Optimal. Leaf size=236 \[ -\frac{293 b^3 \left (a+b x^3\right )^{2/3}}{440 a^4 d x^2}-\frac{49 b^2 \left (a+b x^3\right )^{2/3}}{220 a^3 d x^5}+\frac{b^{11/3} \log \left (a d-b d x^3\right )}{3 \sqrt [3]{2} a^4 d}-\frac{b^{11/3} \log \left (\sqrt [3]{2} \sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} a^4 d}+\frac{2^{2/3} b^{11/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} a^4 d}-\frac{13 b \left (a+b x^3\right )^{2/3}}{88 a^2 d x^8}-\frac{\left (a+b x^3\right )^{2/3}}{11 a d x^{11}} \]
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Rubi [C] time = 17.6006, antiderivative size = 391, normalized size of antiderivative = 1.66, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {511, 510} \[ -\frac{-54 b x^3 \left (a-b x^3\right )^2 \left (5 a+6 b x^3\right ) \, _3F_2\left (\frac{1}{3},2,2;1,\frac{4}{3};\frac{2 b x^3}{b x^3+a}\right )+54 b x^3 \left (a-b x^3\right )^3 \, _4F_3\left (\frac{1}{3},2,2,2;1,1,\frac{4}{3};\frac{2 b x^3}{b x^3+a}\right )-180 a^2 b^2 x^6 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{2 b x^3}{b x^3+a}\right )+198 a^2 b^2 x^6 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{2 b x^3}{b x^3+a}\right )+99 a^2 b^2 x^6-160 a^3 b x^3 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{2 b x^3}{b x^3+a}\right )+396 a^3 b x^3 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{2 b x^3}{b x^3+a}\right )+85 a^3 b x^3+40 a^4-324 b^4 x^{12} \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{2 b x^3}{b x^3+a}\right )-594 b^4 x^{12} \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{2 b x^3}{b x^3+a}\right )-216 a b^3 x^9 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{2 b x^3}{b x^3+a}\right )+135 a b^3 x^9+81 b^4 x^{12}}{440 a^4 d x^{11} \sqrt [3]{a+b x^3}} \]
Warning: Unable to verify antiderivative.
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Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{\left (a+b x^3\right )^{2/3}}{x^{12} \left (a d-b d x^3\right )} \, dx &=\frac{\left (a+b x^3\right )^{2/3} \int \frac{\left (1+\frac{b x^3}{a}\right )^{2/3}}{x^{12} \left (a d-b d x^3\right )} \, dx}{\left (1+\frac{b x^3}{a}\right )^{2/3}}\\ &=-\frac{40 a^4+85 a^3 b x^3+99 a^2 b^2 x^6+135 a b^3 x^9+81 b^4 x^{12}-160 a^3 b x^3 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{2 b x^3}{a+b x^3}\right )-180 a^2 b^2 x^6 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{2 b x^3}{a+b x^3}\right )-216 a b^3 x^9 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{2 b x^3}{a+b x^3}\right )-324 b^4 x^{12} \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{2 b x^3}{a+b x^3}\right )+396 a^3 b x^3 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{2 b x^3}{a+b x^3}\right )+198 a^2 b^2 x^6 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{2 b x^3}{a+b x^3}\right )-594 b^4 x^{12} \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{2 b x^3}{a+b x^3}\right )-54 b x^3 \left (a-b x^3\right )^2 \left (5 a+6 b x^3\right ) \, _3F_2\left (\frac{1}{3},2,2;1,\frac{4}{3};\frac{2 b x^3}{a+b x^3}\right )+54 b x^3 \left (a-b x^3\right )^3 \, _4F_3\left (\frac{1}{3},2,2,2;1,1,\frac{4}{3};\frac{2 b x^3}{a+b x^3}\right )}{440 a^4 d x^{11} \sqrt [3]{a+b x^3}}\\ \end{align*}
Mathematica [A] time = 5.22345, size = 196, normalized size = 0.83 \[ \frac{\frac{2^{2/3} b^{11/3} \left (\log \left (\frac{2^{2/3} b^{2/3} x^2}{\left (a x^3+b\right )^{2/3}}+\frac{\sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a x^3+b}}+1\right )-2 \log \left (1-\frac{\sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a x^3+b}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a x^3+b}}+1}{\sqrt{3}}\right )\right )}{a^4}-\frac{3 \left (a+b x^3\right )^{2/3} \left (65 a^2 b x^3+40 a^3+98 a b^2 x^6+293 b^3 x^9\right )}{220 a^4 x^{11}}}{6 d} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.055, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{12} \left ( -bd{x}^{3}+ad \right ) } \left ( b{x}^{3}+a \right ) ^{{\frac{2}{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{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{{\left (b d x^{3} - a d\right )} x^{12}}\,{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{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{{\left (b d x^{3} - a d\right )} x^{12}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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