Optimal. Leaf size=145 \[ -\frac {\log \left (c+d x^3\right )}{6 \sqrt [3]{d} (b c-a d)^{2/3}}+\frac {\log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{d} (b c-a d)^{2/3}}-\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{d} (b c-a d)^{2/3}} \]
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Rubi [A] time = 0.12, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {444, 58, 617, 204, 31} \[ -\frac {\log \left (c+d x^3\right )}{6 \sqrt [3]{d} (b c-a d)^{2/3}}+\frac {\log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{d} (b c-a d)^{2/3}}-\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{d} (b c-a d)^{2/3}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 58
Rule 204
Rule 444
Rule 617
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a+b x^3\right )^{2/3} \left (c+d x^3\right )} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{(a+b x)^{2/3} (c+d x)} \, dx,x,x^3\right )\\ &=-\frac {\log \left (c+d x^3\right )}{6 \sqrt [3]{d} (b c-a d)^{2/3}}+\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{b c-a d}}{\sqrt [3]{d}}+x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{d} (b c-a d)^{2/3}}+\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {(b c-a d)^{2/3}}{d^{2/3}}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{d}}+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right )}{2 d^{2/3} \sqrt [3]{b c-a d}}\\ &=-\frac {\log \left (c+d x^3\right )}{6 \sqrt [3]{d} (b c-a d)^{2/3}}+\frac {\log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{d} (b c-a d)^{2/3}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}\right )}{\sqrt [3]{d} (b c-a d)^{2/3}}\\ &=-\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{d} (b c-a d)^{2/3}}-\frac {\log \left (c+d x^3\right )}{6 \sqrt [3]{d} (b c-a d)^{2/3}}+\frac {\log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{d} (b c-a d)^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 164, normalized size = 1.13 \[ \frac {-\log \left (-\sqrt [3]{d} \sqrt [3]{a+b x^3} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left (a+b x^3\right )^{2/3}\right )+2 \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}-1}{\sqrt {3}}\right )}{6 \sqrt [3]{d} (b c-a d)^{2/3}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 927, normalized size = 6.39 \[ \left [-\frac {3 \, \sqrt {\frac {1}{3}} {\left (b c d - a d^{2}\right )} \sqrt {-\frac {{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}^{\frac {1}{3}}}{d}} \log \left (\frac {b^{2} c^{2} - 4 \, a b c d + 3 \, a^{2} d^{2} - 2 \, {\left (b^{2} c d - a b d^{2}\right )} x^{3} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (b c d - a d^{2}\right )} - {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}^{\frac {1}{3}} {\left (b c - a d\right )} + {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}^{\frac {2}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )} \sqrt {-\frac {{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}^{\frac {1}{3}}}{d}} + 3 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}^{\frac {1}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (b c - a d\right )}}{d x^{3} + c}\right ) + {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}^{\frac {2}{3}} \log \left (-{\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (b c d - a d^{2}\right )} - {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}^{\frac {1}{3}} {\left (b c - a d\right )} + {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}^{\frac {2}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right ) - 2 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}^{\frac {2}{3}} \log \left (-{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (b c d - a d^{2}\right )} - {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}^{\frac {2}{3}}\right )}{6 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}}, \frac {6 \, \sqrt {\frac {1}{3}} {\left (b c d - a d^{2}\right )} \sqrt {\frac {{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}^{\frac {1}{3}}}{d}} \arctan \left (-\frac {\sqrt {\frac {1}{3}} {\left ({\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}^{\frac {1}{3}} {\left (b c - a d\right )} - 2 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}^{\frac {2}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )} \sqrt {\frac {{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}^{\frac {1}{3}}}{d}}}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}\right ) - {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}^{\frac {2}{3}} \log \left (-{\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (b c d - a d^{2}\right )} - {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}^{\frac {1}{3}} {\left (b c - a d\right )} + {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}^{\frac {2}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right ) + 2 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}^{\frac {2}{3}} \log \left (-{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (b c d - a d^{2}\right )} - {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}^{\frac {2}{3}}\right )}{6 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 221, normalized size = 1.52 \[ \frac {{\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} b c d - \sqrt {3} a d^{2}} + \frac {{\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c d - a d^{2}\right )}} - \frac {\left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (b c - a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.60, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (b \,x^{3}+a \right )^{\frac {2}{3}} \left (d \,x^{3}+c \right )}\, dx \]
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 = 4.85, size = 213, normalized size = 1.47 \[ \frac {\ln \left (3\,d^2\,{\left (b\,x^3+a\right )}^{1/3}-\frac {9\,a\,d^3-9\,b\,c\,d^2}{3\,d^{1/3}\,{\left (a\,d-b\,c\right )}^{2/3}}\right )}{3\,d^{1/3}\,{\left (a\,d-b\,c\right )}^{2/3}}+\frac {\ln \left (3\,d^2\,{\left (b\,x^3+a\right )}^{1/3}-\frac {\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (9\,a\,d^3-9\,b\,c\,d^2\right )}{6\,d^{1/3}\,{\left (a\,d-b\,c\right )}^{2/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,d^{1/3}\,{\left (a\,d-b\,c\right )}^{2/3}}-\frac {\ln \left (3\,d^2\,{\left (b\,x^3+a\right )}^{1/3}+\frac {\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (9\,a\,d^3-9\,b\,c\,d^2\right )}{6\,d^{1/3}\,{\left (a\,d-b\,c\right )}^{2/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,d^{1/3}\,{\left (a\,d-b\,c\right )}^{2/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (a + b x^{3}\right )^{\frac {2}{3}} \left (c + d x^{3}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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