Optimal. Leaf size=107 \[ \frac{d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{256 c^{5/2}}+\frac{d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{256 c^{5/2}}-\frac{d \sqrt{c+d x^3}}{64 c^2 x^3}-\frac{\sqrt{c+d x^3}}{48 c x^6} \]
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Rubi [A] time = 0.0940667, antiderivative size = 107, 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, 99, 151, 156, 63, 208, 206} \[ \frac{d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{256 c^{5/2}}+\frac{d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{256 c^{5/2}}-\frac{d \sqrt{c+d x^3}}{64 c^2 x^3}-\frac{\sqrt{c+d x^3}}{48 c x^6} \]
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 )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{\sqrt{c+d x}}{x^3 (8 c-d x)} \, dx,x,x^3\right )\\ &=-\frac{\sqrt{c+d x^3}}{48 c x^6}+\frac{\operatorname{Subst}\left (\int \frac{6 c d+\frac{3 d^2 x}{2}}{x^2 (8 c-d x) \sqrt{c+d x}} \, dx,x,x^3\right )}{48 c}\\ &=-\frac{\sqrt{c+d x^3}}{48 c x^6}-\frac{d \sqrt{c+d x^3}}{64 c^2 x^3}-\frac{\operatorname{Subst}\left (\int \frac{6 c^2 d^2-3 c 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 c x^6}-\frac{d \sqrt{c+d x^3}}{64 c^2 x^3}-\frac{d^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+d x}} \, dx,x,x^3\right )}{512 c^2}+\frac{\left (3 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{(8 c-d x) \sqrt{c+d x}} \, dx,x,x^3\right )}{512 c^2}\\ &=-\frac{\sqrt{c+d x^3}}{48 c x^6}-\frac{d \sqrt{c+d x^3}}{64 c^2 x^3}-\frac{d \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+d x^3}\right )}{256 c^2}+\frac{\left (3 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{9 c-x^2} \, dx,x,\sqrt{c+d x^3}\right )}{256 c^2}\\ &=-\frac{\sqrt{c+d x^3}}{48 c x^6}-\frac{d \sqrt{c+d x^3}}{64 c^2 x^3}+\frac{d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{256 c^{5/2}}+\frac{d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{256 c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0522811, size = 96, normalized size = 0.9 \[ \frac{3 d^2 x^6 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )+3 d^2 x^6 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )-4 \sqrt{c} \sqrt{c+d x^3} \left (4 c+3 d x^3\right )}{768 c^{5/2} x^6} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.024, size = 574, normalized size = 5.4 \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 )} x^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51803, size = 439, normalized size = 4.1 \begin{align*} \left [\frac{3 \, \sqrt{c} d^{2} x^{6} \log \left (\frac{d^{2} x^{6} + 24 \, c d x^{3} + 8 \,{\left (d x^{3} + 4 \, c\right )} \sqrt{d x^{3} + c} \sqrt{c} + 32 \, c^{2}}{d x^{6} - 8 \, c x^{3}}\right ) - 8 \,{\left (3 \, c d x^{3} + 4 \, c^{2}\right )} \sqrt{d x^{3} + c}}{1536 \, c^{3} x^{6}}, -\frac{3 \, \sqrt{-c} d^{2} x^{6} \arctan \left (\frac{{\left (d x^{3} + 4 \, c\right )} \sqrt{d x^{3} + c} \sqrt{-c}}{4 \,{\left (c d x^{3} + c^{2}\right )}}\right ) + 4 \,{\left (3 \, c d x^{3} + 4 \, c^{2}\right )} \sqrt{d x^{3} + c}}{768 \, c^{3} x^{6}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{\sqrt{c + d x^{3}}}{- 8 c x^{7} + d x^{10}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11543, size = 126, normalized size = 1.18 \begin{align*} -\frac{1}{768} \, d^{2}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{2}} + \frac{3 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} c^{2}} + \frac{4 \,{\left (3 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} + \sqrt{d x^{3} + c} c\right )}}{c^{2} d^{2} x^{6}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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