Optimal. Leaf size=90 \[ \frac {128 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^3}-\frac {128 c^2 \sqrt {c+d x^3}}{3 d^3}-\frac {14 c \left (c+d x^3\right )^{3/2}}{9 d^3}-\frac {2 \left (c+d x^3\right )^{5/2}}{15 d^3} \]
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Rubi [A] time = 0.09, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {446, 88, 50, 63, 206} \[ -\frac {128 c^2 \sqrt {c+d x^3}}{3 d^3}+\frac {128 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^3}-\frac {14 c \left (c+d x^3\right )^{3/2}}{9 d^3}-\frac {2 \left (c+d x^3\right )^{5/2}}{15 d^3} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 88
Rule 206
Rule 446
Rubi steps
\begin {align*} \int \frac {x^8 \sqrt {c+d x^3}}{8 c-d x^3} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^2 \sqrt {c+d x}}{8 c-d x} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (-\frac {7 c \sqrt {c+d x}}{d^2}+\frac {64 c^2 \sqrt {c+d x}}{d^2 (8 c-d x)}-\frac {(c+d x)^{3/2}}{d^2}\right ) \, dx,x,x^3\right )\\ &=-\frac {14 c \left (c+d x^3\right )^{3/2}}{9 d^3}-\frac {2 \left (c+d x^3\right )^{5/2}}{15 d^3}+\frac {\left (64 c^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c+d x}}{8 c-d x} \, dx,x,x^3\right )}{3 d^2}\\ &=-\frac {128 c^2 \sqrt {c+d x^3}}{3 d^3}-\frac {14 c \left (c+d x^3\right )^{3/2}}{9 d^3}-\frac {2 \left (c+d x^3\right )^{5/2}}{15 d^3}+\frac {\left (192 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{d^2}\\ &=-\frac {128 c^2 \sqrt {c+d x^3}}{3 d^3}-\frac {14 c \left (c+d x^3\right )^{3/2}}{9 d^3}-\frac {2 \left (c+d x^3\right )^{5/2}}{15 d^3}+\frac {\left (384 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{d^3}\\ &=-\frac {128 c^2 \sqrt {c+d x^3}}{3 d^3}-\frac {14 c \left (c+d x^3\right )^{3/2}}{9 d^3}-\frac {2 \left (c+d x^3\right )^{5/2}}{15 d^3}+\frac {128 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^3}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 70, normalized size = 0.78 \[ \frac {5760 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )-2 \sqrt {c+d x^3} \left (998 c^2+41 c d x^3+3 d^2 x^6\right )}{45 d^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 147, normalized size = 1.63 \[ \left [\frac {2 \, {\left (1440 \, c^{\frac {5}{2}} \log \left (\frac {d x^{3} + 6 \, \sqrt {d x^{3} + c} \sqrt {c} + 10 \, c}{d x^{3} - 8 \, c}\right ) - {\left (3 \, d^{2} x^{6} + 41 \, c d x^{3} + 998 \, c^{2}\right )} \sqrt {d x^{3} + c}\right )}}{45 \, d^{3}}, -\frac {2 \, {\left (2880 \, \sqrt {-c} c^{2} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{3 \, c}\right ) + {\left (3 \, d^{2} x^{6} + 41 \, c d x^{3} + 998 \, c^{2}\right )} \sqrt {d x^{3} + c}\right )}}{45 \, d^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 83, normalized size = 0.92 \[ -\frac {128 \, c^{3} \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{\sqrt {-c} d^{3}} - \frac {2 \, {\left (3 \, {\left (d x^{3} + c\right )}^{\frac {5}{2}} d^{12} + 35 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} c d^{12} + 960 \, \sqrt {d x^{3} + c} c^{2} d^{12}\right )}}{45 \, d^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.16, size = 507, normalized size = 5.63 \[ -\frac {64 \left (\frac {2 \sqrt {d \,x^{3}+c}}{3 d}+\frac {i \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 )}{3 d^{3} \sqrt {d \,x^{3}+c}}\right ) c^{2}}{d^{2}}-\frac {\left (\frac {2 \sqrt {d \,x^{3}+c}\, x^{6}}{15}+\frac {2 \sqrt {d \,x^{3}+c}\, c \,x^{3}}{45 d}-\frac {4 \sqrt {d \,x^{3}+c}\, c^{2}}{45 d^{2}}\right ) d +\frac {16 \left (d \,x^{3}+c \right )^{\frac {3}{2}} c}{9 d}}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.21, size = 82, normalized size = 0.91 \[ -\frac {2 \, {\left (1440 \, c^{\frac {5}{2}} \log \left (\frac {\sqrt {d x^{3} + c} - 3 \, \sqrt {c}}{\sqrt {d x^{3} + c} + 3 \, \sqrt {c}}\right ) + 3 \, {\left (d x^{3} + c\right )}^{\frac {5}{2}} + 35 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} c + 960 \, \sqrt {d x^{3} + c} c^{2}\right )}}{45 \, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.40, size = 98, normalized size = 1.09 \[ \frac {64\,c^{5/2}\,\ln \left (\frac {10\,c+d\,x^3+6\,\sqrt {c}\,\sqrt {d\,x^3+c}}{8\,c-d\,x^3}\right )}{d^3}-\frac {1996\,c^2\,\sqrt {d\,x^3+c}}{45\,d^3}-\frac {2\,x^6\,\sqrt {d\,x^3+c}}{15\,d}-\frac {82\,c\,x^3\,\sqrt {d\,x^3+c}}{45\,d^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 30.23, size = 82, normalized size = 0.91 \[ \frac {2 \left (- \frac {64 c^{3} \operatorname {atan}{\left (\frac {\sqrt {c + d x^{3}}}{3 \sqrt {- c}} \right )}}{\sqrt {- c}} - \frac {64 c^{2} \sqrt {c + d x^{3}}}{3} - \frac {7 c \left (c + d x^{3}\right )^{\frac {3}{2}}}{9} - \frac {\left (c + d x^{3}\right )^{\frac {5}{2}}}{15}\right )}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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