Optimal. Leaf size=136 \[ -\frac{3 b^7 x^{2/3}}{2 a^8}-\frac{3 b^5 x^{4/3}}{4 a^6}+\frac{3 b^4 x^{5/3}}{5 a^5}-\frac{b^3 x^2}{2 a^4}+\frac{3 b^2 x^{7/3}}{7 a^3}+\frac{3 b^8 \sqrt [3]{x}}{a^9}+\frac{b^6 x}{a^7}-\frac{3 b^9 \log \left (a \sqrt [3]{x}+b\right )}{a^{10}}-\frac{3 b x^{8/3}}{8 a^2}+\frac{x^3}{3 a} \]
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Rubi [A] time = 0.0892607, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {263, 266, 43} \[ -\frac{3 b^7 x^{2/3}}{2 a^8}-\frac{3 b^5 x^{4/3}}{4 a^6}+\frac{3 b^4 x^{5/3}}{5 a^5}-\frac{b^3 x^2}{2 a^4}+\frac{3 b^2 x^{7/3}}{7 a^3}+\frac{3 b^8 \sqrt [3]{x}}{a^9}+\frac{b^6 x}{a^7}-\frac{3 b^9 \log \left (a \sqrt [3]{x}+b\right )}{a^{10}}-\frac{3 b x^{8/3}}{8 a^2}+\frac{x^3}{3 a} \]
Antiderivative was successfully verified.
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Rule 263
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^2}{a+\frac{b}{\sqrt [3]{x}}} \, dx &=\int \frac{x^{7/3}}{b+a \sqrt [3]{x}} \, dx\\ &=3 \operatorname{Subst}\left (\int \frac{x^9}{b+a x} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{b^8}{a^9}-\frac{b^7 x}{a^8}+\frac{b^6 x^2}{a^7}-\frac{b^5 x^3}{a^6}+\frac{b^4 x^4}{a^5}-\frac{b^3 x^5}{a^4}+\frac{b^2 x^6}{a^3}-\frac{b x^7}{a^2}+\frac{x^8}{a}-\frac{b^9}{a^9 (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3 b^8 \sqrt [3]{x}}{a^9}-\frac{3 b^7 x^{2/3}}{2 a^8}+\frac{b^6 x}{a^7}-\frac{3 b^5 x^{4/3}}{4 a^6}+\frac{3 b^4 x^{5/3}}{5 a^5}-\frac{b^3 x^2}{2 a^4}+\frac{3 b^2 x^{7/3}}{7 a^3}-\frac{3 b x^{8/3}}{8 a^2}+\frac{x^3}{3 a}-\frac{3 b^9 \log \left (b+a \sqrt [3]{x}\right )}{a^{10}}\\ \end{align*}
Mathematica [A] time = 0.0952866, size = 146, normalized size = 1.07 \[ -\frac{3 b^7 x^{2/3}}{2 a^8}-\frac{3 b^5 x^{4/3}}{4 a^6}+\frac{3 b^4 x^{5/3}}{5 a^5}-\frac{b^3 x^2}{2 a^4}+\frac{3 b^2 x^{7/3}}{7 a^3}+\frac{3 b^8 \sqrt [3]{x}}{a^9}+\frac{b^6 x}{a^7}-\frac{3 b^9 \log \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^{10}}-\frac{b^9 \log (x)}{a^{10}}-\frac{3 b x^{8/3}}{8 a^2}+\frac{x^3}{3 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 109, normalized size = 0.8 \begin{align*} 3\,{\frac{{b}^{8}\sqrt [3]{x}}{{a}^{9}}}-{\frac{3\,{b}^{7}}{2\,{a}^{8}}{x}^{{\frac{2}{3}}}}+{\frac{{b}^{6}x}{{a}^{7}}}-{\frac{3\,{b}^{5}}{4\,{a}^{6}}{x}^{{\frac{4}{3}}}}+{\frac{3\,{b}^{4}}{5\,{a}^{5}}{x}^{{\frac{5}{3}}}}-{\frac{{b}^{3}{x}^{2}}{2\,{a}^{4}}}+{\frac{3\,{b}^{2}}{7\,{a}^{3}}{x}^{{\frac{7}{3}}}}-{\frac{3\,b}{8\,{a}^{2}}{x}^{{\frac{8}{3}}}}+{\frac{{x}^{3}}{3\,a}}-3\,{\frac{{b}^{9}\ln \left ( b+a\sqrt [3]{x} \right ) }{{a}^{10}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0163, size = 165, normalized size = 1.21 \begin{align*} -\frac{3 \, b^{9} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{a^{10}} - \frac{b^{9} \log \left (x\right )}{a^{10}} + \frac{{\left (280 \, a^{8} - \frac{315 \, a^{7} b}{x^{\frac{1}{3}}} + \frac{360 \, a^{6} b^{2}}{x^{\frac{2}{3}}} - \frac{420 \, a^{5} b^{3}}{x} + \frac{504 \, a^{4} b^{4}}{x^{\frac{4}{3}}} - \frac{630 \, a^{3} b^{5}}{x^{\frac{5}{3}}} + \frac{840 \, a^{2} b^{6}}{x^{2}} - \frac{1260 \, a b^{7}}{x^{\frac{7}{3}}} + \frac{2520 \, b^{8}}{x^{\frac{8}{3}}}\right )} x^{3}}{840 \, a^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4833, size = 267, normalized size = 1.96 \begin{align*} \frac{280 \, a^{9} x^{3} - 420 \, a^{6} b^{3} x^{2} + 840 \, a^{3} b^{6} x - 2520 \, b^{9} \log \left (a x^{\frac{1}{3}} + b\right ) - 63 \,{\left (5 \, a^{8} b x^{2} - 8 \, a^{5} b^{4} x + 20 \, a^{2} b^{7}\right )} x^{\frac{2}{3}} + 90 \,{\left (4 \, a^{7} b^{2} x^{2} - 7 \, a^{4} b^{5} x + 28 \, a b^{8}\right )} x^{\frac{1}{3}}}{840 \, a^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.4587, size = 143, normalized size = 1.05 \begin{align*} \begin{cases} \frac{x^{3}}{3 a} - \frac{3 b x^{\frac{8}{3}}}{8 a^{2}} + \frac{3 b^{2} x^{\frac{7}{3}}}{7 a^{3}} - \frac{b^{3} x^{2}}{2 a^{4}} + \frac{3 b^{4} x^{\frac{5}{3}}}{5 a^{5}} - \frac{3 b^{5} x^{\frac{4}{3}}}{4 a^{6}} + \frac{b^{6} x}{a^{7}} - \frac{3 b^{7} x^{\frac{2}{3}}}{2 a^{8}} + \frac{3 b^{8} \sqrt [3]{x}}{a^{9}} - \frac{3 b^{9} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{a^{10}} & \text{for}\: a \neq 0 \\\frac{3 x^{\frac{10}{3}}}{10 b} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20175, size = 150, normalized size = 1.1 \begin{align*} -\frac{3 \, b^{9} \log \left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{a^{10}} + \frac{280 \, a^{8} x^{3} - 315 \, a^{7} b x^{\frac{8}{3}} + 360 \, a^{6} b^{2} x^{\frac{7}{3}} - 420 \, a^{5} b^{3} x^{2} + 504 \, a^{4} b^{4} x^{\frac{5}{3}} - 630 \, a^{3} b^{5} x^{\frac{4}{3}} + 840 \, a^{2} b^{6} x - 1260 \, a b^{7} x^{\frac{2}{3}} + 2520 \, b^{8} x^{\frac{1}{3}}}{840 \, a^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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