Optimal. Leaf size=162 \[ -\frac {(b c-a d)^{2/3} \log \left (c+d x^3\right )}{6 d^{5/3}}+\frac {(b c-a d)^{2/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{5/3}}+\frac {(b c-a d)^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} d^{5/3}}+\frac {\left (a+b x^3\right )^{2/3}}{2 d} \]
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Rubi [A] time = 0.17, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {444, 50, 56, 617, 204, 31} \begin {gather*} -\frac {(b c-a d)^{2/3} \log \left (c+d x^3\right )}{6 d^{5/3}}+\frac {(b c-a d)^{2/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{5/3}}+\frac {(b c-a d)^{2/3} \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} d^{5/3}}+\frac {\left (a+b x^3\right )^{2/3}}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 50
Rule 56
Rule 204
Rule 444
Rule 617
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b x^3\right )^{2/3}}{c+d x^3} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {(a+b x)^{2/3}}{c+d x} \, dx,x,x^3\right )\\ &=\frac {\left (a+b x^3\right )^{2/3}}{2 d}-\frac {(b c-a d) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a+b x} (c+d x)} \, dx,x,x^3\right )}{3 d}\\ &=\frac {\left (a+b x^3\right )^{2/3}}{2 d}-\frac {(b c-a d)^{2/3} \log \left (c+d x^3\right )}{6 d^{5/3}}+\frac {(b c-a d)^{2/3} \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 d^{5/3}}-\frac {(b c-a d) \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}\\ &=\frac {\left (a+b x^3\right )^{2/3}}{2 d}-\frac {(b c-a d)^{2/3} \log \left (c+d x^3\right )}{6 d^{5/3}}+\frac {(b c-a d)^{2/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{5/3}}-\frac {(b c-a d)^{2/3} \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 )}{d^{5/3}}\\ &=\frac {\left (a+b x^3\right )^{2/3}}{2 d}+\frac {(b c-a d)^{2/3} \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} d^{5/3}}-\frac {(b c-a d)^{2/3} \log \left (c+d x^3\right )}{6 d^{5/3}}+\frac {(b c-a d)^{2/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{5/3}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 47, normalized size = 0.29 \begin {gather*} -\frac {\left (a+b x^3\right )^{2/3} \left (\, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {d \left (b x^3+a\right )}{a d-b c}\right )-1\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.25, size = 218, normalized size = 1.35 \begin {gather*} \frac {(b c-a d)^{2/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{3 d^{5/3}}-\frac {(b c-a d)^{2/3} \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 )}{6 d^{5/3}}+\frac {(b c-a d)^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{b c-a d}}\right )}{\sqrt {3} d^{5/3}}+\frac {\left (a+b x^3\right )^{2/3}}{2 d} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 323, normalized size = 1.99 \begin {gather*} \frac {2 \, \sqrt {3} \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{d^{2}}\right )^{\frac {1}{3}} \arctan \left (-\frac {2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} d \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{d^{2}}\right )^{\frac {1}{3}} - \sqrt {3} {\left (b c - a d\right )}}{3 \, {\left (b c - a d\right )}}\right ) - \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{d^{2}}\right )^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} d \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{d^{2}}\right )^{\frac {2}{3}} - {\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (b c - a d\right )} - {\left (b c - a d\right )} \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{d^{2}}\right )^{\frac {1}{3}}\right ) + 2 \, \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{d^{2}}\right )^{\frac {1}{3}} \log \left (-d \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{d^{2}}\right )^{\frac {2}{3}} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (b c - a d\right )}\right ) + 3 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 259, normalized size = 1.60 \begin {gather*} \frac {{\left (b c d \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} - a d^{2} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}\right )} \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 d^{2} - a d^{3}\right )}} + \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{2 \, d} + \frac {\sqrt {3} {\left (-b c d^{2} + a d^{3}\right )}^{\frac {2}{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 )}{3 \, d^{3}} - \frac {{\left (-b c d^{2} + a d^{3}\right )}^{\frac {2}{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 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.58, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} x^{2}}{d \,x^{3}+c}\, dx \end {gather*}
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.05, size = 238, normalized size = 1.47 \begin {gather*} \frac {{\left (b\,x^3+a\right )}^{2/3}}{2\,d}+\frac {\ln \left (\frac {{\left (b\,x^3+a\right )}^{1/3}\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{d}-\frac {{\left (a\,d-b\,c\right )}^{4/3}\,\left (9\,a\,d^3-9\,b\,c\,d^2\right )}{9\,d^{10/3}}\right )\,{\left (a\,d-b\,c\right )}^{2/3}}{3\,d^{5/3}}-\frac {\ln \left (\frac {{\left (b\,x^3+a\right )}^{1/3}\,{\left (a\,d-b\,c\right )}^2}{d}-\frac {\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (a\,d-b\,c\right )}^{7/3}}{d^{4/3}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (a\,d-b\,c\right )}^{2/3}}{3\,d^{5/3}}+\frac {\ln \left (\frac {{\left (b\,x^3+a\right )}^{1/3}\,{\left (a\,d-b\,c\right )}^2}{d}-\frac {{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,{\left (a\,d-b\,c\right )}^{7/3}}{4\,d^{4/3}}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,{\left (a\,d-b\,c\right )}^{2/3}}{d^{5/3}} \end {gather*}
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 \begin {gather*} \int \frac {x^{2} \left (a + b x^{3}\right )^{\frac {2}{3}}}{c + d x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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