Optimal. Leaf size=100 \[ -\frac{25 d}{216 c^3 \sqrt{c+d x^3}}-\frac{1}{24 c^2 x^3 \sqrt{c+d x^3}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{2592 c^{7/2}}+\frac{11 d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{96 c^{7/2}} \]
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Rubi [A] time = 0.0950825, antiderivative size = 100, 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, 103, 152, 156, 63, 208, 206} \[ -\frac{25 d}{216 c^3 \sqrt{c+d x^3}}-\frac{1}{24 c^2 x^3 \sqrt{c+d x^3}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{2592 c^{7/2}}+\frac{11 d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{96 c^{7/2}} \]
Antiderivative was successfully verified.
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Rule 446
Rule 103
Rule 152
Rule 156
Rule 63
Rule 208
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{x^4 \left (8 c-d x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x^2 (8 c-d x) (c+d x)^{3/2}} \, dx,x,x^3\right )\\ &=-\frac{1}{24 c^2 x^3 \sqrt{c+d x^3}}-\frac{\operatorname{Subst}\left (\int \frac{11 c d-\frac{3 d^2 x}{2}}{x (8 c-d x) (c+d x)^{3/2}} \, dx,x,x^3\right )}{24 c^2}\\ &=-\frac{25 d}{216 c^3 \sqrt{c+d x^3}}-\frac{1}{24 c^2 x^3 \sqrt{c+d x^3}}-\frac{\operatorname{Subst}\left (\int \frac{\frac{99 c^2 d^2}{2}-\frac{25}{4} c d^3 x}{x (8 c-d x) \sqrt{c+d x}} \, dx,x,x^3\right )}{108 c^4 d}\\ &=-\frac{25 d}{216 c^3 \sqrt{c+d x^3}}-\frac{1}{24 c^2 x^3 \sqrt{c+d x^3}}-\frac{(11 d) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+d x}} \, dx,x,x^3\right )}{192 c^3}+\frac{d^2 \operatorname{Subst}\left (\int \frac{1}{(8 c-d x) \sqrt{c+d x}} \, dx,x,x^3\right )}{1728 c^3}\\ &=-\frac{25 d}{216 c^3 \sqrt{c+d x^3}}-\frac{1}{24 c^2 x^3 \sqrt{c+d x^3}}-\frac{11 \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+d x^3}\right )}{96 c^3}+\frac{d \operatorname{Subst}\left (\int \frac{1}{9 c-x^2} \, dx,x,\sqrt{c+d x^3}\right )}{864 c^3}\\ &=-\frac{25 d}{216 c^3 \sqrt{c+d x^3}}-\frac{1}{24 c^2 x^3 \sqrt{c+d x^3}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{2592 c^{7/2}}+\frac{11 d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{96 c^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0288534, size = 77, normalized size = 0.77 \[ \frac{-d x^3 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{d x^3+c}{9 c}\right )-99 d x^3 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{d x^3}{c}+1\right )-36 c}{864 c^3 x^3 \sqrt{c+d x^3}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.024, size = 549, normalized size = 5.5 \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}{{\left (d x^{3} + c\right )}^{\frac{3}{2}}{\left (d x^{3} - 8 \, c\right )} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37141, size = 629, normalized size = 6.29 \begin{align*} \left [\frac{{\left (d^{2} x^{6} + c d x^{3}\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 ) + 297 \,{\left (d^{2} x^{6} + c d x^{3}\right )} \sqrt{c} \log \left (\frac{d x^{3} + 2 \, \sqrt{d x^{3} + c} \sqrt{c} + 2 \, c}{x^{3}}\right ) - 24 \,{\left (25 \, c d x^{3} + 9 \, c^{2}\right )} \sqrt{d x^{3} + c}}{5184 \,{\left (c^{4} d x^{6} + c^{5} x^{3}\right )}}, -\frac{297 \,{\left (d^{2} x^{6} + c d x^{3}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{d x^{3} + c} \sqrt{-c}}{c}\right ) +{\left (d^{2} x^{6} + c d x^{3}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{d x^{3} + c} \sqrt{-c}}{3 \, c}\right ) + 12 \,{\left (25 \, c d x^{3} + 9 \, c^{2}\right )} \sqrt{d x^{3} + c}}{2592 \,{\left (c^{4} d x^{6} + c^{5} x^{3}\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 = 1.1405, size = 128, normalized size = 1.28 \begin{align*} -\frac{1}{2592} \, d{\left (\frac{297 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{3}} + \frac{\arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} c^{3}} + \frac{12 \,{\left (25 \, d x^{3} + 9 \, c\right )}}{{\left ({\left (d x^{3} + c\right )}^{\frac{3}{2}} - \sqrt{d x^{3} + c} c\right )} c^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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