Optimal. Leaf size=169 \[ \frac {(b c-a d)^{2/3} \log \left (c+d x^3\right )}{6 c^{5/3}}-\frac {(b c-a d)^{2/3} \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{5/3}}+\frac {(b c-a d)^{2/3} \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} c^{5/3}}-\frac {\left (a+b x^3\right )^{2/3}}{2 c x^2} \]
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Rubi [C] time = 0.06, antiderivative size = 89, normalized size of antiderivative = 0.53, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {511, 510} \begin {gather*} -\frac {\left (a+b x^3\right )^{2/3} \left (\frac {d x^3}{c}+1\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {2}{3};\frac {1}{3};-\frac {c \left (\frac {b x^3}{a}-\frac {d x^3}{c}\right )}{d x^3+c}\right )}{2 c x^2 \left (\frac {b x^3}{a}+1\right )^{2/3}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 510
Rule 511
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^{2/3}}{x^3 \left (c+d x^3\right )} \, dx &=\frac {\left (a+b x^3\right )^{2/3} \int \frac {\left (1+\frac {b x^3}{a}\right )^{2/3}}{x^3 \left (c+d x^3\right )} \, dx}{\left (1+\frac {b x^3}{a}\right )^{2/3}}\\ &=-\frac {\left (a+b x^3\right )^{2/3} \left (1+\frac {d x^3}{c}\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {2}{3};\frac {1}{3};-\frac {c \left (\frac {b x^3}{a}-\frac {d x^3}{c}\right )}{c+d x^3}\right )}{2 c x^2 \left (1+\frac {b x^3}{a}\right )^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 83, normalized size = 0.49 \begin {gather*} -\frac {\left (a+b x^3\right )^{2/3} \left (\frac {d x^3}{c}+1\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {2}{3};\frac {1}{3};\frac {(a d-b c) x^3}{a \left (d x^3+c\right )}\right )}{2 c x^2 \left (\frac {b x^3}{a}+1\right )^{2/3}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [C] time = 2.38, size = 365, normalized size = 2.16 \begin {gather*} \frac {\left ((b c-a d)^{2/3}+i \sqrt {3} (b c-a d)^{2/3}\right ) \log \left (2 x \sqrt [3]{b c-a d}+\left (1+i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}\right )}{6 c^{5/3}}-\frac {\sqrt {\frac {1}{6} \left (-1+i \sqrt {3}\right )} (b c-a d)^{2/3} \tan ^{-1}\left (\frac {3 x \sqrt [3]{b c-a d}}{\sqrt {3} x \sqrt [3]{b c-a d}-\sqrt {3} \sqrt [3]{c} \sqrt [3]{a+b x^3}-3 i \sqrt [3]{c} \sqrt [3]{a+b x^3}}\right )}{c^{5/3}}-\frac {i \left (\sqrt {3} (b c-a d)^{2/3}-i (b c-a d)^{2/3}\right ) \log \left (\left (\sqrt {3}+i\right ) c^{2/3} \left (a+b x^3\right )^{2/3}+\sqrt [3]{c} \left (-\sqrt {3} x+i x\right ) \sqrt [3]{a+b x^3} \sqrt [3]{b c-a d}-2 i x^2 (b c-a d)^{2/3}\right )}{12 c^{5/3}}-\frac {\left (a+b x^3\right )^{2/3}}{2 c x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{{\left (d x^{3} + c\right )} x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.61, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{\left (d \,x^{3}+c \right ) x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{{\left (d x^{3} + c\right )} x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (b\,x^3+a\right )}^{2/3}}{x^3\,\left (d\,x^3+c\right )} \,d x \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 {\left (a + b x^{3}\right )^{\frac {2}{3}}}{x^{3} \left (c + d x^{3}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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