Optimal. Leaf size=77 \[ \frac {\left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}+\frac {\sqrt {c+d x^3}}{d}-\frac {3 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d} \]
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Rubi [A] time = 0.06, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {444, 47, 50, 63, 206} \[ \frac {\left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}+\frac {\sqrt {c+d x^3}}{d}-\frac {3 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 206
Rule 444
Rubi steps
\begin {align*} \int \frac {x^2 \left (c+d x^3\right )^{3/2}}{\left (8 c-d x^3\right )^2} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {(c+d x)^{3/2}}{(8 c-d x)^2} \, dx,x,x^3\right )\\ &=\frac {\left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {c+d x}}{8 c-d x} \, dx,x,x^3\right )\\ &=\frac {\sqrt {c+d x^3}}{d}+\frac {\left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}-\frac {1}{2} (9 c) \operatorname {Subst}\left (\int \frac {1}{(8 c-d x) \sqrt {c+d x}} \, dx,x,x^3\right )\\ &=\frac {\sqrt {c+d x^3}}{d}+\frac {\left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}-\frac {(9 c) \operatorname {Subst}\left (\int \frac {1}{9 c-x^2} \, dx,x,\sqrt {c+d x^3}\right )}{d}\\ &=\frac {\sqrt {c+d x^3}}{d}+\frac {\left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}-\frac {3 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 43, normalized size = 0.56 \[ \frac {2 \left (c+d x^3\right )^{5/2} \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};\frac {d x^3+c}{9 c}\right )}{1215 c^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 162, normalized size = 2.10 \[ \left [\frac {9 \, {\left (d x^{3} - 8 \, c\right )} \sqrt {c} \log \left (\frac {d x^{3} - 6 \, \sqrt {d x^{3} + c} \sqrt {c} + 10 \, c}{d x^{3} - 8 \, c}\right ) + 2 \, {\left (2 \, d x^{3} - 25 \, c\right )} \sqrt {d x^{3} + c}}{6 \, {\left (d^{2} x^{3} - 8 \, c d\right )}}, \frac {9 \, {\left (d x^{3} - 8 \, c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{3 \, c}\right ) + {\left (2 \, d x^{3} - 25 \, c\right )} \sqrt {d x^{3} + c}}{3 \, {\left (d^{2} x^{3} - 8 \, c d\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 69, normalized size = 0.90 \[ \frac {3 \, c \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{\sqrt {-c} d} + \frac {2 \, \sqrt {d x^{3} + c}}{3 \, d} - \frac {3 \, \sqrt {d x^{3} + c} c}{{\left (d x^{3} - 8 \, c\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.18, size = 451, normalized size = 5.86 \[ -\frac {3 \sqrt {d \,x^{3}+c}\, c}{\left (d \,x^{3}-8 c \right ) d}+\frac {2 \sqrt {d \,x^{3}+c}}{3 d}+\frac {i \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (2 x +\frac {-i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right ) d}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {\left (x -\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right ) d}{-3 \left (-c \,d^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i \left (2 x +\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right ) d}{2 \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \left (2 \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right )^{2} d^{2}+i \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right ) d -\left (-c \,d^{2}\right )^{\frac {1}{3}} \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right ) d -i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {2}{3}}-\left (-c \,d^{2}\right )^{\frac {2}{3}}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) \sqrt {3}\, d}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {2 i \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right )^{2} d +i \sqrt {3}\, c d -3 c d -i \left (-c \,d^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right )-3 \left (-c \,d^{2}\right )^{\frac {2}{3}} \RootOf \left (d \,\textit {\_Z}^{3}-8 c \right )}{18 c d}, \sqrt {\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{\left (-\frac {3 \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) d}}\right )}{2 d^{3} \sqrt {d \,x^{3}+c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 79, normalized size = 1.03 \[ \frac {9 \, \sqrt {c} \log \left (\frac {\sqrt {d x^{3} + c} - 3 \, \sqrt {c}}{\sqrt {d x^{3} + c} + 3 \, \sqrt {c}}\right ) + 4 \, \sqrt {d x^{3} + c} - \frac {18 \, \sqrt {d x^{3} + c} c}{d x^{3} - 8 \, c}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.99, size = 87, normalized size = 1.13 \[ \frac {2\,\sqrt {d\,x^3+c}}{3\,d}+\frac {3\,\sqrt {c}\,\ln \left (\frac {10\,c+d\,x^3-6\,\sqrt {c}\,\sqrt {d\,x^3+c}}{8\,c-d\,x^3}\right )}{2\,d}+\frac {3\,c\,\sqrt {d\,x^3+c}}{d\,\left (8\,c-d\,x^3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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