Optimal. Leaf size=55 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{81 c^{3/2} d}-\frac {2}{27 c d \sqrt {c+d x^3}} \]
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Rubi [A] time = 0.05, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {444, 51, 63, 206} \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{81 c^{3/2} d}-\frac {2}{27 c d \sqrt {c+d x^3}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 206
Rule 444
Rubi steps
\begin {align*} \int \frac {x^2}{\left (8 c-d x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{(8 c-d x) (c+d x)^{3/2}} \, dx,x,x^3\right )\\ &=-\frac {2}{27 c d \sqrt {c+d x^3}}+\frac {\operatorname {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )}{27 c}\\ &=-\frac {2}{27 c d \sqrt {c+d x^3}}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{27 c d}\\ &=-\frac {2}{27 c d \sqrt {c+d x^3}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{81 c^{3/2} d}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 43, normalized size = 0.78 \[ -\frac {2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {d x^3+c}{9 c}\right )}{27 c d \sqrt {c+d x^3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 147, normalized size = 2.67 \[ \left [\frac {{\left (d x^{3} + c\right )} \sqrt {c} \log \left (\frac {d x^{3} + 6 \, \sqrt {d x^{3} + c} \sqrt {c} + 10 \, c}{d x^{3} - 8 \, c}\right ) - 6 \, \sqrt {d x^{3} + c} c}{81 \, {\left (c^{2} d^{2} x^{3} + c^{3} d\right )}}, -\frac {2 \, {\left ({\left (d x^{3} + c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{3 \, c}\right ) + 3 \, \sqrt {d x^{3} + c} c\right )}}{81 \, {\left (c^{2} d^{2} x^{3} + c^{3} d\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 48, normalized size = 0.87 \[ -\frac {2 \, \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{81 \, \sqrt {-c} c d} - \frac {2}{27 \, \sqrt {d x^{3} + c} c d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.21, size = 435, normalized size = 7.91 \[ -\frac {2}{27 \sqrt {\left (x^{3}+\frac {c}{d}\right ) d}\, c d}-\frac {i \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (2 x +\frac {-i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right ) d}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {\left (x -\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right ) d}{-3 \left (-c \,d^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i \left (2 x +\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right ) d}{2 \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \left (2 \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right )^{2} d^{2}+i \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right ) d -\left (-c \,d^{2}\right )^{\frac {1}{3}} \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right ) d -i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {2}{3}}-\left (-c \,d^{2}\right )^{\frac {2}{3}}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) \sqrt {3}\, d}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {2 i \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right )^{2} d +i \sqrt {3}\, c d -3 c d -i \left (-c \,d^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right )-3 \left (-c \,d^{2}\right )^{\frac {2}{3}} \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right )}{18 c d}, \sqrt {\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{\left (-\frac {3 \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) d}}\right )}{243 c^{2} d^{3} \sqrt {d \,x^{3}+c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.23, size = 58, normalized size = 1.05 \[ -\frac {\frac {\log \left (\frac {\sqrt {d x^{3} + c} - 3 \, \sqrt {c}}{\sqrt {d x^{3} + c} + 3 \, \sqrt {c}}\right )}{c^{\frac {3}{2}}} + \frac {6}{\sqrt {d x^{3} + c} c}}{81 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.63, size = 63, normalized size = 1.15 \[ \frac {\ln \left (\frac {10\,c+d\,x^3+6\,\sqrt {c}\,\sqrt {d\,x^3+c}}{8\,c-d\,x^3}\right )}{81\,c^{3/2}\,d}-\frac {2}{27\,c\,d\,\sqrt {d\,x^3+c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 25.53, size = 51, normalized size = 0.93 \[ - \frac {2}{27 c d \sqrt {c + d x^{3}}} - \frac {2 \operatorname {atan}{\left (\frac {\sqrt {c + d x^{3}}}{3 \sqrt {- c}} \right )}}{81 c d \sqrt {- c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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