Optimal. Leaf size=233 \[ \frac{a \log \left (a d-b d x^3\right )}{3\ 2^{2/3} b^{5/3} d}+\frac{2 a \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{3 b^{5/3} d}-\frac{a \log \left (\sqrt [3]{2} \sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2^{2/3} b^{5/3} d}+\frac{4 a \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} b^{5/3} d}-\frac{\sqrt [3]{2} a \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} b^{5/3} d}-\frac{x^2 \sqrt [3]{a+b x^3}}{3 b d} \]
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Rubi [C] time = 0.0625171, antiderivative size = 66, normalized size of antiderivative = 0.28, 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} \[ \frac{x^5 \sqrt [3]{a+b x^3} F_1\left (\frac{5}{3};-\frac{1}{3},1;\frac{8}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )}{5 a d \sqrt [3]{\frac{b x^3}{a}+1}} \]
Warning: Unable to verify antiderivative.
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Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{x^4 \sqrt [3]{a+b x^3}}{a d-b d x^3} \, dx &=\frac{\sqrt [3]{a+b x^3} \int \frac{x^4 \sqrt [3]{1+\frac{b x^3}{a}}}{a d-b d x^3} \, dx}{\sqrt [3]{1+\frac{b x^3}{a}}}\\ &=\frac{x^5 \sqrt [3]{a+b x^3} F_1\left (\frac{5}{3};-\frac{1}{3},1;\frac{8}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )}{5 a d \sqrt [3]{1+\frac{b x^3}{a}}}\\ \end{align*}
Mathematica [C] time = 0.149134, size = 160, normalized size = 0.69 \[ \frac{4 b x^5 \left (1-\frac{b^2 x^6}{a^2}\right )^{2/3} F_1\left (\frac{5}{3};\frac{2}{3},1;\frac{8}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )-5 x^2 \left (\left (a+b x^3\right ) \left (1-\frac{b x^3}{a}\right )^{2/3}-a \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{2 b x^3}{a-b x^3}\right )\right )}{15 b d \left (a+b x^3\right )^{2/3} \left (1-\frac{b x^3}{a}\right )^{2/3}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.049, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{4}}{-bd{x}^{3}+ad}\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}} x^{4}}{b d x^{3} - a d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45501, size = 933, normalized size = 4. \begin{align*} -\frac{6 \, \sqrt{3} 2^{\frac{1}{3}} a b^{2} \left (-\frac{1}{b^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3} 2^{\frac{2}{3}}{\left (b x^{3} + a\right )}^{\frac{1}{3}} b \left (-\frac{1}{b^{2}}\right )^{\frac{2}{3}} + \sqrt{3} x}{3 \, x}\right ) - 6 \cdot 2^{\frac{1}{3}} a b^{2} \left (-\frac{1}{b^{2}}\right )^{\frac{1}{3}} \log \left (\frac{2^{\frac{1}{3}} b x \left (-\frac{1}{b^{2}}\right )^{\frac{1}{3}} +{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{x}\right ) + 3 \cdot 2^{\frac{1}{3}} a b^{2} \left (-\frac{1}{b^{2}}\right )^{\frac{1}{3}} \log \left (\frac{2^{\frac{2}{3}} b^{2} x^{2} \left (-\frac{1}{b^{2}}\right )^{\frac{2}{3}} - 2^{\frac{1}{3}}{\left (b x^{3} + a\right )}^{\frac{1}{3}} b x \left (-\frac{1}{b^{2}}\right )^{\frac{1}{3}} +{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{x^{2}}\right ) + 6 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} b^{2} x^{2} + 8 \, \sqrt{3} a{\left (b^{2}\right )}^{\frac{1}{6}} b \arctan \left (\frac{{\left (\sqrt{3}{\left (b^{2}\right )}^{\frac{1}{3}} b x + 2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (b^{2}\right )}^{\frac{2}{3}}\right )}{\left (b^{2}\right )}^{\frac{1}{6}}}{3 \, b^{2} x}\right ) - 8 \, a{\left (b^{2}\right )}^{\frac{2}{3}} \log \left (-\frac{{\left (b^{2}\right )}^{\frac{2}{3}} x -{\left (b x^{3} + a\right )}^{\frac{1}{3}} b}{x}\right ) + 4 \, a{\left (b^{2}\right )}^{\frac{2}{3}} \log \left (\frac{{\left (b^{2}\right )}^{\frac{1}{3}} b x^{2} +{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (b^{2}\right )}^{\frac{2}{3}} x +{\left (b x^{3} + a\right )}^{\frac{2}{3}} b}{x^{2}}\right )}{18 \, b^{3} d} \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{x^{4} \sqrt [3]{a + b x^{3}}}{- a + b x^{3}}\, 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}} x^{4}}{b d x^{3} - a d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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