Optimal. Leaf size=1513 \[ \frac {2 a^2 \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 ) d^6}{9 \sqrt {3} c^2 \left (b c^3-a d^3\right )^{7/3}}+\frac {a^2 \log \left (c^3+d^3 x^3\right ) d^6}{27 c^2 \left (b c^3-a d^3\right )^{7/3}}-\frac {a^2 \log \left (\frac {\sqrt [3]{b c^3-a d^3} x}{c}-\sqrt [3]{b x^3+a}\right ) d^6}{9 c^2 \left (b c^3-a d^3\right )^{7/3}}+\frac {6 x^5 \sqrt [3]{\frac {b x^3}{a}+1} F_1\left (\frac {5}{3};\frac {1}{3},3;\frac {8}{3};-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right ) d^4}{5 c^7 \sqrt [3]{b x^3+a}}+\frac {7 a \left (3 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 ) d^3}{9 \sqrt {3} c^2 \left (b c^3-a d^3\right )^{7/3}}+\frac {7 a \left (3 b c^3-a d^3\right ) \log \left (c^3+d^3 x^3\right ) d^3}{54 c^2 \left (b c^3-a d^3\right )^{7/3}}-\frac {7 a \left (3 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 ) d^3}{18 c^2 \left (b c^3-a d^3\right )^{7/3}}-\frac {7 \left (3 b c^3+a d^3\right ) x \left (b x^3+a\right )^{2/3} d^3}{18 \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}+\frac {\left (3 b c^3-7 a d^3\right ) x \left (b x^3+a\right )^{2/3} d^3}{18 \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}-\frac {\left (9 b c^3-5 a d^3\right ) x \left (b x^3+a\right )^{2/3} d^3}{18 \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}-\frac {3 c^3 x \left (b x^3+a\right )^{2/3} d^3}{2 \left (b c^3-a d^3\right ) \left (c^3+d^3 x^3\right )^2}+\frac {4 b c^4 \left (b x^3+a\right )^{2/3} d^2}{3 \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}-\frac {c \left (b c^3-3 a d^3\right ) \left (b x^3+a\right )^{2/3} d^2}{3 \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}+\frac {3 c^4 \left (b x^3+a\right )^{2/3} d^2}{2 \left (b c^3-a d^3\right ) \left (c^3+d^3 x^3\right )^2}-\frac {3 x^2 \sqrt [3]{\frac {b x^3}{a}+1} F_1\left (\frac {2}{3};\frac {1}{3},3;\frac {5}{3};-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right ) d}{2 c^4 \sqrt [3]{b x^3+a}}+\frac {\left (9 b^2 c^6-12 a b d^3 c^3+5 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 )}{9 \sqrt {3} c^2 \left (b c^3-a d^3\right )^{7/3}}-\frac {4 b^2 c^4 \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 )^{7/3}}+\frac {b c \left (b c^3-3 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 )^{7/3}}+\frac {\left (9 b^2 c^6-12 a b d^3 c^3+5 a^2 d^6\right ) \log \left (c^3+d^3 x^3\right )}{54 c^2 \left (b c^3-a d^3\right )^{7/3}}+\frac {2 b^2 c^4 \log \left (c^3+d^3 x^3\right )}{9 \left (b c^3-a d^3\right )^{7/3}}-\frac {b c \left (b c^3-3 a d^3\right ) \log \left (c^3+d^3 x^3\right )}{18 \left (b c^3-a d^3\right )^{7/3}}-\frac {\left (9 b^2 c^6-12 a b d^3 c^3+5 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 )}{18 c^2 \left (b c^3-a d^3\right )^{7/3}}-\frac {2 b^2 c^4 \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 )^{7/3}}+\frac {b c \left (b c^3-3 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 )^{7/3}} \]
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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 = {} \[ \int \frac {1}{(c+d x)^3 \sqrt [3]{a+b x^3}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{(c+d x)^3 \sqrt [3]{a+b x^3}} \, dx &=\int \frac {1}{(c+d x)^3 \sqrt [3]{a+b x^3}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.48, size = 0, normalized size = 0.00 \[ \int \frac {1}{(c+d x)^3 \sqrt [3]{a+b x^3}} \, dx \]
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 \[ \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{3}}\,{d x} \]
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 \[ \int \frac {1}{\left (d x +c \right )^{3} \left (b \,x^{3}+a \right )^{\frac {1}{3}}}\, dx \]
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 \[ \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{3}}\,{d x} \]
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 \[ \int \frac {1}{{\left (b\,x^3+a\right )}^{1/3}\,{\left (c+d\,x\right )}^3} \,d x \]
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 \[ \int \frac {1}{\sqrt [3]{a + b x^{3}} \left (c + d x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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