Optimal. Leaf size=26 \[ \frac {2 \log \left (c \sqrt {a+b x^3}+d\right )}{3 b c} \]
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Rubi [A] time = 0.11, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2155, 31} \[ \frac {2 \log \left (c \sqrt {a+b x^3}+d\right )}{3 b c} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2155
Rubi steps
\begin {align*} \int \frac {x^2}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{a c+b c x+d \sqrt {a+b x}} \, dx,x,x^3\right )\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {1}{d+c x} \, dx,x,\sqrt {a+b x^3}\right )}{3 b}\\ &=\frac {2 \log \left (d+c \sqrt {a+b x^3}\right )}{3 b c}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 26, normalized size = 1.00 \[ \frac {2 \log \left (c \sqrt {a+b x^3}+d\right )}{3 b c} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 61, normalized size = 2.35 \[ \frac {\log \left (b c^{2} x^{3} + a c^{2} - d^{2}\right ) + \log \left (\sqrt {b x^{3} + a} c + d\right ) - \log \left (\sqrt {b x^{3} + a} c - d\right )}{3 \, b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 23, normalized size = 0.88 \[ \frac {2 \, \log \left ({\left | \sqrt {b x^{3} + a} c + d \right |}\right )}{3 \, b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 455, normalized size = 17.50 \[ \frac {\ln \left (b \,c^{2} x^{3}+a \,c^{2}-d^{2}\right )}{3 b c}-\frac {i \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (2 x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}-i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right ) b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {\left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right ) b}{-3 \left (-a \,b^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i \left (2 x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right ) b}{2 \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (2 \RootOf \left (b \,c^{2} \textit {\_Z}^{3}+a \,c^{2}-d^{2}\right )^{2} b^{2}+i \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \RootOf \left (b \,c^{2} \textit {\_Z}^{3}+a \,c^{2}-d^{2}\right ) b -\left (-a \,b^{2}\right )^{\frac {1}{3}} \RootOf \left (b \,c^{2} \textit {\_Z}^{3}+a \,c^{2}-d^{2}\right ) b -i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}-\left (-a \,b^{2}\right )^{\frac {2}{3}}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {\left (2 i \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \RootOf \left (b \,c^{2} \textit {\_Z}^{3}+a \,c^{2}-d^{2}\right )^{2} b +i \sqrt {3}\, a b -3 a b -i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \RootOf \left (b \,c^{2} \textit {\_Z}^{3}+a \,c^{2}-d^{2}\right )-3 \left (-a \,b^{2}\right )^{\frac {2}{3}} \RootOf \left (b \,c^{2} \textit {\_Z}^{3}+a \,c^{2}-d^{2}\right )\right ) c^{2}}{2 b \,d^{2}}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{\left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) b}}\right )}{3 b^{3} d \sqrt {b \,x^{3}+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 22, normalized size = 0.85 \[ \frac {2 \, \log \left (\sqrt {b x^{3} + a} c + d\right )}{3 \, b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.51, size = 60, normalized size = 2.31 \[ \frac {\ln \left (\frac {d+c\,\sqrt {b\,x^3+a}}{d-c\,\sqrt {b\,x^3+a}}\right )+\ln \left (b\,c^2\,x^3+a\,c^2-d^2\right )}{3\,b\,c} \]
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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