Optimal. Leaf size=90 \[ \frac {2 c^2}{27 d^4 \sqrt {c+d x^3}}-\frac {4 c \sqrt {c+d x^3}}{d^4}-\frac {2 \left (c+d x^3\right )^{3/2}}{9 d^4}+\frac {1024 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{81 d^4} \]
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Rubi [A]
time = 0.07, antiderivative size = 90, normalized size of antiderivative = 1.00, 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 = {457, 89, 45, 65,
212} \begin {gather*} \frac {1024 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{81 d^4}+\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 65
Rule 89
Rule 212
Rule 457
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} \text {Subst}\left (\int \frac {x^3}{(8 c-d x) (c+d x)^{3/2}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {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 ) \text {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{27 d^3}-\frac {\text {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 ) \text {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{27 d^4}-\frac {\text {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*}
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Mathematica [A]
time = 0.08, size = 70, normalized size = 0.78 \begin {gather*} \frac {2 \left (-\frac {3 \left (56 c^2+60 c d x^3+3 d^2 x^6\right )}{\sqrt {c+d x^3}}+512 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )\right )}{81 d^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.37, size = 560, normalized size = 6.22 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 82, normalized size = 0.91 \begin {gather*} -\frac {2 \, {\left (256 \, c^{\frac {3}{2}} \log \left (\frac {\sqrt {d x^{3} + c} - 3 \, \sqrt {c}}{\sqrt {d x^{3} + c} + 3 \, \sqrt {c}}\right ) + 9 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} + 162 \, \sqrt {d x^{3} + c} c - \frac {3 \, c^{2}}{\sqrt {d x^{3} + c}}\right )}}{81 \, d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.49, size = 189, normalized size = 2.10 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.63, size = 82, normalized size = 0.91 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.78, size = 95, normalized size = 1.06 \begin {gather*} \frac {512\,c^{3/2}\,\ln \left (\frac {10\,c+d\,x^3+6\,\sqrt {c}\,\sqrt {d\,x^3+c}}{8\,c-d\,x^3}\right )}{81\,d^4}-\frac {38\,c\,\sqrt {d\,x^3+c}}{9\,d^4}+\frac {2\,c^2}{27\,d^4\,\sqrt {d\,x^3+c}}-\frac {2\,x^3\,\sqrt {d\,x^3+c}}{9\,d^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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