Optimal. Leaf size=183 \[ \frac{b^{4/3} \log \left (a d-b d x^3\right )}{3\ 2^{2/3} a^2 d}-\frac{b^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2^{2/3} a^2 d}-\frac{\sqrt [3]{2} b^{4/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} a^2 d}-\frac{5 b \sqrt [3]{a+b x^3}}{4 a^2 d x}-\frac{\sqrt [3]{a+b x^3}}{4 a d x^4} \]
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Rubi [C] time = 0.422017, antiderivative size = 117, normalized size of antiderivative = 0.64, 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} \[ -\frac{a^2-b x^3 \left (a+3 b x^3\right ) \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{2 b x^3}{b x^3+a}\right )+3 b x^3 \left (a-b x^3\right ) \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{2 b x^3}{b x^3+a}\right )+4 a b x^3+3 b^2 x^6}{4 a^2 d x^4 \left (a+b x^3\right )^{2/3}} \]
Warning: Unable to verify antiderivative.
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Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{\sqrt [3]{a+b x^3}}{x^5 \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^5 \left (a d-b d x^3\right )} \, dx}{\sqrt [3]{1+\frac{b x^3}{a}}}\\ &=-\frac{a^2+4 a b x^3+3 b^2 x^6-b x^3 \left (a+3 b x^3\right ) \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{2 b x^3}{a+b x^3}\right )+3 b x^3 \left (a-b x^3\right ) \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{2 b x^3}{a+b x^3}\right )}{4 a^2 d x^4 \left (a+b x^3\right )^{2/3}}\\ \end{align*}
Mathematica [C] time = 4.85741, size = 112, normalized size = 0.61 \[ -\frac{\left (a+b x^3\right )^{4/3} \left (\left (a^2-4 a b x^3+3 b^2 x^6\right ) \, _2F_1\left (1,1;\frac{2}{3};-\frac{2 b x^3}{a-b x^3}\right )+9 b x^3 \left (a+b x^3\right ) \, _2F_1\left (2,2;\frac{5}{3};-\frac{2 b x^3}{a-b x^3}\right )\right )}{4 a^2 d x^4 \left (a-b x^3\right )^2} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{5} \left ( -bd{x}^{3}+ad \right ) }\sqrt [3]{b{x}^{3}+a}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{{\left (b d x^{3} - a d\right )} x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{\sqrt [3]{a + b x^{3}}}{- a x^{5} + b x^{8}}\, dx}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{{\left (b d x^{3} - a d\right )} x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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