Optimal. Leaf size=124 \[ \frac{3 a^6 x^{2/3}}{2 b^7}+\frac{3 a^4 x^{4/3}}{4 b^5}-\frac{3 a^3 x^{5/3}}{5 b^4}+\frac{a^2 x^2}{2 b^3}-\frac{3 a^7 \sqrt [3]{x}}{b^8}-\frac{a^5 x}{b^6}+\frac{3 a^8 \log \left (a+b \sqrt [3]{x}\right )}{b^9}-\frac{3 a x^{7/3}}{7 b^2}+\frac{3 x^{8/3}}{8 b} \]
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Rubi [A] time = 0.0704568, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{3 a^6 x^{2/3}}{2 b^7}+\frac{3 a^4 x^{4/3}}{4 b^5}-\frac{3 a^3 x^{5/3}}{5 b^4}+\frac{a^2 x^2}{2 b^3}-\frac{3 a^7 \sqrt [3]{x}}{b^8}-\frac{a^5 x}{b^6}+\frac{3 a^8 \log \left (a+b \sqrt [3]{x}\right )}{b^9}-\frac{3 a x^{7/3}}{7 b^2}+\frac{3 x^{8/3}}{8 b} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^2}{a+b \sqrt [3]{x}} \, dx &=3 \operatorname{Subst}\left (\int \frac{x^8}{a+b x} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (-\frac{a^7}{b^8}+\frac{a^6 x}{b^7}-\frac{a^5 x^2}{b^6}+\frac{a^4 x^3}{b^5}-\frac{a^3 x^4}{b^4}+\frac{a^2 x^5}{b^3}-\frac{a x^6}{b^2}+\frac{x^7}{b}+\frac{a^8}{b^8 (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{3 a^7 \sqrt [3]{x}}{b^8}+\frac{3 a^6 x^{2/3}}{2 b^7}-\frac{a^5 x}{b^6}+\frac{3 a^4 x^{4/3}}{4 b^5}-\frac{3 a^3 x^{5/3}}{5 b^4}+\frac{a^2 x^2}{2 b^3}-\frac{3 a x^{7/3}}{7 b^2}+\frac{3 x^{8/3}}{8 b}+\frac{3 a^8 \log \left (a+b \sqrt [3]{x}\right )}{b^9}\\ \end{align*}
Mathematica [A] time = 0.0741907, size = 124, normalized size = 1. \[ \frac{3 a^6 x^{2/3}}{2 b^7}+\frac{3 a^4 x^{4/3}}{4 b^5}-\frac{3 a^3 x^{5/3}}{5 b^4}+\frac{a^2 x^2}{2 b^3}-\frac{3 a^7 \sqrt [3]{x}}{b^8}-\frac{a^5 x}{b^6}+\frac{3 a^8 \log \left (a+b \sqrt [3]{x}\right )}{b^9}-\frac{3 a x^{7/3}}{7 b^2}+\frac{3 x^{8/3}}{8 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 99, normalized size = 0.8 \begin{align*} -3\,{\frac{{a}^{7}\sqrt [3]{x}}{{b}^{8}}}+{\frac{3\,{a}^{6}}{2\,{b}^{7}}{x}^{{\frac{2}{3}}}}-{\frac{x{a}^{5}}{{b}^{6}}}+{\frac{3\,{a}^{4}}{4\,{b}^{5}}{x}^{{\frac{4}{3}}}}-{\frac{3\,{a}^{3}}{5\,{b}^{4}}{x}^{{\frac{5}{3}}}}+{\frac{{a}^{2}{x}^{2}}{2\,{b}^{3}}}-{\frac{3\,a}{7\,{b}^{2}}{x}^{{\frac{7}{3}}}}+{\frac{3}{8\,b}{x}^{{\frac{8}{3}}}}+3\,{\frac{{a}^{8}\ln \left ( a+b\sqrt [3]{x} \right ) }{{b}^{9}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01248, size = 197, normalized size = 1.59 \begin{align*} \frac{3 \, a^{8} \log \left (b x^{\frac{1}{3}} + a\right )}{b^{9}} + \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{8}}{8 \, b^{9}} - \frac{24 \,{\left (b x^{\frac{1}{3}} + a\right )}^{7} a}{7 \, b^{9}} + \frac{14 \,{\left (b x^{\frac{1}{3}} + a\right )}^{6} a^{2}}{b^{9}} - \frac{168 \,{\left (b x^{\frac{1}{3}} + a\right )}^{5} a^{3}}{5 \, b^{9}} + \frac{105 \,{\left (b x^{\frac{1}{3}} + a\right )}^{4} a^{4}}{2 \, b^{9}} - \frac{56 \,{\left (b x^{\frac{1}{3}} + a\right )}^{3} a^{5}}{b^{9}} + \frac{42 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} a^{6}}{b^{9}} - \frac{24 \,{\left (b x^{\frac{1}{3}} + a\right )} a^{7}}{b^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44679, size = 240, normalized size = 1.94 \begin{align*} \frac{140 \, a^{2} b^{6} x^{2} - 280 \, a^{5} b^{3} x + 840 \, a^{8} \log \left (b x^{\frac{1}{3}} + a\right ) + 21 \,{\left (5 \, b^{8} x^{2} - 8 \, a^{3} b^{5} x + 20 \, a^{6} b^{2}\right )} x^{\frac{2}{3}} - 30 \,{\left (4 \, a b^{7} x^{2} - 7 \, a^{4} b^{4} x + 28 \, a^{7} b\right )} x^{\frac{1}{3}}}{280 \, b^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 25.1611, size = 122, normalized size = 0.98 \begin{align*} \frac{3 a^{8} \log{\left (1 + \frac{b \sqrt [3]{x}}{a} \right )}}{b^{9}} - \frac{3 a^{7} \sqrt [3]{x}}{b^{8}} + \frac{3 a^{6} x^{\frac{2}{3}}}{2 b^{7}} - \frac{a^{5} x}{b^{6}} + \frac{3 a^{4} x^{\frac{4}{3}}}{4 b^{5}} - \frac{3 a^{3} x^{\frac{5}{3}}}{5 b^{4}} + \frac{a^{2} x^{2}}{2 b^{3}} - \frac{3 a x^{\frac{7}{3}}}{7 b^{2}} + \frac{3 x^{\frac{8}{3}}}{8 b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16368, size = 135, normalized size = 1.09 \begin{align*} \frac{3 \, a^{8} \log \left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{b^{9}} + \frac{105 \, b^{7} x^{\frac{8}{3}} - 120 \, a b^{6} x^{\frac{7}{3}} + 140 \, a^{2} b^{5} x^{2} - 168 \, a^{3} b^{4} x^{\frac{5}{3}} + 210 \, a^{4} b^{3} x^{\frac{4}{3}} - 280 \, a^{5} b^{2} x + 420 \, a^{6} b x^{\frac{2}{3}} - 840 \, a^{7} x^{\frac{1}{3}}}{280 \, b^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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