Optimal. Leaf size=108 \[ -\frac {d}{\sqrt {c+d x^3} (b c-a d)^2}-\frac {1}{3 \left (a+b x^3\right ) \sqrt {c+d x^3} (b c-a d)}+\frac {\sqrt {b} d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{5/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {444, 51, 63, 208} \[ -\frac {d}{\sqrt {c+d x^3} (b c-a d)^2}-\frac {1}{3 \left (a+b x^3\right ) \sqrt {c+d x^3} (b c-a d)}+\frac {\sqrt {b} d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 444
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a+b x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{(a+b x)^2 (c+d x)^{3/2}} \, dx,x,x^3\right )\\ &=-\frac {1}{3 (b c-a d) \left (a+b x^3\right ) \sqrt {c+d x^3}}-\frac {d \operatorname {Subst}\left (\int \frac {1}{(a+b x) (c+d x)^{3/2}} \, dx,x,x^3\right )}{2 (b c-a d)}\\ &=-\frac {d}{(b c-a d)^2 \sqrt {c+d x^3}}-\frac {1}{3 (b c-a d) \left (a+b x^3\right ) \sqrt {c+d x^3}}-\frac {(b d) \operatorname {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^3\right )}{2 (b c-a d)^2}\\ &=-\frac {d}{(b c-a d)^2 \sqrt {c+d x^3}}-\frac {1}{3 (b c-a d) \left (a+b x^3\right ) \sqrt {c+d x^3}}-\frac {b \operatorname {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^3}\right )}{(b c-a d)^2}\\ &=-\frac {d}{(b c-a d)^2 \sqrt {c+d x^3}}-\frac {1}{3 (b c-a d) \left (a+b x^3\right ) \sqrt {c+d x^3}}+\frac {\sqrt {b} d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 54, normalized size = 0.50 \[ -\frac {2 d \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};-\frac {b \left (d x^3+c\right )}{a d-b c}\right )}{3 \sqrt {c+d x^3} (a d-b c)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.42, size = 450, normalized size = 4.17 \[ \left [\frac {3 \, {\left (b d^{2} x^{6} + {\left (b c d + a d^{2}\right )} x^{3} + a c d\right )} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b d x^{3} + 2 \, b c - a d + 2 \, \sqrt {d x^{3} + c} {\left (b c - a d\right )} \sqrt {\frac {b}{b c - a d}}}{b x^{3} + a}\right ) - 2 \, {\left (3 \, b d x^{3} + b c + 2 \, a d\right )} \sqrt {d x^{3} + c}}{6 \, {\left ({\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{6} + a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{3}\right )}}, \frac {3 \, {\left (b d^{2} x^{6} + {\left (b c d + a d^{2}\right )} x^{3} + a c d\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (-\frac {\sqrt {d x^{3} + c} {\left (b c - a d\right )} \sqrt {-\frac {b}{b c - a d}}}{b d x^{3} + b c}\right ) - {\left (3 \, b d x^{3} + b c + 2 \, a d\right )} \sqrt {d x^{3} + c}}{3 \, {\left ({\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{6} + a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 153, normalized size = 1.42 \[ -\frac {b d \arctan \left (\frac {\sqrt {d x^{3} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-b^{2} c + a b d}} - \frac {3 \, {\left (d x^{3} + c\right )} b d - 2 \, b c d + 2 \, a d^{2}}{3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} {\left ({\left (d x^{3} + c\right )}^{\frac {3}{2}} b - \sqrt {d x^{3} + c} b c + \sqrt {d x^{3} + c} a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.25, size = 485, normalized size = 4.49 \[ \frac {i b \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 (\textit {\_Z}^{3} b +a \right )^{2} d^{2}+i \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \RootOf \left (\textit {\_Z}^{3} b +a \right ) d -\left (-c \,d^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3} b +a \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 {\left (2 i \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \RootOf \left (\textit {\_Z}^{3} b +a \right )^{2} d +i \sqrt {3}\, c d -3 c d -i \left (-c \,d^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \RootOf \left (\textit {\_Z}^{3} b +a \right )-3 \left (-c \,d^{2}\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3} b +a \right )\right ) b}{2 \left (a d -b c \right ) 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 \left (a d -b c \right )^{3} \sqrt {d \,x^{3}+c}}-\frac {\sqrt {d \,x^{3}+c}\, b}{3 \left (a d -b c \right )^{2} \left (b \,x^{3}+a \right )}-\frac {2 d}{3 \left (a d -b c \right )^{2} \sqrt {\left (x^{3}+\frac {c}{d}\right ) d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.43, size = 199, normalized size = 1.84 \[ -\frac {\left (\frac {3\,b\,d\,\left (a\,d+b\,c\right )-b\,d\,\left (a\,d+2\,b\,c\right )}{3\,\left (a^2\,b\,d^3-2\,a\,b^2\,c\,d^2+b^3\,c^2\,d\right )}+\frac {b^2\,d^2\,x^3}{a^2\,b\,d^3-2\,a\,b^2\,c\,d^2+b^3\,c^2\,d}\right )\,\sqrt {d\,x^3+c}}{b\,d\,x^6+\left (a\,d+b\,c\right )\,x^3+a\,c}+\frac {\sqrt {b}\,d\,\ln \left (\frac {a\,d-2\,b\,c-b\,d\,x^3+\sqrt {b}\,\sqrt {d\,x^3+c}\,\sqrt {a\,d-b\,c}\,2{}\mathrm {i}}{b\,x^3+a}\right )\,1{}\mathrm {i}}{2\,{\left (a\,d-b\,c\right )}^{5/2}} \]
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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