Optimal. Leaf size=209 \[ -\frac{5 b^2 \left (a+b x^3\right )^{2/3}}{8 a^3 d x^2}+\frac{b^{8/3} \log \left (a d-b d x^3\right )}{3 \sqrt [3]{2} a^3 d}-\frac{b^{8/3} \log \left (\sqrt [3]{2} \sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} a^3 d}+\frac{2^{2/3} b^{8/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} a^3 d}-\frac{b \left (a+b x^3\right )^{2/3}}{4 a^2 d x^5}-\frac{\left (a+b x^3\right )^{2/3}}{8 a d x^8} \]
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Rubi [C] time = 10.6764, antiderivative size = 244, normalized size of antiderivative = 1.17, 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} \[ -\frac{-18 b x^3 \left (a-b x^3\right )^2 \, _3F_2\left (\frac{1}{3},2,2;1,\frac{4}{3};\frac{2 b x^3}{b x^3+a}\right )-4 b x^3 \left (5 a^2+6 a b x^3+9 b^2 x^6\right ) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{2 b x^3}{b x^3+a}\right )+42 a^2 b x^3 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{2 b x^3}{b x^3+a}\right )+11 a^2 b x^3+5 a^3-54 b^3 x^9 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{2 b x^3}{b x^3+a}\right )+12 a b^2 x^6 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{2 b x^3}{b x^3+a}\right )+15 a b^2 x^6+9 b^3 x^9}{40 a^3 d x^8 \sqrt [3]{a+b x^3}} \]
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^9 \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^9 \left (a d-b d x^3\right )} \, dx}{\left (1+\frac{b x^3}{a}\right )^{2/3}}\\ &=-\frac{5 a^3+11 a^2 b x^3+15 a b^2 x^6+9 b^3 x^9-4 b x^3 \left (5 a^2+6 a b x^3+9 b^2 x^6\right ) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{2 b x^3}{a+b x^3}\right )+42 a^2 b x^3 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{2 b x^3}{a+b x^3}\right )+12 a b^2 x^6 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{2 b x^3}{a+b x^3}\right )-54 b^3 x^9 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{2 b x^3}{a+b x^3}\right )-18 b x^3 \left (a-b x^3\right )^2 \, _3F_2\left (\frac{1}{3},2,2;1,\frac{4}{3};\frac{2 b x^3}{a+b x^3}\right )}{40 a^3 d x^8 \sqrt [3]{a+b x^3}}\\ \end{align*}
Mathematica [A] time = 5.21285, size = 179, normalized size = 0.86 \[ \frac{4\ 2^{2/3} b^{8/3} \left (\log \left (\frac{2^{2/3} b^{2/3} x^2}{\left (a x^3+b\right )^{2/3}}+\frac{\sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a x^3+b}}+1\right )-2 \log \left (1-\frac{\sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a x^3+b}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a x^3+b}}+1}{\sqrt{3}}\right )\right )-\frac{3 \left (a+b x^3\right )^{2/3} \left (a^2+2 a b x^3+5 b^2 x^6\right )}{x^8}}{24 a^3 d} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.05, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{9} \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^{9}}\,{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^{9}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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