Optimal. Leaf size=71 \[ \frac {128 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{9 d^3}-\frac {14 c \sqrt {c+d x^3}}{3 d^3}-\frac {2 \left (c+d x^3\right )^{3/2}}{9 d^3} \]
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Rubi [A] time = 0.07, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {446, 88, 63, 206} \begin {gather*} \frac {128 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{9 d^3}-\frac {14 c \sqrt {c+d x^3}}{3 d^3}-\frac {2 \left (c+d x^3\right )^{3/2}}{9 d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 88
Rule 206
Rule 446
Rubi steps
\begin {align*} \int \frac {x^8}{\left (8 c-d x^3\right ) \sqrt {c+d x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^2}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (-\frac {7 c}{d^2 \sqrt {c+d x}}+\frac {64 c^2}{d^2 (8 c-d x) \sqrt {c+d x}}-\frac {\sqrt {c+d x}}{d^2}\right ) \, dx,x,x^3\right )\\ &=-\frac {14 c \sqrt {c+d x^3}}{3 d^3}-\frac {2 \left (c+d x^3\right )^{3/2}}{9 d^3}+\frac {\left (64 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{3 d^2}\\ &=-\frac {14 c \sqrt {c+d x^3}}{3 d^3}-\frac {2 \left (c+d x^3\right )^{3/2}}{9 d^3}+\frac {\left (128 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{3 d^3}\\ &=-\frac {14 c \sqrt {c+d x^3}}{3 d^3}-\frac {2 \left (c+d x^3\right )^{3/2}}{9 d^3}+\frac {128 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{9 d^3}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 58, normalized size = 0.82 \begin {gather*} \frac {128 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )-2 \sqrt {c+d x^3} \left (22 c+d x^3\right )}{9 d^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 61, normalized size = 0.86 \begin {gather*} \frac {128 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{9 d^3}-\frac {2 \sqrt {c+d x^3} \left (22 c+d x^3\right )}{9 d^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.10, size = 121, normalized size = 1.70 \begin {gather*} \left [\frac {2 \, {\left (32 \, c^{\frac {3}{2}} \log \left (\frac {d x^{3} + 6 \, \sqrt {d x^{3} + c} \sqrt {c} + 10 \, c}{d x^{3} - 8 \, c}\right ) - {\left (d x^{3} + 22 \, c\right )} \sqrt {d x^{3} + c}\right )}}{9 \, d^{3}}, -\frac {2 \, {\left (64 \, \sqrt {-c} c \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{3 \, c}\right ) + {\left (d x^{3} + 22 \, c\right )} \sqrt {d x^{3} + c}\right )}}{9 \, d^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 65, normalized size = 0.92 \begin {gather*} -\frac {128 \, c^{2} \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{9 \, \sqrt {-c} d^{3}} - \frac {2 \, {\left ({\left (d x^{3} + c\right )}^{\frac {3}{2}} d^{6} + 21 \, \sqrt {d x^{3} + c} c d^{6}\right )}}{9 \, d^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.16, size = 468, normalized size = 6.59 \begin {gather*} -\frac {\left (\frac {2 \sqrt {d \,x^{3}+c}\, x^{3}}{9 d}-\frac {4 \sqrt {d \,x^{3}+c}\, c}{9 d^{2}}\right ) d +\frac {16 \sqrt {d \,x^{3}+c}\, c}{3 d}}{d^{2}}-\frac {64 i c \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \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 ) d}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {\left (x -\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right ) d}{-3 \left (-c \,d^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i \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 ) d}{2 \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \left (2 \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right )^{2} d^{2}+i \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right ) d -\left (-c \,d^{2}\right )^{\frac {1}{3}} \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right ) d -i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {2}{3}}-\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}\, \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right )^{2} d +i \sqrt {3}\, c d -3 c d -i \left (-c \,d^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right )-3 \left (-c \,d^{2}\right )^{\frac {2}{3}} \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right )}{18 c d}, \sqrt {\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{\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 ) d}}\right )}{27 d^{5} \sqrt {d \,x^{3}+c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 66, normalized size = 0.93 \begin {gather*} -\frac {2 \, {\left (32 \, c^{\frac {3}{2}} \log \left (\frac {\sqrt {d x^{3} + c} - 3 \, \sqrt {c}}{\sqrt {d x^{3} + c} + 3 \, \sqrt {c}}\right ) + {\left (d x^{3} + c\right )}^{\frac {3}{2}} + 21 \, \sqrt {d x^{3} + c} c\right )}}{9 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.39, size = 78, normalized size = 1.10 \begin {gather*} \frac {64\,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 )}{9\,d^3}-\frac {44\,c\,\sqrt {d\,x^3+c}}{9\,d^3}-\frac {2\,x^3\,\sqrt {d\,x^3+c}}{9\,d^2} \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}}{- 8 c \sqrt {c + d x^{3}} + d x^{3} \sqrt {c + d x^{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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