Optimal. Leaf size=416 \[ -\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3\ 2^{2/3} \sqrt [3]{a} \sqrt [3]{b} d}+\frac {\log \left (\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3\ 2^{2/3} \sqrt [3]{a} \sqrt [3]{b} d}-\frac {\sqrt [3]{2} \log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3 \sqrt [3]{a} \sqrt [3]{b} d}+\frac {\log \left (\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+2 \sqrt [3]{2}\right )}{6\ 2^{2/3} \sqrt [3]{a} \sqrt [3]{b} d}-\frac {\sqrt [3]{2} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{b} d}-\frac {\tan ^{-1}\left (\frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} \sqrt [3]{a} \sqrt [3]{b} d} \]
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Rubi [C] time = 0.03, antiderivative size = 61, normalized size of antiderivative = 0.15, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {430, 429} \begin {gather*} \frac {x \sqrt [3]{a+b x^3} F_1\left (\frac {1}{3};-\frac {1}{3},1;\frac {4}{3};-\frac {b x^3}{a},\frac {b x^3}{a}\right )}{a d \sqrt [3]{\frac {b x^3}{a}+1}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 429
Rule 430
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{a+b x^3}}{a d-b d x^3} \, dx &=\frac {\sqrt [3]{a+b x^3} \int \frac {\sqrt [3]{1+\frac {b x^3}{a}}}{a d-b d x^3} \, dx}{\sqrt [3]{1+\frac {b x^3}{a}}}\\ &=\frac {x \sqrt [3]{a+b x^3} F_1\left (\frac {1}{3};-\frac {1}{3},1;\frac {4}{3};-\frac {b x^3}{a},\frac {b x^3}{a}\right )}{a d \sqrt [3]{1+\frac {b x^3}{a}}}\\ \end {align*}
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Mathematica [C] time = 0.15, size = 154, normalized size = 0.37 \begin {gather*} \frac {4 a x \sqrt [3]{a+b x^3} F_1\left (\frac {1}{3};-\frac {1}{3},1;\frac {4}{3};-\frac {b x^3}{a},\frac {b x^3}{a}\right )}{d \left (a-b x^3\right ) \left (b x^3 \left (3 F_1\left (\frac {4}{3};-\frac {1}{3},2;\frac {7}{3};-\frac {b x^3}{a},\frac {b x^3}{a}\right )+F_1\left (\frac {4}{3};\frac {2}{3},1;\frac {7}{3};-\frac {b x^3}{a},\frac {b x^3}{a}\right )\right )+4 a F_1\left (\frac {1}{3};-\frac {1}{3},1;\frac {4}{3};-\frac {b x^3}{a},\frac {b x^3}{a}\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 3.09, size = 533, normalized size = 1.28 \begin {gather*} \frac {\log \left (2^{2/3} a^{2/3}-\sqrt [3]{2} \sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a+b x^3}+2\ 2^{2/3} \sqrt [3]{a} \sqrt [3]{b} x+2^{2/3} b^{2/3} x^2\right )}{3\ 2^{2/3} \sqrt [3]{a} \sqrt [3]{b} d}+\frac {\log \left (2^{2/3} a^{2/3}+2 \sqrt [3]{2} \sqrt [3]{b} x \sqrt [3]{a+b x^3}+4 \left (a+b x^3\right )^{2/3}+2 \sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a+b x^3}+2\ 2^{2/3} \sqrt [3]{a} \sqrt [3]{b} x+2^{2/3} b^{2/3} x^2\right )}{6\ 2^{2/3} \sqrt [3]{a} \sqrt [3]{b} d}-\frac {\sqrt [3]{2} \log \left (\sqrt [3]{a+b x^3}+\sqrt [3]{2} \sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{b} x\right )}{3 \sqrt [3]{a} \sqrt [3]{b} d}-\frac {\log \left (2 \sqrt [3]{a+b x^3}-\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{b} x\right )}{3\ 2^{2/3} \sqrt [3]{a} \sqrt [3]{b} d}+\frac {\sqrt [3]{2} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a+b x^3}-2 \sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{2} \sqrt [3]{b} x}\right )}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{b} d}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a+b x^3}+\sqrt [3]{2} \sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{b} x}\right )}{2^{2/3} \sqrt {3} \sqrt [3]{a} \sqrt [3]{b} d} \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 {1}{3}}}{b d x^{3} - a d}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.63, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{-b d \,x^{3}+a d}\, 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 {1}{3}}}{b d x^{3} - a d}\,{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 = -0.00 \begin {gather*} \int \frac {{\left (b\,x^3+a\right )}^{1/3}}{a\,d-b\,d\,x^3} \,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*} - \frac {\int \frac {\sqrt [3]{a + b x^{3}}}{- a + b x^{3}}\, dx}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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