Optimal. Leaf size=187 \[ -\frac {(b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 d^{7/3}}+\frac {(b c-a d)^{4/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{7/3}}-\frac {(b c-a d)^{4/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} d^{7/3}}-\frac {\sqrt [3]{a+b x^3} (b c-a d)}{d^2}+\frac {\left (a+b x^3\right )^{4/3}}{4 d} \]
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Rubi [A] time = 0.21, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {444, 50, 58, 617, 204, 31} \[ -\frac {\sqrt [3]{a+b x^3} (b c-a d)}{d^2}-\frac {(b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 d^{7/3}}+\frac {(b c-a d)^{4/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{7/3}}-\frac {(b c-a d)^{4/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} d^{7/3}}+\frac {\left (a+b x^3\right )^{4/3}}{4 d} \]
Antiderivative was successfully verified.
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Rule 31
Rule 50
Rule 58
Rule 204
Rule 444
Rule 617
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b x^3\right )^{4/3}}{c+d x^3} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {(a+b x)^{4/3}}{c+d x} \, dx,x,x^3\right )\\ &=\frac {\left (a+b x^3\right )^{4/3}}{4 d}-\frac {(b c-a d) \operatorname {Subst}\left (\int \frac {\sqrt [3]{a+b x}}{c+d x} \, dx,x,x^3\right )}{3 d}\\ &=-\frac {(b c-a d) \sqrt [3]{a+b x^3}}{d^2}+\frac {\left (a+b x^3\right )^{4/3}}{4 d}+\frac {(b c-a d)^2 \operatorname {Subst}\left (\int \frac {1}{(a+b x)^{2/3} (c+d x)} \, dx,x,x^3\right )}{3 d^2}\\ &=-\frac {(b c-a d) \sqrt [3]{a+b x^3}}{d^2}+\frac {\left (a+b x^3\right )^{4/3}}{4 d}-\frac {(b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 d^{7/3}}+\frac {(b c-a d)^{4/3} \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^{7/3}}+\frac {(b c-a d)^{5/3} \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^{8/3}}\\ &=-\frac {(b c-a d) \sqrt [3]{a+b x^3}}{d^2}+\frac {\left (a+b x^3\right )^{4/3}}{4 d}-\frac {(b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 d^{7/3}}+\frac {(b c-a d)^{4/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{7/3}}+\frac {(b c-a d)^{4/3} \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^{7/3}}\\ &=-\frac {(b c-a d) \sqrt [3]{a+b x^3}}{d^2}+\frac {\left (a+b x^3\right )^{4/3}}{4 d}-\frac {(b c-a d)^{4/3} \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^{7/3}}-\frac {(b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 d^{7/3}}+\frac {(b c-a d)^{4/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{7/3}}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 232, normalized size = 1.24 \[ \frac {(a d-b c) \left (\sqrt [3]{b c-a d} \log \left (-\sqrt [3]{d} \sqrt [3]{a+b x^3} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left (a+b x^3\right )^{2/3}\right )-2 \sqrt [3]{b c-a d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )-2 \sqrt {3} \sqrt [3]{b c-a d} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}-1}{\sqrt {3}}\right )+6 \sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{6 d^{7/3}}+\frac {\left (a+b x^3\right )^{4/3}}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 246, normalized size = 1.32 \[ \frac {4 \, \sqrt {3} {\left (b c - a d\right )} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \arctan \left (-\frac {2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} d \left (-\frac {b c - a d}{d}\right )^{\frac {2}{3}} - \sqrt {3} {\left (b c - a d\right )}}{3 \, {\left (b c - a d\right )}}\right ) + 2 \, {\left (b c - a d\right )} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {2}{3}}\right ) - 4 \, {\left (b c - a d\right )} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} - \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}\right ) + 3 \, {\left (b d x^{3} - 4 \, b c + 5 \, a d\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{12 \, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 297, normalized size = 1.59 \[ -\frac {{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (b c d^{4} - a d^{5}\right )}} + \frac {\sqrt {3} {\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} {\left (b c - a d\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}}\right )}{3 \, d^{3}} + \frac {{\left (-b c d^{2} + a d^{3}\right )}^{\frac {1}{3}} {\left (b c - a d\right )} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {2}{3}}\right )}{6 \, d^{3}} - \frac {4 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} b c d^{2} - {\left (b x^{3} + a\right )}^{\frac {4}{3}} d^{3} - 4 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} a d^{3}}{4 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.65, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{3}+a \right )^{\frac {4}{3}} x^{2}}{d \,x^{3}+c}\, dx \]
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 = 4.72, size = 304, normalized size = 1.63 \[ \frac {{\left (b\,x^3+a\right )}^{4/3}}{4\,d}+\frac {\ln \left ({\left (b\,x^3+a\right )}^{1/3}\,\left (3\,a^2\,d^2-6\,a\,b\,c\,d+3\,b^2\,c^2\right )-\frac {{\left (a\,d-b\,c\right )}^{4/3}\,\left (9\,a\,d^3-9\,b\,c\,d^2\right )}{3\,d^{7/3}}\right )\,{\left (a\,d-b\,c\right )}^{4/3}}{3\,d^{7/3}}+\frac {{\left (b\,x^3+a\right )}^{1/3}\,\left (a\,d-b\,c\right )}{d^2}-\frac {\ln \left ({\left (b\,x^3+a\right )}^{1/3}\,\left (3\,a^2\,d^2-6\,a\,b\,c\,d+3\,b^2\,c^2\right )+\frac {\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (a\,d-b\,c\right )}^{4/3}\,\left (9\,a\,d^3-9\,b\,c\,d^2\right )}{3\,d^{7/3}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (a\,d-b\,c\right )}^{4/3}}{3\,d^{7/3}}+\frac {\ln \left ({\left (b\,x^3+a\right )}^{1/3}\,\left (3\,a^2\,d^2-6\,a\,b\,c\,d+3\,b^2\,c^2\right )-\frac {\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,{\left (a\,d-b\,c\right )}^{4/3}\,\left (9\,a\,d^3-9\,b\,c\,d^2\right )}{d^{7/3}}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,{\left (a\,d-b\,c\right )}^{4/3}}{d^{7/3}} \]
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 \[ \int \frac {x^{2} \left (a + b x^{3}\right )^{\frac {4}{3}}}{c + d x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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