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 = 0.42, 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} \begin {gather*} -\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}} \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^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*}
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Mathematica [C] time = 5.10, size = 125, normalized size = 0.68 \begin {gather*} \frac {b^2 x^2 \left (\frac {a+b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac {2}{3},\frac {2}{3};\frac {5}{3};-\frac {2 b x^3}{a \left (1-\frac {b x^3}{a}\right )}\right )}{a^2 d \left (a+b x^3\right )^{2/3} \left (1-\frac {b x^3}{a}\right )^{2/3}}-\frac {\left (\frac {5 b}{4 a^2 x}+\frac {1}{4 a x^4}\right ) \sqrt [3]{a+b x^3}}{d} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.48, size = 217, normalized size = 1.19 \begin {gather*} -\frac {\sqrt [3]{2} b^{4/3} \log \left (2^{2/3} \sqrt [3]{a+b x^3}-2 \sqrt [3]{b} x\right )}{3 a^2 d}-\frac {\sqrt [3]{2} b^{4/3} \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^2 d}+\frac {b^{4/3} \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^2 d}+\frac {\left (-a-5 b x^3\right ) \sqrt [3]{a+b x^3}}{4 a^2 d x^4} \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^{5}}\,{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^{5}}\, 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^{5}}\,{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 = -0.01 \begin {gather*} \int \frac {{\left (b\,x^3+a\right )}^{1/3}}{x^5\,\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] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {\int \frac {\sqrt [3]{a + b x^{3}}}{- a x^{5} + b x^{8}}\, dx}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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