Optimal. Leaf size=209 \[ \frac {b^{8/3} \log \left (a d-b d x^3\right )}{3 \sqrt [3]{2} a^3 d}-\frac {b^{8/3} \log \left (\sqrt [3]{2} \sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} a^3 d}+\frac {2^{2/3} b^{8/3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} a^3 d}-\frac {5 b^2 \left (a+b x^3\right )^{2/3}}{8 a^3 d x^2}-\frac {b \left (a+b x^3\right )^{2/3}}{4 a^2 d x^5}-\frac {\left (a+b x^3\right )^{2/3}}{8 a d x^8} \]
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Rubi [C] time = 10.68, 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} \begin {gather*} -\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}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 510
Rule 511
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*}
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Mathematica [A] time = 5.27, size = 179, normalized size = 0.86 \begin {gather*} \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} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.55, size = 227, normalized size = 1.09 \begin {gather*} -\frac {2^{2/3} b^{8/3} \log \left (2^{2/3} \sqrt [3]{a+b x^3}-2 \sqrt [3]{b} x\right )}{3 a^3 d}+\frac {2^{2/3} b^{8/3} \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^3 d}+\frac {b^{8/3} \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 \sqrt [3]{2} a^3 d}+\frac {\left (a+b x^3\right )^{2/3} \left (-a^2-2 a b x^3-5 b^2 x^6\right )}{8 a^3 d x^8} \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 {2}{3}}}{{\left (b d x^{3} - a d\right )} x^{9}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.64, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{\left (-b d \,x^{3}+a d \right ) x^{9}}\, 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 {2}{3}}}{{\left (b d x^{3} - a d\right )} x^{9}}\,{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 )}^{2/3}}{x^9\,\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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