Optimal. Leaf size=164 \[ -\frac{35 d^2 \sqrt{c+d x^3}}{13824 c^4 \left (8 c-d x^3\right )}+\frac{31 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{165888 c^{9/2}}-\frac{19 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{6144 c^{9/2}}+\frac{3 d \sqrt{c+d x^3}}{128 c^3 x^3 \left (8 c-d x^3\right )}-\frac{\sqrt{c+d x^3}}{48 c^2 x^6 \left (8 c-d x^3\right )} \]
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Rubi [A] time = 0.128709, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {446, 103, 151, 156, 63, 208, 206} \[ -\frac{35 d^2 \sqrt{c+d x^3}}{13824 c^4 \left (8 c-d x^3\right )}+\frac{31 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{165888 c^{9/2}}-\frac{19 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{6144 c^{9/2}}+\frac{3 d \sqrt{c+d x^3}}{128 c^3 x^3 \left (8 c-d x^3\right )}-\frac{\sqrt{c+d x^3}}{48 c^2 x^6 \left (8 c-d x^3\right )} \]
Antiderivative was successfully verified.
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Rule 446
Rule 103
Rule 151
Rule 156
Rule 63
Rule 208
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{x^7 \left (8 c-d x^3\right )^2 \sqrt{c+d x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x^3 (8 c-d x)^2 \sqrt{c+d x}} \, dx,x,x^3\right )\\ &=-\frac{\sqrt{c+d x^3}}{48 c^2 x^6 \left (8 c-d x^3\right )}-\frac{\operatorname{Subst}\left (\int \frac{9 c d-\frac{5 d^2 x}{2}}{x^2 (8 c-d x)^2 \sqrt{c+d x}} \, dx,x,x^3\right )}{48 c^2}\\ &=-\frac{\sqrt{c+d x^3}}{48 c^2 x^6 \left (8 c-d x^3\right )}+\frac{3 d \sqrt{c+d x^3}}{128 c^3 x^3 \left (8 c-d x^3\right )}+\frac{\operatorname{Subst}\left (\int \frac{38 c^2 d^2-\frac{27}{2} c d^3 x}{x (8 c-d x)^2 \sqrt{c+d x}} \, dx,x,x^3\right )}{384 c^4}\\ &=-\frac{35 d^2 \sqrt{c+d x^3}}{13824 c^4 \left (8 c-d x^3\right )}-\frac{\sqrt{c+d x^3}}{48 c^2 x^6 \left (8 c-d x^3\right )}+\frac{3 d \sqrt{c+d x^3}}{128 c^3 x^3 \left (8 c-d x^3\right )}-\frac{\operatorname{Subst}\left (\int \frac{-342 c^3 d^3+35 c^2 d^4 x}{x (8 c-d x) \sqrt{c+d x}} \, dx,x,x^3\right )}{27648 c^6 d}\\ &=-\frac{35 d^2 \sqrt{c+d x^3}}{13824 c^4 \left (8 c-d x^3\right )}-\frac{\sqrt{c+d x^3}}{48 c^2 x^6 \left (8 c-d x^3\right )}+\frac{3 d \sqrt{c+d x^3}}{128 c^3 x^3 \left (8 c-d x^3\right )}+\frac{\left (19 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+d x}} \, dx,x,x^3\right )}{12288 c^4}+\frac{\left (31 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{(8 c-d x) \sqrt{c+d x}} \, dx,x,x^3\right )}{110592 c^4}\\ &=-\frac{35 d^2 \sqrt{c+d x^3}}{13824 c^4 \left (8 c-d x^3\right )}-\frac{\sqrt{c+d x^3}}{48 c^2 x^6 \left (8 c-d x^3\right )}+\frac{3 d \sqrt{c+d x^3}}{128 c^3 x^3 \left (8 c-d x^3\right )}+\frac{(19 d) \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+d x^3}\right )}{6144 c^4}+\frac{\left (31 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{9 c-x^2} \, dx,x,\sqrt{c+d x^3}\right )}{55296 c^4}\\ &=-\frac{35 d^2 \sqrt{c+d x^3}}{13824 c^4 \left (8 c-d x^3\right )}-\frac{\sqrt{c+d x^3}}{48 c^2 x^6 \left (8 c-d x^3\right )}+\frac{3 d \sqrt{c+d x^3}}{128 c^3 x^3 \left (8 c-d x^3\right )}+\frac{31 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{165888 c^{9/2}}-\frac{19 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{6144 c^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.180005, size = 112, normalized size = 0.68 \[ \frac{\frac{12 \sqrt{c} \sqrt{c+d x^3} \left (288 c^2-324 c d x^3+35 d^2 x^6\right )}{d x^9-8 c x^6}+31 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )-513 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{165888 c^{9/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.013, size = 989, normalized size = 6. \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{1}{\sqrt{d x^{3} + c}{\left (d x^{3} - 8 \, c\right )}^{2} x^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73586, size = 722, normalized size = 4.4 \begin{align*} \left [\frac{31 \,{\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\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 ) + 513 \,{\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\right )} \sqrt{c} \log \left (\frac{d x^{3} - 2 \, \sqrt{d x^{3} + c} \sqrt{c} + 2 \, c}{x^{3}}\right ) + 24 \,{\left (35 \, c d^{2} x^{6} - 324 \, c^{2} d x^{3} + 288 \, c^{3}\right )} \sqrt{d x^{3} + c}}{331776 \,{\left (c^{5} d x^{9} - 8 \, c^{6} x^{6}\right )}}, \frac{513 \,{\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{d x^{3} + c} \sqrt{-c}}{c}\right ) - 31 \,{\left (d^{3} x^{9} - 8 \, c d^{2} x^{6}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{d x^{3} + c} \sqrt{-c}}{3 \, c}\right ) + 12 \,{\left (35 \, c d^{2} x^{6} - 324 \, c^{2} d x^{3} + 288 \, c^{3}\right )} \sqrt{d x^{3} + c}}{165888 \,{\left (c^{5} d x^{9} - 8 \, c^{6} x^{6}\right )}}\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 = 1.09662, size = 158, normalized size = 0.96 \begin{align*} \frac{1}{165888} \, d^{2}{\left (\frac{513 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{4}} - \frac{31 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} c^{4}} - \frac{12 \, \sqrt{d x^{3} + c}}{{\left (d x^{3} - 8 \, c\right )} c^{4}} + \frac{432 \,{\left ({\left (d x^{3} + c\right )}^{\frac{3}{2}} - 2 \, \sqrt{d x^{3} + c} c\right )}}{c^{4} d^{2} x^{6}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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