Optimal. Leaf size=1357 \[ \frac {d^6 \left (\frac {b x^3}{a}+1\right )^{2/3} F_1\left (\frac {7}{3};\frac {2}{3},3;\frac {10}{3};-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right ) x^7}{7 c^9 \left (b x^3+a\right )^{2/3}}-\frac {7 d^3 \left (\frac {b x^3}{a}+1\right )^{2/3} F_1\left (\frac {4}{3};\frac {2}{3},3;\frac {7}{3};-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right ) x^4}{4 c^6 \left (b x^3+a\right )^{2/3}}+\frac {d^4 \left (3 b c^3+2 a d^3\right ) \sqrt [3]{b x^3+a} x^2}{3 c \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}+\frac {d^4 \left (9 b c^3-4 a d^3\right ) \sqrt [3]{b x^3+a} x^2}{6 c \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}+\frac {3 c^2 d^4 \sqrt [3]{b x^3+a} x^2}{2 \left (b c^3-a d^3\right ) \left (c^3+d^3 x^3\right )^2}+\frac {\left (\frac {b x^3}{a}+1\right )^{2/3} F_1\left (\frac {1}{3};\frac {2}{3},3;\frac {4}{3};-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right ) x}{c^3 \left (b x^3+a\right )^{2/3}}+\frac {2 a d^4 \left (6 b c^3-a d^3\right ) \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b c^3-a d^3} x}{c \sqrt [3]{b x^3+a}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} c^2 \left (b c^3-a d^3\right )^{8/3}}+\frac {d \left (9 b^2 c^6-6 a b d^3 c^3+2 a^2 d^6\right ) \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b c^3-a d^3} x}{c \sqrt [3]{b x^3+a}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} c^2 \left (b c^3-a d^3\right )^{8/3}}-\frac {10 b^2 c^4 d \tan ^{-1}\left (\frac {1-\frac {2 d \sqrt [3]{b x^3+a}}{\sqrt [3]{b c^3-a d^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} \left (b c^3-a d^3\right )^{8/3}}+\frac {b c d \left (b c^3-6 a d^3\right ) \tan ^{-1}\left (\frac {1-\frac {2 d \sqrt [3]{b x^3+a}}{\sqrt [3]{b c^3-a d^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} \left (b c^3-a d^3\right )^{8/3}}-\frac {a d^4 \left (6 b c^3-a d^3\right ) \log \left (c^3+d^3 x^3\right )}{9 c^2 \left (b c^3-a d^3\right )^{8/3}}-\frac {d \left (9 b^2 c^6-6 a b d^3 c^3+2 a^2 d^6\right ) \log \left (c^3+d^3 x^3\right )}{18 c^2 \left (b c^3-a d^3\right )^{8/3}}-\frac {5 b^2 c^4 d \log \left (c^3+d^3 x^3\right )}{9 \left (b c^3-a d^3\right )^{8/3}}+\frac {b c d \left (b c^3-6 a d^3\right ) \log \left (c^3+d^3 x^3\right )}{18 \left (b c^3-a d^3\right )^{8/3}}+\frac {a d^4 \left (6 b c^3-a d^3\right ) \log \left (\frac {\sqrt [3]{b c^3-a d^3} x}{c}-\sqrt [3]{b x^3+a}\right )}{3 c^2 \left (b c^3-a d^3\right )^{8/3}}+\frac {d \left (9 b^2 c^6-6 a b d^3 c^3+2 a^2 d^6\right ) \log \left (\frac {\sqrt [3]{b c^3-a d^3} x}{c}-\sqrt [3]{b x^3+a}\right )}{6 c^2 \left (b c^3-a d^3\right )^{8/3}}+\frac {5 b^2 c^4 d \log \left (\sqrt [3]{b x^3+a} d+\sqrt [3]{b c^3-a d^3}\right )}{3 \left (b c^3-a d^3\right )^{8/3}}-\frac {b c d \left (b c^3-6 a d^3\right ) \log \left (\sqrt [3]{b x^3+a} d+\sqrt [3]{b c^3-a d^3}\right )}{6 \left (b c^3-a d^3\right )^{8/3}}+\frac {5 b c^4 d^2 \sqrt [3]{b x^3+a}}{3 \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}-\frac {c d^2 \left (b c^3-6 a d^3\right ) \sqrt [3]{b x^3+a}}{6 \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}+\frac {3 c^4 d^2 \sqrt [3]{b x^3+a}}{2 \left (b c^3-a d^3\right ) \left (c^3+d^3 x^3\right )^2} \]
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Rubi [F] time = 0.08, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{(c+d x)^3 \left (a+b x^3\right )^{2/3}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{(c+d x)^3 \left (a+b x^3\right )^{2/3}} \, dx &=\int \frac {1}{(c+d x)^3 \left (a+b x^3\right )^{2/3}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.44, size = 0, normalized size = 0.00 \[ \int \frac {1}{(c+d x)^3 \left (a+b x^3\right )^{2/3}} \, dx \]
Verification is Not applicable to the result.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d x +c \right )^{3} \left (b \,x^{3}+a \right )^{\frac {2}{3}}}\, dx \]
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 \[ \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (b\,x^3+a\right )}^{2/3}\,{\left (c+d\,x\right )}^3} \,d x \]
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 \[ \int \frac {1}{\left (a + b x^{3}\right )^{\frac {2}{3}} \left (c + d x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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