Optimal. Leaf size=159 \[ \frac {\sqrt [3]{a+b x^3}}{d}+\frac {\sqrt [3]{b c-a d} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} d^{4/3}}+\frac {\sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 d^{4/3}}-\frac {\sqrt [3]{b c-a d} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{4/3}} \]
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Rubi [A]
time = 0.11, antiderivative size = 159, 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 = {455, 52, 60,
631, 210, 31} \begin {gather*} \frac {\sqrt [3]{b c-a d} \text {ArcTan}\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^{4/3}}+\frac {\sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 d^{4/3}}-\frac {\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 d^{4/3}}+\frac {\sqrt [3]{a+b x^3}}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 52
Rule 60
Rule 210
Rule 455
Rule 631
Rubi steps
\begin {align*} \int \frac {x^2 \sqrt [3]{a+b x^3}}{c+d x^3} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {\sqrt [3]{a+b x}}{c+d x} \, dx,x,x^3\right )\\ &=\frac {\sqrt [3]{a+b x^3}}{d}-\frac {(b c-a d) \text {Subst}\left (\int \frac {1}{(a+b x)^{2/3} (c+d x)} \, dx,x,x^3\right )}{3 d}\\ &=\frac {\sqrt [3]{a+b x^3}}{d}+\frac {\sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 d^{4/3}}-\frac {\sqrt [3]{b c-a d} \text {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^{4/3}}-\frac {(b c-a d)^{2/3} \text {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^{5/3}}\\ &=\frac {\sqrt [3]{a+b x^3}}{d}+\frac {\sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 d^{4/3}}-\frac {\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 d^{4/3}}-\frac {\sqrt [3]{b c-a d} \text {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^{4/3}}\\ &=\frac {\sqrt [3]{a+b x^3}}{d}+\frac {\sqrt [3]{b c-a d} \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^{4/3}}+\frac {\sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 d^{4/3}}-\frac {\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 d^{4/3}}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 205, normalized size = 1.29 \begin {gather*} \frac {6 \sqrt [3]{d} \sqrt [3]{a+b x^3}+2 \sqrt {3} \sqrt [3]{b c-a d} \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 )-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 )+\sqrt [3]{b c-a d} \log \left ((b c-a d)^{2/3}-\sqrt [3]{d} \sqrt [3]{b c-a d} \sqrt [3]{a+b x^3}+d^{2/3} \left (a+b x^3\right )^{2/3}\right )}{6 d^{4/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{d \,x^{3}+c}\, dx\]
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.63, size = 206, normalized size = 1.30 \begin {gather*} -\frac {2 \, \sqrt {3} \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 ) + \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 ) - 2 \, \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 ) - 6 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{6 \, d} \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} \sqrt [3]{a + b x^{3}}}{c + d x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.73, size = 223, normalized size = 1.40 \begin {gather*} \frac {{\left (b c - a d\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 - a d^{2}\right )}} - \frac {\sqrt {3} {\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 )}{3 \, d^{2}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{d} - \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 \, d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.62, size = 249, normalized size = 1.57 \begin {gather*} \frac {{\left (b\,x^3+a\right )}^{1/3}}{d}+\frac {\ln \left ({\left (b\,x^3+a\right )}^{1/3}\,\left (3\,a\,d^2-3\,b\,c\,d\right )-\frac {{\left (a\,d-b\,c\right )}^{1/3}\,\left (9\,a\,d^3-9\,b\,c\,d^2\right )}{3\,d^{4/3}}\right )\,{\left (a\,d-b\,c\right )}^{1/3}}{3\,d^{4/3}}-\frac {\ln \left ({\left (b\,x^3+a\right )}^{1/3}\,\left (3\,a\,d^2-3\,b\,c\,d\right )+\frac {\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (a\,d-b\,c\right )}^{1/3}\,\left (9\,a\,d^3-9\,b\,c\,d^2\right )}{3\,d^{4/3}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (a\,d-b\,c\right )}^{1/3}}{3\,d^{4/3}}+\frac {\ln \left ({\left (b\,x^3+a\right )}^{1/3}\,\left (3\,a\,d^2-3\,b\,c\,d\right )-\frac {\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,{\left (a\,d-b\,c\right )}^{1/3}\,\left (9\,a\,d^3-9\,b\,c\,d^2\right )}{d^{4/3}}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,{\left (a\,d-b\,c\right )}^{1/3}}{d^{4/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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