Optimal. Leaf size=333 \[ -\frac {d x^2 \sqrt [3]{1+\frac {b x^3}{a}} F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right )}{2 c^2 \sqrt [3]{a+b x^3}}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b c^3-a d^3} x}{c \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b c^3-a d^3}}-\frac {\tan ^{-1}\left (\frac {1-\frac {2 d \sqrt [3]{a+b x^3}}{\sqrt [3]{b c^3-a d^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b c^3-a d^3}}+\frac {\log \left (c^3+d^3 x^3\right )}{3 \sqrt [3]{b c^3-a d^3}}-\frac {\log \left (\frac {\sqrt [3]{b c^3-a d^3} x}{c}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{b c^3-a d^3}}-\frac {\log \left (\sqrt [3]{b c^3-a d^3}+d \sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{b c^3-a d^3}} \]
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Rubi [A]
time = 0.21, antiderivative size = 333, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 9, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {2181, 384,
525, 524, 455, 58, 631, 210, 31} \begin {gather*} -\frac {d x^2 \sqrt [3]{\frac {b x^3}{a}+1} F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right )}{2 c^2 \sqrt [3]{a+b x^3}}+\frac {\text {ArcTan}\left (\frac {\frac {2 x \sqrt [3]{b c^3-a d^3}}{c \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b c^3-a d^3}}-\frac {\text {ArcTan}\left (\frac {1-\frac {2 d \sqrt [3]{a+b x^3}}{\sqrt [3]{b c^3-a d^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b c^3-a d^3}}+\frac {\log \left (c^3+d^3 x^3\right )}{3 \sqrt [3]{b c^3-a d^3}}-\frac {\log \left (\frac {x \sqrt [3]{b c^3-a d^3}}{c}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{b c^3-a d^3}}-\frac {\log \left (\sqrt [3]{b c^3-a d^3}+d \sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{b c^3-a d^3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 58
Rule 210
Rule 384
Rule 455
Rule 524
Rule 525
Rule 631
Rule 2181
Rubi steps
\begin {align*} \int \frac {1}{(c+d x) \sqrt [3]{a+b x^3}} \, dx &=\int \frac {1}{(c+d x) \sqrt [3]{a+b x^3}} \, dx\\ \end {align*}
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Mathematica [F]
time = 7.03, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(c+d x) \sqrt [3]{a+b x^3}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (d x +c \right ) \left (b \,x^{3}+a \right )^{\frac {1}{3}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{a + b x^{3}} \left (c + d x\right )}\, dx \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*} \text {could not integrate} \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 = -0.00 \begin {gather*} \int \frac {1}{{\left (b\,x^3+a\right )}^{1/3}\,\left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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