Optimal. Leaf size=67 \[ \frac {18 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d}-\frac {6 c \sqrt {c+d x^3}}{d}-\frac {2 \left (c+d x^3\right )^{3/2}}{9 d} \]
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Rubi [A] time = 0.06, antiderivative size = 67, 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 = {444, 50, 63, 206} \begin {gather*} \frac {18 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d}-\frac {6 c \sqrt {c+d x^3}}{d}-\frac {2 \left (c+d x^3\right )^{3/2}}{9 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 206
Rule 444
Rubi steps
\begin {align*} \int \frac {x^2 \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {(c+d x)^{3/2}}{8 c-d x} \, dx,x,x^3\right )\\ &=-\frac {2 \left (c+d x^3\right )^{3/2}}{9 d}+(3 c) \operatorname {Subst}\left (\int \frac {\sqrt {c+d x}}{8 c-d x} \, dx,x,x^3\right )\\ &=-\frac {6 c \sqrt {c+d x^3}}{d}-\frac {2 \left (c+d x^3\right )^{3/2}}{9 d}+\left (27 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )\\ &=-\frac {6 c \sqrt {c+d x^3}}{d}-\frac {2 \left (c+d x^3\right )^{3/2}}{9 d}+\frac {\left (54 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{d}\\ &=-\frac {6 c \sqrt {c+d x^3}}{d}-\frac {2 \left (c+d x^3\right )^{3/2}}{9 d}+\frac {18 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 58, normalized size = 0.87 \begin {gather*} \frac {162 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )-2 \sqrt {c+d x^3} \left (28 c+d x^3\right )}{9 d} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 59, normalized size = 0.88 \begin {gather*} \frac {18 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d}-\frac {2 \sqrt {c+d x^3} \left (28 c+d x^3\right )}{9 d} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 121, normalized size = 1.81 \begin {gather*} \left [\frac {81 \, 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 ) - 2 \, {\left (d x^{3} + 28 \, c\right )} \sqrt {d x^{3} + c}}{9 \, d}, -\frac {2 \, {\left (81 \, \sqrt {-c} c \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{3 \, c}\right ) + {\left (d x^{3} + 28 \, c\right )} \sqrt {d x^{3} + c}\right )}}{9 \, d}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 65, normalized size = 0.97 \begin {gather*} -\frac {18 \, c^{2} \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{\sqrt {-c} d} - \frac {2 \, {\left ({\left (d x^{3} + c\right )}^{\frac {3}{2}} d^{2} + 27 \, \sqrt {d x^{3} + c} c d^{2}\right )}}{9 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.16, size = 441, normalized size = 6.58 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 68, normalized size = 1.01 \begin {gather*} -\frac {81 \, c^{\frac {3}{2}} \log \left (\frac {\sqrt {d x^{3} + c} - 3 \, \sqrt {c}}{\sqrt {d x^{3} + c} + 3 \, \sqrt {c}}\right ) + 2 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} + 54 \, \sqrt {d x^{3} + c} c}{9 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.45, size = 75, normalized size = 1.12 \begin {gather*} \frac {9\,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 )}{d}-\frac {56\,c\,\sqrt {d\,x^3+c}}{9\,d}-\frac {2\,x^3\,\sqrt {d\,x^3+c}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 29.25, size = 65, normalized size = 0.97 \begin {gather*} - \frac {18 c^{2} \operatorname {atan}{\left (\frac {\sqrt {c + d x^{3}}}{3 \sqrt {- c}} \right )}}{d \sqrt {- c}} - \frac {6 c \sqrt {c + d x^{3}}}{d} - \frac {2 \left (c + d x^{3}\right )^{\frac {3}{2}}}{9 d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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