Optimal. Leaf size=109 \[ \frac {1152 c^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^3}-\frac {384 c^3 \sqrt {c+d x^3}}{d^3}-\frac {128 c^2 \left (c+d x^3\right )^{3/2}}{9 d^3}-\frac {14 c \left (c+d x^3\right )^{5/2}}{15 d^3}-\frac {2 \left (c+d x^3\right )^{7/2}}{21 d^3} \]
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Rubi [A] time = 0.10, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \begin {gather*} -\frac {384 c^3 \sqrt {c+d x^3}}{d^3}-\frac {128 c^2 \left (c+d x^3\right )^{3/2}}{9 d^3}+\frac {1152 c^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^3}-\frac {14 c \left (c+d x^3\right )^{5/2}}{15 d^3}-\frac {2 \left (c+d x^3\right )^{7/2}}{21 d^3} \end {gather*}
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 \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^2 (c+d x)^{3/2}}{8 c-d x} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (-\frac {7 c (c+d x)^{3/2}}{d^2}+\frac {64 c^2 (c+d x)^{3/2}}{d^2 (8 c-d x)}-\frac {(c+d x)^{5/2}}{d^2}\right ) \, dx,x,x^3\right )\\ &=-\frac {14 c \left (c+d x^3\right )^{5/2}}{15 d^3}-\frac {2 \left (c+d x^3\right )^{7/2}}{21 d^3}+\frac {\left (64 c^2\right ) \operatorname {Subst}\left (\int \frac {(c+d x)^{3/2}}{8 c-d x} \, dx,x,x^3\right )}{3 d^2}\\ &=-\frac {128 c^2 \left (c+d x^3\right )^{3/2}}{9 d^3}-\frac {14 c \left (c+d x^3\right )^{5/2}}{15 d^3}-\frac {2 \left (c+d x^3\right )^{7/2}}{21 d^3}+\frac {\left (192 c^3\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c+d x}}{8 c-d x} \, dx,x,x^3\right )}{d^2}\\ &=-\frac {384 c^3 \sqrt {c+d x^3}}{d^3}-\frac {128 c^2 \left (c+d x^3\right )^{3/2}}{9 d^3}-\frac {14 c \left (c+d x^3\right )^{5/2}}{15 d^3}-\frac {2 \left (c+d x^3\right )^{7/2}}{21 d^3}+\frac {\left (1728 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{d^2}\\ &=-\frac {384 c^3 \sqrt {c+d x^3}}{d^3}-\frac {128 c^2 \left (c+d x^3\right )^{3/2}}{9 d^3}-\frac {14 c \left (c+d x^3\right )^{5/2}}{15 d^3}-\frac {2 \left (c+d x^3\right )^{7/2}}{21 d^3}+\frac {\left (3456 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{d^3}\\ &=-\frac {384 c^3 \sqrt {c+d x^3}}{d^3}-\frac {128 c^2 \left (c+d x^3\right )^{3/2}}{9 d^3}-\frac {14 c \left (c+d x^3\right )^{5/2}}{15 d^3}-\frac {2 \left (c+d x^3\right )^{7/2}}{21 d^3}+\frac {1152 c^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^3}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 81, normalized size = 0.74 \begin {gather*} \frac {362880 c^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )-2 \sqrt {c+d x^3} \left (62882 c^3+2579 c^2 d x^3+192 c d^2 x^6+15 d^3 x^9\right )}{315 d^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 82, normalized size = 0.75 \begin {gather*} \frac {1152 c^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^3}-\frac {2 \sqrt {c+d x^3} \left (62882 c^3+2579 c^2 d x^3+192 c d^2 x^6+15 d^3 x^9\right )}{315 d^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 169, normalized size = 1.55 \begin {gather*} \left [\frac {2 \, {\left (90720 \, c^{\frac {7}{2}} \log \left (\frac {d x^{3} + 6 \, \sqrt {d x^{3} + c} \sqrt {c} + 10 \, c}{d x^{3} - 8 \, c}\right ) - {\left (15 \, d^{3} x^{9} + 192 \, c d^{2} x^{6} + 2579 \, c^{2} d x^{3} + 62882 \, c^{3}\right )} \sqrt {d x^{3} + c}\right )}}{315 \, d^{3}}, -\frac {2 \, {\left (181440 \, \sqrt {-c} c^{3} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{3 \, c}\right ) + {\left (15 \, d^{3} x^{9} + 192 \, c d^{2} x^{6} + 2579 \, c^{2} d x^{3} + 62882 \, c^{3}\right )} \sqrt {d x^{3} + c}\right )}}{315 \, d^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 100, normalized size = 0.92 \begin {gather*} -\frac {1152 \, c^{4} \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{\sqrt {-c} d^{3}} - \frac {2 \, {\left (15 \, {\left (d x^{3} + c\right )}^{\frac {7}{2}} d^{18} + 147 \, {\left (d x^{3} + c\right )}^{\frac {5}{2}} c d^{18} + 2240 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} c^{2} d^{18} + 60480 \, \sqrt {d x^{3} + c} c^{3} d^{18}\right )}}{315 \, d^{21}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.16, size = 541, normalized size = 4.96 \begin {gather*} -\frac {64 \left (\frac {2 \sqrt {d \,x^{3}+c}\, x^{3}}{9}+\frac {56 \sqrt {d \,x^{3}+c}\, c}{9 d}+\frac {3 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 )}{d^{3} \sqrt {d \,x^{3}+c}}\right ) c^{2}}{d^{2}}-\frac {\left (\frac {2 \sqrt {d \,x^{3}+c}\, d \,x^{9}}{21}+\frac {16 \sqrt {d \,x^{3}+c}\, c \,x^{6}}{105}+\frac {2 \sqrt {d \,x^{3}+c}\, c^{2} x^{3}}{105 d}-\frac {4 \sqrt {d \,x^{3}+c}\, c^{3}}{105 d^{2}}\right ) d +\frac {16 \left (d \,x^{3}+c \right )^{\frac {5}{2}} c}{15 d}}{d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 96, normalized size = 0.88 \begin {gather*} -\frac {2 \, {\left (90720 \, c^{\frac {7}{2}} \log \left (\frac {\sqrt {d x^{3} + c} - 3 \, \sqrt {c}}{\sqrt {d x^{3} + c} + 3 \, \sqrt {c}}\right ) + 15 \, {\left (d x^{3} + c\right )}^{\frac {7}{2}} + 147 \, {\left (d x^{3} + c\right )}^{\frac {5}{2}} c + 2240 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} c^{2} + 60480 \, \sqrt {d x^{3} + c} c^{3}\right )}}{315 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.50, size = 115, normalized size = 1.06 \begin {gather*} \frac {576\,c^{7/2}\,\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 {2\,x^9\,\sqrt {d\,x^3+c}}{21}-\frac {125764\,c^3\,\sqrt {d\,x^3+c}}{315\,d^3}-\frac {128\,c\,x^6\,\sqrt {d\,x^3+c}}{105\,d}-\frac {5158\,c^2\,x^3\,\sqrt {d\,x^3+c}}{315\,d^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 108.79, size = 110, normalized size = 1.01 \begin {gather*} - \frac {1152 c^{4} \operatorname {atan}{\left (\frac {\sqrt {c + d x^{3}}}{3 \sqrt {- c}} \right )}}{d^{3} \sqrt {- c}} - \frac {384 c^{3} \sqrt {c + d x^{3}}}{d^{3}} - \frac {128 c^{2} \left (c + d x^{3}\right )^{\frac {3}{2}}}{9 d^{3}} - \frac {14 c \left (c + d x^{3}\right )^{\frac {5}{2}}}{15 d^{3}} - \frac {2 \left (c + d x^{3}\right )^{\frac {7}{2}}}{21 d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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