Optimal. Leaf size=83 \[ \frac {2 \sqrt {c+d x^3}}{3 d^3}+\frac {64 c \sqrt {c+d x^3}}{27 d^3 \left (8 c-d x^3\right )}-\frac {224 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{81 d^3} \]
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Rubi [A]
time = 0.04, antiderivative size = 83, 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, 91, 81, 65,
212} \begin {gather*} \frac {64 c \sqrt {c+d x^3}}{27 d^3 \left (8 c-d x^3\right )}+\frac {2 \sqrt {c+d x^3}}{3 d^3}-\frac {224 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{81 d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 81
Rule 91
Rule 212
Rule 457
Rubi steps
\begin {align*} \int \frac {x^8}{\left (8 c-d x^3\right )^2 \sqrt {c+d x^3}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^2}{(8 c-d x)^2 \sqrt {c+d x}} \, dx,x,x^3\right )\\ &=\frac {64 c \sqrt {c+d x^3}}{27 d^3 \left (8 c-d x^3\right )}-\frac {\text {Subst}\left (\int \frac {40 c^2 d+9 c d^2 x}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{27 c d^3}\\ &=\frac {2 \sqrt {c+d x^3}}{3 d^3}+\frac {64 c \sqrt {c+d x^3}}{27 d^3 \left (8 c-d x^3\right )}-\frac {(112 c) \text {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{27 d^2}\\ &=\frac {2 \sqrt {c+d x^3}}{3 d^3}+\frac {64 c \sqrt {c+d x^3}}{27 d^3 \left (8 c-d x^3\right )}-\frac {(224 c) \text {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{27 d^3}\\ &=\frac {2 \sqrt {c+d x^3}}{3 d^3}+\frac {64 c \sqrt {c+d x^3}}{27 d^3 \left (8 c-d x^3\right )}-\frac {224 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{81 d^3}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 70, normalized size = 0.84 \begin {gather*} \frac {2 \left (\frac {3 \sqrt {c+d x^3} \left (-104 c+9 d x^3\right )}{-8 c+d x^3}-112 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )\right )}{81 d^3} \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 = 0.42, size = 875, normalized size = 10.54
method | result | size |
elliptic | \(\frac {64 c \sqrt {d \,x^{3}+c}}{27 d^{3} \left (-d \,x^{3}+8 c \right )}+\frac {2 \sqrt {d \,x^{3}+c}}{3 d^{3}}+\frac {112 i \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^{5}}\) | \(452\) |
default | \(\text {Expression too large to display}\) | \(875\) |
risch | \(\text {Expression too large to display}\) | \(879\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 79, normalized size = 0.95 \begin {gather*} \frac {2 \, {\left (56 \, \sqrt {c} \log \left (\frac {\sqrt {d x^{3} + c} - 3 \, \sqrt {c}}{\sqrt {d x^{3} + c} + 3 \, \sqrt {c}}\right ) + 27 \, \sqrt {d x^{3} + c} - \frac {96 \, \sqrt {d x^{3} + c} c}{d x^{3} - 8 \, c}\right )}}{81 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.23, size = 167, normalized size = 2.01 \begin {gather*} \left [\frac {2 \, {\left (56 \, {\left (d x^{3} - 8 \, c\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 (9 \, d x^{3} - 104 \, c\right )} \sqrt {d x^{3} + c}\right )}}{81 \, {\left (d^{4} x^{3} - 8 \, c d^{3}\right )}}, \frac {2 \, {\left (112 \, {\left (d x^{3} - 8 \, c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{3 \, c}\right ) + 3 \, {\left (9 \, d x^{3} - 104 \, c\right )} \sqrt {d x^{3} + c}\right )}}{81 \, {\left (d^{4} x^{3} - 8 \, c d^{3}\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^{8}}{\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 = 0.87, size = 69, normalized size = 0.83 \begin {gather*} \frac {224 \, c \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{81 \, \sqrt {-c} d^{3}} + \frac {2 \, \sqrt {d x^{3} + c}}{3 \, d^{3}} - \frac {64 \, \sqrt {d x^{3} + c} c}{27 \, {\left (d x^{3} - 8 \, c\right )} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.00, size = 87, normalized size = 1.05 \begin {gather*} \frac {2\,\sqrt {d\,x^3+c}}{3\,d^3}+\frac {112\,\sqrt {c}\,\ln \left (\frac {10\,c+d\,x^3-6\,\sqrt {c}\,\sqrt {d\,x^3+c}}{8\,c-d\,x^3}\right )}{81\,d^3}+\frac {64\,c\,\sqrt {d\,x^3+c}}{27\,d^3\,\left (8\,c-d\,x^3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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