Optimal. Leaf size=182 \[ -\frac {\left (a+b x^3\right )^{2/3}}{5 a d x^5}-\frac {7 b \left (a+b x^3\right )^{2/3}}{10 a^2 d x^2}+\frac {2^{2/3} b^{5/3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} a^2 d}+\frac {b^{5/3} \log \left (a d-b d x^3\right )}{3 \sqrt [3]{2} a^2 d}-\frac {b^{5/3} \log \left (\sqrt [3]{2} \sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} a^2 d} \]
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Rubi [A]
time = 0.10, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {486, 597, 12,
384} \begin {gather*} \frac {2^{2/3} b^{5/3} \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^2 d}+\frac {b^{5/3} \log \left (a d-b d x^3\right )}{3 \sqrt [3]{2} a^2 d}-\frac {b^{5/3} \log \left (\sqrt [3]{2} \sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} a^2 d}-\frac {7 b \left (a+b x^3\right )^{2/3}}{10 a^2 d x^2}-\frac {\left (a+b x^3\right )^{2/3}}{5 a d x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 384
Rule 486
Rule 597
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^{2/3}}{x^6 \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^6 \left (a d-b d x^3\right )} \, dx}{\left (1+\frac {b x^3}{a}\right )^{2/3}}\\ &=-\frac {2 a^2+5 a b x^3+3 b^2 x^6-4 b x^3 \left (2 a+3 b x^3\right ) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {2 b x^3}{a+b x^3}\right )+12 b x^3 \left (a-b x^3\right ) \, _2F_1\left (\frac {1}{3},2;\frac {4}{3};\frac {2 b x^3}{a+b x^3}\right )}{10 a^2 d x^5 \sqrt [3]{a+b x^3}}\\ \end {align*}
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Mathematica [A]
time = 0.36, size = 216, normalized size = 1.19 \begin {gather*} \frac {\left (-2 a-7 b x^3\right ) \left (a+b x^3\right )^{2/3}}{10 a^2 d x^5}+\frac {2^{2/3} b^{5/3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2^{2/3} \sqrt [3]{a+b x^3}}\right )}{\sqrt {3} a^2 d}-\frac {2^{2/3} b^{5/3} \log \left (-2 \sqrt [3]{b} x+2^{2/3} \sqrt [3]{a+b x^3}\right )}{3 a^2 d}+\frac {b^{5/3} \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 )}{3 \sqrt [3]{2} a^2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{6} \left (-b d \,x^{3}+a d \right )}\, dx\]
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 {\left (a + b x^{3}\right )^{\frac {2}{3}}}{- a x^{6} + b x^{9}}\, 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 = -0.01 \begin {gather*} \int \frac {{\left (b\,x^3+a\right )}^{2/3}}{x^6\,\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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