\right )} \sqrt{d x^{3} + c}}{768 \, c^{2} x^{6}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10344, size = 127, normalized size = 1.22 \begin{align*} \frac{1}{768} \, d^{2}{\left (\frac{37 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c} - \frac{27 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} c} - \frac{4 \,{\left (11 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} - 7 \, \sqrt{d x^{3} + c} c\right )}}{c d^{2} x^{6}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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