Optimal. Leaf size=124 \[ \frac{5 d \sqrt{c+d x^3}}{864 c^3 \left (8 c-d x^3\right )}-\frac{\sqrt{c+d x^3}}{24 c^2 x^3 \left (8 c-d x^3\right )}+\frac{11 d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{10368 c^{7/2}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{384 c^{7/2}} \]
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Rubi [A] time = 0.10125, antiderivative size = 124, 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, 151, 156, 63, 208, 206} \[ \frac{5 d \sqrt{c+d x^3}}{864 c^3 \left (8 c-d x^3\right )}-\frac{\sqrt{c+d x^3}}{24 c^2 x^3 \left (8 c-d x^3\right )}+\frac{11 d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{10368 c^{7/2}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{384 c^{7/2}} \]
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^4 \left (8 c-d x^3\right )^2 \sqrt{c+d x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x^2 (8 c-d x)^2 \sqrt{c+d x}} \, dx,x,x^3\right )\\ &=-\frac{\sqrt{c+d x^3}}{24 c^2 x^3 \left (8 c-d x^3\right )}-\frac{\operatorname{Subst}\left (\int \frac{2 c d-\frac{3 d^2 x}{2}}{x (8 c-d x)^2 \sqrt{c+d x}} \, dx,x,x^3\right )}{24 c^2}\\ &=\frac{5 d \sqrt{c+d x^3}}{864 c^3 \left (8 c-d x^3\right )}-\frac{\sqrt{c+d x^3}}{24 c^2 x^3 \left (8 c-d x^3\right )}+\frac{\operatorname{Subst}\left (\int \frac{-18 c^2 d^2+5 c d^3 x}{x (8 c-d x) \sqrt{c+d x}} \, dx,x,x^3\right )}{1728 c^4 d}\\ &=\frac{5 d \sqrt{c+d x^3}}{864 c^3 \left (8 c-d x^3\right )}-\frac{\sqrt{c+d x^3}}{24 c^2 x^3 \left (8 c-d x^3\right )}-\frac{d \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+d x}} \, dx,x,x^3\right )}{768 c^3}+\frac{\left (11 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{(8 c-d x) \sqrt{c+d x}} \, dx,x,x^3\right )}{6912 c^3}\\ &=\frac{5 d \sqrt{c+d x^3}}{864 c^3 \left (8 c-d x^3\right )}-\frac{\sqrt{c+d x^3}}{24 c^2 x^3 \left (8 c-d x^3\right )}-\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+d x^3}\right )}{384 c^3}+\frac{(11 d) \operatorname{Subst}\left (\int \frac{1}{9 c-x^2} \, dx,x,\sqrt{c+d x^3}\right )}{3456 c^3}\\ &=\frac{5 d \sqrt{c+d x^3}}{864 c^3 \left (8 c-d x^3\right )}-\frac{\sqrt{c+d x^3}}{24 c^2 x^3 \left (8 c-d x^3\right )}+\frac{11 d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{10368 c^{7/2}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{384 c^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.133074, size = 97, normalized size = 0.78 \[ \frac{\frac{12 \sqrt{c} \sqrt{c+d x^3} \left (36 c-5 d x^3\right )}{d x^6-8 c x^3}+11 d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )+27 d \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{10368 c^{7/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.014, size = 926, normalized size = 7.5 \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}{\sqrt{d x^{3} + c}{\left (d x^{3} - 8 \, c\right )}^{2} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78352, size = 653, normalized size = 5.27 \begin{align*} \left [\frac{11 \,{\left (d^{2} x^{6} - 8 \, 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 ) + 27 \,{\left (d^{2} x^{6} - 8 \, 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 (5 \, c d x^{3} - 36 \, c^{2}\right )} \sqrt{d x^{3} + c}}{20736 \,{\left (c^{4} d x^{6} - 8 \, c^{5} x^{3}\right )}}, -\frac{27 \,{\left (d^{2} x^{6} - 8 \, c d x^{3}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{d x^{3} + c} \sqrt{-c}}{c}\right ) + 11 \,{\left (d^{2} x^{6} - 8 \, c d x^{3}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{d x^{3} + c} \sqrt{-c}}{3 \, c}\right ) + 12 \,{\left (5 \, c d x^{3} - 36 \, c^{2}\right )} \sqrt{d x^{3} + c}}{10368 \,{\left (c^{4} d x^{6} - 8 \, 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 = 1.13434, size = 153, normalized size = 1.23 \begin{align*} -\frac{1}{10368} \, d{\left (\frac{27 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{3}} + \frac{11 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} c^{3}} + \frac{12 \,{\left (5 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} - 41 \, \sqrt{d x^{3} + c} c\right )}}{{\left ({\left (d x^{3} + c\right )}^{2} - 10 \,{\left (d x^{3} + c\right )} c + 9 \, c^{2}\right )} c^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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