Optimal. Leaf size=95 \[ -\frac {2944 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{81 d^4}+\frac {2 \sqrt {c+d x^3} \left (170 c+7 d x^3\right )}{27 d^4}+\frac {8 x^6 \sqrt {c+d x^3}}{27 d^2 \left (8 c-d x^3\right )} \]
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Rubi [A] time = 0.07, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {446, 98, 147, 63, 206} \begin {gather*} -\frac {2944 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{81 d^4}+\frac {8 x^6 \sqrt {c+d x^3}}{27 d^2 \left (8 c-d x^3\right )}+\frac {2 \sqrt {c+d x^3} \left (170 c+7 d x^3\right )}{27 d^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 98
Rule 147
Rule 206
Rule 446
Rubi steps
\begin {align*} \int \frac {x^{11}}{\left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^3}{(8 c-d x)^2 \sqrt {c+d x}} \, dx,x,x^3\right )\\ &=\frac {8 x^6 \sqrt {c+d x^3}}{27 d^2 \left (8 c-d x^3\right )}-\frac {\operatorname {Subst}\left (\int \frac {x \left (16 c^2+21 c d x\right )}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{27 c d^2}\\ &=\frac {8 x^6 \sqrt {c+d x^3}}{27 d^2 \left (8 c-d x^3\right )}+\frac {2 \sqrt {c+d x^3} \left (170 c+7 d x^3\right )}{27 d^4}-\frac {\left (1472 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 {8 x^6 \sqrt {c+d x^3}}{27 d^2 \left (8 c-d x^3\right )}+\frac {2 \sqrt {c+d x^3} \left (170 c+7 d x^3\right )}{27 d^4}-\frac {\left (2944 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 {8 x^6 \sqrt {c+d x^3}}{27 d^2 \left (8 c-d x^3\right )}+\frac {2 \sqrt {c+d x^3} \left (170 c+7 d x^3\right )}{27 d^4}-\frac {2944 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 = 0.05, size = 91, normalized size = 0.96 \begin {gather*} \frac {2944 c^{3/2} \left (8 c-d x^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )+6 \sqrt {c+d x^3} \left (-1360 c^2+114 c d x^3+3 d^2 x^6\right )}{81 d^4 \left (d x^3-8 c\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 84, normalized size = 0.88 \begin {gather*} -\frac {2944 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{81 d^4}-\frac {2 \sqrt {c+d x^3} \left (1360 c^2-114 c d x^3-3 d^2 x^6\right )}{27 d^4 \left (d x^3-8 c\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 195, normalized size = 2.05 \begin {gather*} \left [\frac {2 \, {\left (736 \, {\left (c d x^{3} - 8 \, 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} + 114 \, c d x^{3} - 1360 \, c^{2}\right )} \sqrt {d x^{3} + c}\right )}}{81 \, {\left (d^{5} x^{3} - 8 \, c d^{4}\right )}}, \frac {2 \, {\left (1472 \, {\left (c d x^{3} - 8 \, 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} + 114 \, c d x^{3} - 1360 \, c^{2}\right )} \sqrt {d x^{3} + c}\right )}}{81 \, {\left (d^{5} x^{3} - 8 \, c d^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 93, normalized size = 0.98 \begin {gather*} \frac {2944 \, c^{2} \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{81 \, \sqrt {-c} d^{4}} - \frac {512 \, \sqrt {d x^{3} + c} c^{2}}{27 \, {\left (d x^{3} - 8 \, c\right )} d^{4}} + \frac {2 \, {\left ({\left (d x^{3} + c\right )}^{\frac {3}{2}} d^{8} + 45 \, \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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maple [C] time = 0.30, size = 916, normalized size = 9.64
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 93, normalized size = 0.98 \begin {gather*} \frac {2 \, {\left (736 \, 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}} + 405 \, \sqrt {d x^{3} + c} c - \frac {768 \, \sqrt {d x^{3} + c} c^{2}}{d x^{3} - 8 \, c}\right )}}{81 \, d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.06, size = 107, normalized size = 1.13 \begin {gather*} \frac {92\,c\,\sqrt {d\,x^3+c}}{9\,d^4}+\frac {1472\,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 {2\,x^3\,\sqrt {d\,x^3+c}}{9\,d^3}+\frac {512\,c^2\,\sqrt {d\,x^3+c}}{27\,d^4\,\left (8\,c-d\,x^3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] 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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