Optimal. Leaf size=95 \[ \frac {2 \left (38 c+39 d x^3\right )}{81 d^4 \sqrt {c+d x^3}}-\frac {640 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{243 d^4}+\frac {8 x^6}{27 d^2 \left (8 c-d x^3\right ) \sqrt {c+d x^3}} \]
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Rubi [A] time = 0.08, 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, 146, 63, 206} \begin {gather*} \frac {8 x^6}{27 d^2 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}+\frac {2 \left (38 c+39 d x^3\right )}{81 d^4 \sqrt {c+d x^3}}-\frac {640 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{243 d^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 98
Rule 146
Rule 206
Rule 446
Rubi steps
\begin {align*} \int \frac {x^{11}}{\left (8 c-d x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^3}{(8 c-d x)^2 (c+d x)^{3/2}} \, dx,x,x^3\right )\\ &=\frac {8 x^6}{27 d^2 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}-\frac {\operatorname {Subst}\left (\int \frac {x \left (16 c^2+13 c d x\right )}{(8 c-d x) (c+d x)^{3/2}} \, dx,x,x^3\right )}{27 c d^2}\\ &=\frac {8 x^6}{27 d^2 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}+\frac {2 \left (38 c+39 d x^3\right )}{81 d^4 \sqrt {c+d x^3}}-\frac {(320 c) \operatorname {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{81 d^3}\\ &=\frac {8 x^6}{27 d^2 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}+\frac {2 \left (38 c+39 d x^3\right )}{81 d^4 \sqrt {c+d x^3}}-\frac {(640 c) \operatorname {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{81 d^4}\\ &=\frac {8 x^6}{27 d^2 \left (8 c-d x^3\right ) \sqrt {c+d x^3}}+\frac {2 \left (38 c+39 d x^3\right )}{81 d^4 \sqrt {c+d x^3}}-\frac {640 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{243 d^4}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 88, normalized size = 0.93 \begin {gather*} \frac {6 \left (752 c^2-198 c d x^3+9 d^2 x^6\right )-640 c \left (8 c-d x^3\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {d x^3+c}{9 c}\right )}{81 d^4 \left (d x^3-8 c\right ) \sqrt {c+d x^3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 127, normalized size = 1.34 \begin {gather*} \frac {\left (\frac {5120 c^{3/2}}{243 d^4}-\frac {640 \sqrt {c} x^3}{243 d^3}\right ) \sqrt {c+d x^3} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )-\frac {608 c^2}{81 d^4}-\frac {548 c x^3}{81 d^3}+\frac {2 x^6}{3 d^2}}{d x^3 \sqrt {c+d x^3}-8 c \sqrt {c+d x^3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 233, normalized size = 2.45 \begin {gather*} \left [\frac {2 \, {\left (160 \, {\left (d^{2} x^{6} - 7 \, 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 (27 \, d^{2} x^{6} - 274 \, c d x^{3} - 304 \, c^{2}\right )} \sqrt {d x^{3} + c}\right )}}{243 \, {\left (d^{6} x^{6} - 7 \, c d^{5} x^{3} - 8 \, c^{2} d^{4}\right )}}, \frac {2 \, {\left (320 \, {\left (d^{2} x^{6} - 7 \, 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 (27 \, d^{2} x^{6} - 274 \, c d x^{3} - 304 \, c^{2}\right )} \sqrt {d x^{3} + c}\right )}}{243 \, {\left (d^{6} x^{6} - 7 \, c d^{5} x^{3} - 8 \, c^{2} 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 = 88, normalized size = 0.93 \begin {gather*} \frac {640 \, c \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{243 \, \sqrt {-c} d^{4}} + \frac {2 \, \sqrt {d x^{3} + c}}{3 \, d^{4}} - \frac {2 \, {\left (85 \, {\left (d x^{3} + c\right )} c + 3 \, c^{2}\right )}}{81 \, {\left ({\left (d x^{3} + c\right )}^{\frac {3}{2}} - 9 \, \sqrt {d x^{3} + c} c\right )} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.29, size = 970, normalized size = 10.21
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.21, size = 98, normalized size = 1.03 \begin {gather*} \frac {2 \, {\left (160 \, \sqrt {c} \log \left (\frac {\sqrt {d x^{3} + c} - 3 \, \sqrt {c}}{\sqrt {d x^{3} + c} + 3 \, \sqrt {c}}\right ) + 81 \, \sqrt {d x^{3} + c} - \frac {3 \, {\left (85 \, {\left (d x^{3} + c\right )} c + 3 \, c^{2}\right )}}{{\left (d x^{3} + c\right )}^{\frac {3}{2}} - 9 \, \sqrt {d x^{3} + c} c}\right )}}{243 \, d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.38, size = 111, normalized size = 1.17 \begin {gather*} \frac {2\,\sqrt {d\,x^3+c}}{3\,d^4}+\frac {320\,\sqrt {c}\,\ln \left (\frac {10\,c+d\,x^3-6\,\sqrt {c}\,\sqrt {d\,x^3+c}}{8\,c-d\,x^3}\right )}{243\,d^4}+\frac {\sqrt {d\,x^3+c}\,\left (\frac {176\,c^2}{81\,d^4}+\frac {170\,c\,x^3}{81\,d^3}\right )}{8\,c^2+7\,c\,d\,x^3-d^2\,x^6} \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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