Optimal. Leaf size=210 \[ \frac {b^{7/3} \log \left (a d-b d x^3\right )}{3\ 2^{2/3} a^3 d}-\frac {b^{7/3} \log \left (\sqrt [3]{2} \sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2^{2/3} a^3 d}-\frac {\sqrt [3]{2} b^{7/3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} a^3 d}-\frac {8 b^2 \sqrt [3]{a+b x^3}}{7 a^3 d x}-\frac {2 b \sqrt [3]{a+b x^3}}{7 a^2 d x^4}-\frac {\sqrt [3]{a+b x^3}}{7 a d x^7} \]
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Rubi [C] time = 19.78, antiderivative size = 244, normalized size of antiderivative = 1.16, 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} \[ -\frac {-9 b x^3 \left (a-b x^3\right )^2 \, _3F_2\left (\frac {2}{3},2,2;1,\frac {5}{3};\frac {2 b x^3}{b x^3+a}\right )-2 b x^3 \left (2 a^2+3 a b x^3+9 b^2 x^6\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {2 b x^3}{b x^3+a}\right )+15 a^2 b x^3 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {2 b x^3}{b x^3+a}\right )+10 a^2 b x^3+4 a^3-27 b^3 x^9 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {2 b x^3}{b x^3+a}\right )+12 a b^2 x^6 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {2 b x^3}{b x^3+a}\right )+24 a b^2 x^6+18 b^3 x^9}{28 a^3 d x^7 \left (a+b x^3\right )^{2/3}} \]
Warning: Unable to verify antiderivative.
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Rule 510
Rule 511
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{a+b x^3}}{x^8 \left (a d-b d x^3\right )} \, dx &=\frac {\sqrt [3]{a+b x^3} \int \frac {\sqrt [3]{1+\frac {b x^3}{a}}}{x^8 \left (a d-b d x^3\right )} \, dx}{\sqrt [3]{1+\frac {b x^3}{a}}}\\ &=-\frac {4 a^3+10 a^2 b x^3+24 a b^2 x^6+18 b^3 x^9-2 b x^3 \left (2 a^2+3 a b x^3+9 b^2 x^6\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {2 b x^3}{a+b x^3}\right )+15 a^2 b x^3 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {2 b x^3}{a+b x^3}\right )+12 a b^2 x^6 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {2 b x^3}{a+b x^3}\right )-27 b^3 x^9 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {2 b x^3}{a+b x^3}\right )-9 b x^3 \left (a-b x^3\right )^2 \, _3F_2\left (\frac {2}{3},2,2;1,\frac {5}{3};\frac {2 b x^3}{a+b x^3}\right )}{28 a^3 d x^7 \left (a+b x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [C] time = 5.12, size = 135, normalized size = 0.64 \[ \frac {7 b^3 x^9 \left (\frac {b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac {2}{3},\frac {2}{3};\frac {5}{3};-\frac {2 b x^3}{a-b x^3}\right )-\left (1-\frac {b x^3}{a}\right )^{2/3} \left (a^3+3 a^2 b x^3+10 a b^2 x^6+8 b^3 x^9\right )}{7 a^3 d x^7 \left (a+b x^3\right )^{2/3} \left (1-\frac {b x^3}{a}\right )^{2/3}} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int -\frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{{\left (b d x^{3} - a d\right )} x^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.61, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{\left (-b d \,x^{3}+a d \right ) x^{8}}\, dx \]
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 \[ -\int \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{{\left (b d x^{3} - a d\right )} x^{8}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^3+a\right )}^{1/3}}{x^8\,\left (a\,d-b\,d\,x^3\right )} \,d x \]
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 \[ - \frac {\int \frac {\sqrt [3]{a + b x^{3}}}{- a x^{8} + b x^{11}}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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