Optimal. Leaf size=88 \[ \frac {144 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^2}-\frac {48 c^2 \sqrt {c+d x^3}}{d^2}-\frac {16 c \left (c+d x^3\right )^{3/2}}{9 d^2}-\frac {2 \left (c+d x^3\right )^{5/2}}{15 d^2} \]
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Rubi [A] time = 0.07, antiderivative size = 88, 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, 80, 50, 63, 206} \begin {gather*} -\frac {48 c^2 \sqrt {c+d x^3}}{d^2}+\frac {144 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^2}-\frac {16 c \left (c+d x^3\right )^{3/2}}{9 d^2}-\frac {2 \left (c+d x^3\right )^{5/2}}{15 d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 206
Rule 446
Rubi steps
\begin {align*} \int \frac {x^5 \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x (c+d x)^{3/2}}{8 c-d x} \, dx,x,x^3\right )\\ &=-\frac {2 \left (c+d x^3\right )^{5/2}}{15 d^2}+\frac {(8 c) \operatorname {Subst}\left (\int \frac {(c+d x)^{3/2}}{8 c-d x} \, dx,x,x^3\right )}{3 d}\\ &=-\frac {16 c \left (c+d x^3\right )^{3/2}}{9 d^2}-\frac {2 \left (c+d x^3\right )^{5/2}}{15 d^2}+\frac {\left (24 c^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c+d x}}{8 c-d x} \, dx,x,x^3\right )}{d}\\ &=-\frac {48 c^2 \sqrt {c+d x^3}}{d^2}-\frac {16 c \left (c+d x^3\right )^{3/2}}{9 d^2}-\frac {2 \left (c+d x^3\right )^{5/2}}{15 d^2}+\frac {\left (216 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{d}\\ &=-\frac {48 c^2 \sqrt {c+d x^3}}{d^2}-\frac {16 c \left (c+d x^3\right )^{3/2}}{9 d^2}-\frac {2 \left (c+d x^3\right )^{5/2}}{15 d^2}+\frac {\left (432 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{d^2}\\ &=-\frac {48 c^2 \sqrt {c+d x^3}}{d^2}-\frac {16 c \left (c+d x^3\right )^{3/2}}{9 d^2}-\frac {2 \left (c+d x^3\right )^{5/2}}{15 d^2}+\frac {144 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 70, normalized size = 0.80 \begin {gather*} \frac {6480 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )-2 \sqrt {c+d x^3} \left (1123 c^2+46 c d x^3+3 d^2 x^6\right )}{45 d^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.07, size = 71, normalized size = 0.81 \begin {gather*} \frac {144 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^2}-\frac {2 \sqrt {c+d x^3} \left (1123 c^2+46 c d x^3+3 d^2 x^6\right )}{45 d^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 147, normalized size = 1.67 \begin {gather*} \left [\frac {2 \, {\left (1620 \, 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} + 46 \, c d x^{3} + 1123 \, c^{2}\right )} \sqrt {d x^{3} + c}\right )}}{45 \, d^{2}}, -\frac {2 \, {\left (3240 \, \sqrt {-c} c^{2} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{3 \, c}\right ) + {\left (3 \, d^{2} x^{6} + 46 \, c d x^{3} + 1123 \, c^{2}\right )} \sqrt {d x^{3} + c}\right )}}{45 \, d^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 83, normalized size = 0.94 \begin {gather*} -\frac {144 \, c^{3} \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{\sqrt {-c} d^{2}} - \frac {2 \, {\left (3 \, {\left (d x^{3} + c\right )}^{\frac {5}{2}} d^{8} + 40 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} c d^{8} + 1080 \, \sqrt {d x^{3} + c} c^{2} d^{8}\right )}}{45 \, d^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.18, size = 462, normalized size = 5.25 \begin {gather*} -\frac {8 \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}{d}-\frac {2 \left (d \,x^{3}+c \right )^{\frac {5}{2}}}{15 d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.44, size = 82, normalized size = 0.93 \begin {gather*} -\frac {2 \, {\left (1620 \, 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}} + 40 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} c + 1080 \, \sqrt {d x^{3} + c} c^{2}\right )}}{45 \, d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.52, size = 95, normalized size = 1.08 \begin {gather*} \frac {72\,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^2}-\frac {2\,x^6\,\sqrt {d\,x^3+c}}{15}-\frac {2246\,c^2\,\sqrt {d\,x^3+c}}{45\,d^2}-\frac {92\,c\,x^3\,\sqrt {d\,x^3+c}}{45\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 62.74, size = 90, normalized size = 1.02 \begin {gather*} - \frac {144 c^{3} \operatorname {atan}{\left (\frac {\sqrt {c + d x^{3}}}{3 \sqrt {- c}} \right )}}{d^{2} \sqrt {- c}} - \frac {48 c^{2} \sqrt {c + d x^{3}}}{d^{2}} - \frac {16 c \left (c + d x^{3}\right )^{\frac {3}{2}}}{9 d^{2}} - \frac {2 \left (c + d x^{3}\right )^{\frac {5}{2}}}{15 d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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