Optimal. Leaf size=210 \[ -\frac {\sqrt [3]{a+b x^3}}{7 a d x^7}-\frac {2 b \sqrt [3]{a+b x^3}}{7 a^2 d x^4}-\frac {8 b^2 \sqrt [3]{a+b x^3}}{7 a^3 d x}-\frac {\sqrt [3]{2} b^{7/3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} a^3 d}+\frac {b^{7/3} \log \left (a d-b d x^3\right )}{3\ 2^{2/3} a^3 d}-\frac {b^{7/3} \log \left (\sqrt [3]{2} \sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2^{2/3} a^3 d} \]
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Rubi [A]
time = 0.17, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {486, 597, 12,
503} \begin {gather*} -\frac {\sqrt [3]{2} b^{7/3} \text {ArcTan}\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^{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 {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} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 486
Rule 503
Rule 597
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 [A]
time = 0.43, size = 206, normalized size = 0.98 \begin {gather*} -\frac {\frac {6 \sqrt [3]{a+b x^3} \left (a^2+2 a b x^3+8 b^2 x^6\right )}{x^7}+14 \sqrt [3]{2} \sqrt {3} b^{7/3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2^{2/3} \sqrt [3]{a+b x^3}}\right )+14 \sqrt [3]{2} b^{7/3} \log \left (-2 \sqrt [3]{b} x+2^{2/3} \sqrt [3]{a+b x^3}\right )-7 \sqrt [3]{2} b^{7/3} \log \left (2 b^{2/3} x^2+2^{2/3} \sqrt [3]{b} x \sqrt [3]{a+b x^3}+\sqrt [3]{2} \left (a+b x^3\right )^{2/3}\right )}{42 a^3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x^{8} \left (-b d \,x^{3}+a d \right )}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {\int \frac {\sqrt [3]{a + b x^{3}}}{- a x^{8} + b x^{11}}\, dx}{d} \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*} \text {could not integrate} \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^8\,\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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