Optimal. Leaf size=257 \[ \frac{\left (a+b x^3\right )^{2/3} \left (-20 a^2 d^2+8 a b c d+3 b^2 c^2\right )}{40 a^2 c^3 x^2}+\frac{d^2 (b c-a d)^{2/3} \log \left (c+d x^3\right )}{6 c^{11/3}}-\frac{d^2 (b c-a d)^{2/3} \log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{11/3}}+\frac{d^2 (b c-a d)^{2/3} \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} c^{11/3}}-\frac{\left (a+b x^3\right )^{2/3} (b c-4 a d)}{20 a c^2 x^5}-\frac{\left (a+b x^3\right )^{2/3}}{8 c x^8} \]
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Rubi [C] time = 0.970983, antiderivative size = 451, normalized size of antiderivative = 1.75, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {511, 510} \[ -\frac{-9 x^3 \left (c+d x^3\right )^2 (b c-a d) \, _3F_2\left (\frac{1}{3},2,2;1,\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-2 x^3 \left (5 c^2-6 c d x^3+9 d^2 x^6\right ) (b c-a d) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-6 b c^2 d x^6 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+21 b c^3 x^3 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-21 a c^2 d x^3 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+27 a d^3 x^9 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-27 b c d^2 x^9 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+6 a c d^2 x^6 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-6 a c^2 d x^3+5 a c^3+9 a c d^2 x^6-6 b c^2 d x^6+5 b c^3 x^3+9 b c d^2 x^9}{40 c^4 x^8 \sqrt [3]{a+b x^3}} \]
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 (c+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 (c+d x^3\right )} \, dx}{\left (1+\frac{b x^3}{a}\right )^{2/3}}\\ &=-\frac{5 a c^3+5 b c^3 x^3-6 a c^2 d x^3-6 b c^2 d x^6+9 a c d^2 x^6+9 b c d^2 x^9-2 (b c-a d) x^3 \left (5 c^2-6 c d x^3+9 d^2 x^6\right ) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+21 b c^3 x^3 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-21 a c^2 d x^3 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-6 b c^2 d x^6 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+6 a c d^2 x^6 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-27 b c d^2 x^9 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+27 a d^3 x^9 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-9 (b c-a d) x^3 \left (c+d x^3\right )^2 \, _3F_2\left (\frac{1}{3},2,2;1,\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{40 c^4 x^8 \sqrt [3]{a+b x^3}}\\ \end{align*}
Mathematica [C] time = 1.62771, size = 451, normalized size = 1.75 \[ -\frac{-9 x^3 \left (c+d x^3\right )^2 (b c-a d) \text{HypergeometricPFQ}\left (\left \{\frac{1}{3},2,2\right \},\left \{1,\frac{4}{3}\right \},\frac{x^3 (b c-a d)}{c \left (a+b x^3\right )}\right )-2 x^3 \left (5 c^2-6 c d x^3+9 d^2 x^6\right ) (b c-a d) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-6 b c^2 d x^6 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+21 b c^3 x^3 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-21 a c^2 d x^3 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+27 a d^3 x^9 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-27 b c d^2 x^9 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+6 a c d^2 x^6 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-6 a c^2 d x^3+5 a c^3+9 a c d^2 x^6-6 b c^2 d x^6+5 b c^3 x^3+9 b c d^2 x^9}{40 c^4 x^8 \sqrt [3]{a+b x^3}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{9} \left ( d{x}^{3}+c \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 (d x^{3} + c\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x^{3}\right )^{\frac{2}{3}}}{x^{9} \left (c + d x^{3}\right )}\, dx \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 (d x^{3} + c\right )} x^{9}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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