Optimal. Leaf size=164 \[ \frac{5 d^2 \sqrt{c+d x^3}}{1536 c^3 \left (8 c-d x^3\right )}+\frac{23 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{18432 c^{7/2}}-\frac{d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{2048 c^{7/2}}-\frac{7 d \sqrt{c+d x^3}}{384 c^2 x^3 \left (8 c-d x^3\right )}-\frac{\sqrt{c+d x^3}}{48 c x^6 \left (8 c-d x^3\right )} \]
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Rubi [A] time = 0.131336, 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, 99, 151, 156, 63, 208, 206} \[ \frac{5 d^2 \sqrt{c+d x^3}}{1536 c^3 \left (8 c-d x^3\right )}+\frac{23 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{18432 c^{7/2}}-\frac{d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{2048 c^{7/2}}-\frac{7 d \sqrt{c+d x^3}}{384 c^2 x^3 \left (8 c-d x^3\right )}-\frac{\sqrt{c+d x^3}}{48 c x^6 \left (8 c-d x^3\right )} \]
Antiderivative was successfully verified.
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Rule 446
Rule 99
Rule 151
Rule 156
Rule 63
Rule 208
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{c+d x^3}}{x^7 \left (8 c-d x^3\right )^2} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{\sqrt{c+d x}}{x^3 (8 c-d x)^2} \, dx,x,x^3\right )\\ &=-\frac{\sqrt{c+d x^3}}{48 c x^6 \left (8 c-d x^3\right )}+\frac{\operatorname{Subst}\left (\int \frac{7 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}\\ &=-\frac{\sqrt{c+d x^3}}{48 c x^6 \left (8 c-d x^3\right )}-\frac{7 d \sqrt{c+d x^3}}{384 c^2 x^3 \left (8 c-d x^3\right )}-\frac{\operatorname{Subst}\left (\int \frac{-6 c^2 d^2-\frac{21}{2} c d^3 x}{x (8 c-d x)^2 \sqrt{c+d x}} \, dx,x,x^3\right )}{384 c^3}\\ &=\frac{5 d^2 \sqrt{c+d x^3}}{1536 c^3 \left (8 c-d x^3\right )}-\frac{\sqrt{c+d x^3}}{48 c x^6 \left (8 c-d x^3\right )}-\frac{7 d \sqrt{c+d x^3}}{384 c^2 x^3 \left (8 c-d x^3\right )}+\frac{\operatorname{Subst}\left (\int \frac{54 c^3 d^3+45 c^2 d^4 x}{x (8 c-d x) \sqrt{c+d x}} \, dx,x,x^3\right )}{27648 c^5 d}\\ &=\frac{5 d^2 \sqrt{c+d x^3}}{1536 c^3 \left (8 c-d x^3\right )}-\frac{\sqrt{c+d x^3}}{48 c x^6 \left (8 c-d x^3\right )}-\frac{7 d \sqrt{c+d x^3}}{384 c^2 x^3 \left (8 c-d x^3\right )}+\frac{d^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+d x}} \, dx,x,x^3\right )}{4096 c^3}+\frac{\left (23 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{(8 c-d x) \sqrt{c+d x}} \, dx,x,x^3\right )}{12288 c^3}\\ &=\frac{5 d^2 \sqrt{c+d x^3}}{1536 c^3 \left (8 c-d x^3\right )}-\frac{\sqrt{c+d x^3}}{48 c x^6 \left (8 c-d x^3\right )}-\frac{7 d \sqrt{c+d x^3}}{384 c^2 x^3 \left (8 c-d x^3\right )}+\frac{d \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+d x^3}\right )}{2048 c^3}+\frac{\left (23 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{9 c-x^2} \, dx,x,\sqrt{c+d x^3}\right )}{6144 c^3}\\ &=\frac{5 d^2 \sqrt{c+d x^3}}{1536 c^3 \left (8 c-d x^3\right )}-\frac{\sqrt{c+d x^3}}{48 c x^6 \left (8 c-d x^3\right )}-\frac{7 d \sqrt{c+d x^3}}{384 c^2 x^3 \left (8 c-d x^3\right )}+\frac{23 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{18432 c^{7/2}}-\frac{d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{2048 c^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.171532, size = 112, normalized size = 0.68 \[ \frac{\frac{12 \sqrt{c} \sqrt{c+d x^3} \left (32 c^2+28 c d x^3-5 d^2 x^6\right )}{d x^9-8 c x^6}+23 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )-9 d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{18432 c^{7/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.015, size = 1020, normalized size = 6.2 \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{\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 = 2.03926, size = 706, normalized size = 4.3 \begin{align*} \left [\frac{23 \,{\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 ) + 9 \,{\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 (5 \, c d^{2} x^{6} - 28 \, c^{2} d x^{3} - 32 \, c^{3}\right )} \sqrt{d x^{3} + c}}{36864 \,{\left (c^{4} d x^{9} - 8 \, c^{5} x^{6}\right )}}, \frac{9 \,{\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 ) - 23 \,{\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 (5 \, c d^{2} x^{6} - 28 \, c^{2} d x^{3} - 32 \, c^{3}\right )} \sqrt{d x^{3} + c}}{18432 \,{\left (c^{4} d x^{9} - 8 \, c^{5} 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 = 1.11387, size = 140, normalized size = 0.85 \begin{align*} \frac{1}{18432} \, d^{2}{\left (\frac{9 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{3}} - \frac{23 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} c^{3}} - \frac{12 \, \sqrt{d x^{3} + c}}{{\left (d x^{3} - 8 \, c\right )} c^{3}} - \frac{48 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}}}{c^{3} d^{2} x^{6}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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