Optimal. Leaf size=90 \[ \frac{2 c^2}{27 d^4 \sqrt{c+d x^3}}+\frac{1024 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{81 d^4}-\frac{4 c \sqrt{c+d x^3}}{d^4}-\frac{2 \left (c+d x^3\right )^{3/2}}{9 d^4} \]
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Rubi [A] time = 0.10356, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {446, 87, 43, 63, 206} \[ \frac{2 c^2}{27 d^4 \sqrt{c+d x^3}}+\frac{1024 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{81 d^4}-\frac{4 c \sqrt{c+d x^3}}{d^4}-\frac{2 \left (c+d x^3\right )^{3/2}}{9 d^4} \]
Antiderivative was successfully verified.
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Rule 446
Rule 87
Rule 43
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{x^{11}}{\left (8 c-d x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^3}{(8 c-d x) (c+d x)^{3/2}} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{c^2}{9 d^3 (c+d x)^{3/2}}-\frac{7 c}{d^3 \sqrt{c+d x}}-\frac{x}{d^2 \sqrt{c+d x}}+\frac{512 c^2}{9 d^3 (8 c-d x) \sqrt{c+d x}}\right ) \, dx,x,x^3\right )\\ &=\frac{2 c^2}{27 d^4 \sqrt{c+d x^3}}-\frac{14 c \sqrt{c+d x^3}}{3 d^4}+\frac{\left (512 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{(8 c-d x) \sqrt{c+d x}} \, dx,x,x^3\right )}{27 d^3}-\frac{\operatorname{Subst}\left (\int \frac{x}{\sqrt{c+d x}} \, dx,x,x^3\right )}{3 d^2}\\ &=\frac{2 c^2}{27 d^4 \sqrt{c+d x^3}}-\frac{14 c \sqrt{c+d x^3}}{3 d^4}+\frac{\left (1024 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{9 c-x^2} \, dx,x,\sqrt{c+d x^3}\right )}{27 d^4}-\frac{\operatorname{Subst}\left (\int \left (-\frac{c}{d \sqrt{c+d x}}+\frac{\sqrt{c+d x}}{d}\right ) \, dx,x,x^3\right )}{3 d^2}\\ &=\frac{2 c^2}{27 d^4 \sqrt{c+d x^3}}-\frac{4 c \sqrt{c+d x^3}}{d^4}-\frac{2 \left (c+d x^3\right )^{3/2}}{9 d^4}+\frac{1024 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{81 d^4}\\ \end{align*}
Mathematica [C] time = 0.0447, size = 66, normalized size = 0.73 \[ -\frac{2 \left (512 c^2 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{d x^3+c}{9 c}\right )-456 c^2+60 c d x^3+3 d^2 x^6\right )}{27 d^4 \sqrt{c+d x^3}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.053, size = 560, normalized size = 6.2 \begin{align*} -{\frac{1}{d} \left ( -{\frac{2\,{c}^{2}}{3\,{d}^{3}}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{c}{d}} \right ) d}}}}+{\frac{2\,{x}^{3}}{9\,{d}^{2}}\sqrt{d{x}^{3}+c}}-{\frac{10\,c}{9\,{d}^{3}}\sqrt{d{x}^{3}+c}} \right ) }-8\,{\frac{c}{{d}^{2}} \left ( 2/3\,{\frac{c}{{d}^{2}}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{c}{d}} \right ) d}}}}+2/3\,{\frac{\sqrt{d{x}^{3}+c}}{{d}^{2}}} \right ) }+{\frac{128\,{c}^{2}}{3\,{d}^{4}}{\frac{1}{\sqrt{d{x}^{3}+c}}}}-512\,{\frac{{c}^{3}}{{d}^{3}} \left ({\frac{2}{27\,cd}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{c}{d}} \right ) d}}}}+{\frac{{\frac{i}{243}}\sqrt{2}}{{d}^{3}{c}^{2}}\sum _{{\it \_alpha}={\it RootOf} \left ( d{{\it \_Z}}^{3}-8\,c \right ) }{\frac{\sqrt [3]{-{d}^{2}c} \left ( i\sqrt [3]{-{d}^{2}c}{\it \_alpha}\,\sqrt{3}d-i\sqrt{3} \left ( -{d}^{2}c \right ) ^{2/3}+2\,{{\it \_alpha}}^{2}{d}^{2}-\sqrt [3]{-{d}^{2}c}{\it \_alpha}\,d- \left ( -{d}^{2}c \right ) ^{2/3} \right ) }{\sqrt{d{x}^{3}+c}}\sqrt{{\frac{i/2d}{\sqrt [3]{-{d}^{2}c}} \left ( 2\,x+{\frac{-i\sqrt{3}\sqrt [3]{-{d}^{2}c}+\sqrt [3]{-{d}^{2}c}}{d}} \right ) }}\sqrt{{\frac{d}{-3\,\sqrt [3]{-{d}^{2}c}+i\sqrt{3}\sqrt [3]{-{d}^{2}c}} \left ( x-{\frac{\sqrt [3]{-{d}^{2}c}}{d}} \right ) }}\sqrt{{\frac{-i/2d}{\sqrt [3]{-{d}^{2}c}} \left ( 2\,x+{\frac{i\sqrt{3}\sqrt [3]{-{d}^{2}c}+\sqrt [3]{-{d}^{2}c}}{d}} \right ) }}{\it EllipticPi} \left ( 1/3\,\sqrt{3}\sqrt{{\frac{id\sqrt{3}}{\sqrt [3]{-{d}^{2}c}} \left ( x+1/2\,{\frac{\sqrt [3]{-{d}^{2}c}}{d}}-{\frac{i/2\sqrt{3}\sqrt [3]{-{d}^{2}c}}{d}} \right ) }},-1/18\,{\frac{2\,i\sqrt [3]{-{d}^{2}c}\sqrt{3}{{\it \_alpha}}^{2}d-i \left ( -{d}^{2}c \right ) ^{2/3}\sqrt{3}{\it \_alpha}+i\sqrt{3}cd-3\, \left ( -{d}^{2}c \right ) ^{2/3}{\it \_alpha}-3\,cd}{cd}},\sqrt{{\frac{i\sqrt{3}\sqrt [3]{-{d}^{2}c}}{d} \left ( -3/2\,{\frac{\sqrt [3]{-{d}^{2}c}}{d}}+{\frac{i/2\sqrt{3}\sqrt [3]{-{d}^{2}c}}{d}} \right ) ^{-1}}} \right ) }} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.584, size = 436, normalized size = 4.84 \begin{align*} \left [\frac{2 \,{\left (256 \,{\left (c d x^{3} + c^{2}\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 ) - 3 \,{\left (3 \, d^{2} x^{6} + 60 \, c d x^{3} + 56 \, c^{2}\right )} \sqrt{d x^{3} + c}\right )}}{81 \,{\left (d^{5} x^{3} + c d^{4}\right )}}, -\frac{2 \,{\left (512 \,{\left (c d x^{3} + c^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{d x^{3} + c} \sqrt{-c}}{3 \, c}\right ) + 3 \,{\left (3 \, d^{2} x^{6} + 60 \, c d x^{3} + 56 \, c^{2}\right )} \sqrt{d x^{3} + c}\right )}}{81 \,{\left (d^{5} x^{3} + c d^{4}\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 = 1.53924, size = 111, normalized size = 1.23 \begin{align*} -\frac{1024 \, c^{2} \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{81 \, \sqrt{-c} d^{4}} + \frac{2 \, c^{2}}{27 \, \sqrt{d x^{3} + c} d^{4}} - \frac{2 \,{\left ({\left (d x^{3} + c\right )}^{\frac{3}{2}} d^{8} + 18 \, \sqrt{d x^{3} + c} c d^{8}\right )}}{9 \, d^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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