Optimal. Leaf size=492 \[ \frac{9 x \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{20 a^3 \left (a+b x^3\right )^{2/3}}+\frac{13 x}{40 a^3 \left (a+b x^3\right )^{2/3}}+\frac{x}{10 a^2 \left (a+b x^3\right )^{5/3}}-\frac{\log \left (2^{2/3}-\frac{\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{24\ 2^{2/3} a^{10/3} \sqrt [3]{b}}+\frac{\log \left (\frac{2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac{\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{24\ 2^{2/3} a^{10/3} \sqrt [3]{b}}-\frac{\log \left (\frac{\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{12\ 2^{2/3} a^{10/3} \sqrt [3]{b}}+\frac{\log \left (\frac{\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}+\frac{2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+2 \sqrt [3]{2}\right )}{48\ 2^{2/3} a^{10/3} \sqrt [3]{b}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt{3}}\right )}{4\ 2^{2/3} \sqrt{3} a^{10/3} \sqrt [3]{b}}-\frac{\tan ^{-1}\left (\frac{\frac{\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{8\ 2^{2/3} \sqrt{3} a^{10/3} \sqrt [3]{b}} \]
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Rubi [C] time = 0.026983, antiderivative size = 58, normalized size of antiderivative = 0.12, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {430, 429} \[ \frac{x \left (\frac{b x^3}{a}+1\right )^{2/3} F_1\left (\frac{1}{3};1,\frac{8}{3};\frac{4}{3};\frac{b x^3}{a},-\frac{b x^3}{a}\right )}{a^3 \left (a+b x^3\right )^{2/3}} \]
Warning: Unable to verify antiderivative.
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Rule 430
Rule 429
Rubi steps
\begin{align*} \int \frac{1}{\left (a-b x^3\right ) \left (a+b x^3\right )^{8/3}} \, dx &=\frac{\left (1+\frac{b x^3}{a}\right )^{2/3} \int \frac{1}{\left (a-b x^3\right ) \left (1+\frac{b x^3}{a}\right )^{8/3}} \, dx}{a^2 \left (a+b x^3\right )^{2/3}}\\ &=\frac{x \left (1+\frac{b x^3}{a}\right )^{2/3} F_1\left (\frac{1}{3};1,\frac{8}{3};\frac{4}{3};\frac{b x^3}{a},-\frac{b x^3}{a}\right )}{a^3 \left (a+b x^3\right )^{2/3}}\\ \end{align*}
Mathematica [C] time = 0.159652, size = 240, normalized size = 0.49 \[ \frac{x \left (\frac{368 a^3 \left (a+b x^3\right ) F_1\left (\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )}{\left (a-b x^3\right ) \left (b x^3 \left (3 F_1\left (\frac{4}{3};\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )-2 F_1\left (\frac{4}{3};\frac{5}{3},1;\frac{7}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )\right )+4 a F_1\left (\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )\right )}+16 a^2-13 b x^3 \left (a+b x^3\right ) \left (\frac{b x^3}{a}+1\right )^{2/3} F_1\left (\frac{4}{3};\frac{2}{3},1;\frac{7}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )+52 a \left (a+b x^3\right )\right )}{160 a^4 \left (a+b x^3\right )^{5/3}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.423, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-b{x}^{3}+a} \left ( b{x}^{3}+a \right ) ^{-{\frac{8}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{1}{{\left (b x^{3} + a\right )}^{\frac{8}{3}}{\left (b x^{3} - a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{1}{{\left (b x^{3} + a\right )}^{\frac{8}{3}}{\left (b x^{3} - a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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