Optimal. Leaf size=95 \[ \frac {8 x^6 \sqrt {c+d x^3}}{27 d^2 \left (8 c-d x^3\right )}+\frac {2 \sqrt {c+d x^3} \left (170 c+7 d x^3\right )}{27 d^4}-\frac {2944 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{81 d^4} \]
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Rubi [A]
time = 0.06, 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 = {457, 100, 152,
65, 212} \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 (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 )} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 100
Rule 152
Rule 212
Rule 457
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} \text {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 {\text {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 ) \text {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 ) \text {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 = 0.11, size = 81, normalized size = 0.85 \begin {gather*} \frac {2 \left (\frac {3 \sqrt {c+d x^3} \left (-1360 c^2+114 c d x^3+3 d^2 x^6\right )}{-8 c+d x^3}-1472 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 = 917, normalized size = 9.65
method | result | size |
elliptic | \(\frac {512 c^{2} \sqrt {d \,x^{3}+c}}{27 d^{4} \left (-d \,x^{3}+8 c \right )}+\frac {2 x^{3} \sqrt {d \,x^{3}+c}}{9 d^{3}}+\frac {92 c \sqrt {d \,x^{3}+c}}{9 d^{4}}+\frac {1472 i c \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{3}-8 c \right )}{\sum }\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i d \left (2 x +\frac {-i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {d \left (x -\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{-3 \left (-c \,d^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i d \left (2 x +\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{2 \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \left (i \left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, d -i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {2}{3}}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} d^{2}-\left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha d -\left (-c \,d^{2}\right )^{\frac {2}{3}}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) \sqrt {3}\, d}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {2 i \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d -i \left (-c \,d^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, c d -3 \left (-c \,d^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -3 c d}{18 d c}, \sqrt {\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{d \left (-\frac {3 \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right )}}\right )}{2 \sqrt {d \,x^{3}+c}}\right )}{243 d^{6}}\) | \(473\) |
risch | \(\text {Expression too large to display}\) | \(890\) |
default | \(\text {Expression too large to display}\) | \(917\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, 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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Fricas [A]
time = 3.21, 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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{11}}{\left (- 8 c + d x^{3}\right )^{2} \sqrt {c + d x^{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.35, 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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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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