Optimal. Leaf size=237 \[ \frac {b^{10/3} \log \left (a d-b d x^3\right )}{3\ 2^{2/3} a^4 d}-\frac {b^{10/3} \log \left (\sqrt [3]{2} \sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2^{2/3} a^4 d}-\frac {\sqrt [3]{2} b^{10/3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} a^4 d}-\frac {169 b^3 \sqrt [3]{a+b x^3}}{140 a^4 d x}-\frac {37 b^2 \sqrt [3]{a+b x^3}}{140 a^3 d x^4}-\frac {11 b \sqrt [3]{a+b x^3}}{70 a^2 d x^7}-\frac {\sqrt [3]{a+b x^3}}{10 a d x^{10}} \]
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Rubi [C] time = 29.21, antiderivative size = 423, normalized size of antiderivative = 1.78, 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} \begin {gather*} -\frac {-54 b x^3 \left (a-b x^3\right )^2 \left (2 a+3 b x^3\right ) \, _3F_2\left (\frac {2}{3},2,2;1,\frac {5}{3};\frac {2 b x^3}{b x^3+a}\right )+27 b x^3 \left (a-b x^3\right )^3 \, _4F_3\left (\frac {2}{3},2,2,2;1,1,\frac {5}{3};\frac {2 b x^3}{b x^3+a}\right )-36 a^2 b^2 x^6 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {2 b x^3}{b x^3+a}\right )+99 a^2 b^2 x^6 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {2 b x^3}{b x^3+a}\right )+90 a^2 b^2 x^6-28 a^3 b x^3 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {2 b x^3}{b x^3+a}\right )+117 a^3 b x^3 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {2 b x^3}{b x^3+a}\right )+64 a^3 b x^3+28 a^4-162 b^4 x^{12} \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {2 b x^3}{b x^3+a}\right )-297 b^4 x^{12} \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {2 b x^3}{b x^3+a}\right )-54 a b^3 x^9 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {2 b x^3}{b x^3+a}\right )+81 a b^3 x^9 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {2 b x^3}{b x^3+a}\right )+216 a b^3 x^9+162 b^4 x^{12}}{280 a^4 d x^{10} \left (a+b x^3\right )^{2/3}} \end {gather*}
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^{11} \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^{11} \left (a d-b d x^3\right )} \, dx}{\sqrt [3]{1+\frac {b x^3}{a}}}\\ &=-\frac {28 a^4+64 a^3 b x^3+90 a^2 b^2 x^6+216 a b^3 x^9+162 b^4 x^{12}-28 a^3 b x^3 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {2 b x^3}{a+b x^3}\right )-36 a^2 b^2 x^6 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {2 b x^3}{a+b x^3}\right )-54 a b^3 x^9 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {2 b x^3}{a+b x^3}\right )-162 b^4 x^{12} \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {2 b x^3}{a+b x^3}\right )+117 a^3 b x^3 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {2 b x^3}{a+b x^3}\right )+99 a^2 b^2 x^6 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {2 b x^3}{a+b x^3}\right )+81 a b^3 x^9 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {2 b x^3}{a+b x^3}\right )-297 b^4 x^{12} \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {2 b x^3}{a+b x^3}\right )-54 b x^3 \left (a-b x^3\right )^2 \left (2 a+3 b x^3\right ) \, _3F_2\left (\frac {2}{3},2,2;1,\frac {5}{3};\frac {2 b x^3}{a+b x^3}\right )+27 b x^3 \left (a-b x^3\right )^3 \, _4F_3\left (\frac {2}{3},2,2,2;1,1,\frac {5}{3};\frac {2 b x^3}{a+b x^3}\right )}{280 a^4 d x^{10} \left (a+b x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [C] time = 5.20, size = 130, normalized size = 0.55 \begin {gather*} \frac {-14 a^4-36 a^3 b x^3-59 a^2 b^2 x^6+\frac {140 b^4 x^{12} \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}}-206 a b^3 x^9-169 b^4 x^{12}}{140 a^4 d x^{10} \left (a+b x^3\right )^{2/3}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.57, size = 239, normalized size = 1.01 \begin {gather*} -\frac {\sqrt [3]{2} b^{10/3} \log \left (2^{2/3} \sqrt [3]{a+b x^3}-2 \sqrt [3]{b} x\right )}{3 a^4 d}-\frac {\sqrt [3]{2} b^{10/3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} x}{2^{2/3} \sqrt [3]{a+b x^3}+\sqrt [3]{b} x}\right )}{\sqrt {3} a^4 d}+\frac {b^{10/3} \log \left (2^{2/3} \sqrt [3]{b} x \sqrt [3]{a+b x^3}+\sqrt [3]{2} \left (a+b x^3\right )^{2/3}+2 b^{2/3} x^2\right )}{3\ 2^{2/3} a^4 d}+\frac {\sqrt [3]{a+b x^3} \left (-14 a^3-22 a^2 b x^3-37 a b^2 x^6-169 b^3 x^9\right )}{140 a^4 d x^{10}} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \int -\frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{{\left (b d x^{3} - a d\right )} x^{11}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.62, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{\left (-b d \,x^{3}+a d \right ) x^{11}}\, dx \end {gather*}
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 \begin {gather*} -\int \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{{\left (b d x^{3} - a d\right )} x^{11}}\,{d x} \end {gather*}
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 \begin {gather*} \int \frac {{\left (b\,x^3+a\right )}^{1/3}}{x^{11}\,\left (a\,d-b\,d\,x^3\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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