Optimal. Leaf size=77 \[ \frac{15}{4} a^3 b^2 x^{8/3}+\frac{10}{3} a^2 b^3 x^3+\frac{15}{7} a^4 b x^{7/3}+\frac{a^5 x^2}{2}+\frac{3}{2} a b^4 x^{10/3}+\frac{3}{11} b^5 x^{11/3} \]
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Rubi [A] time = 0.0408262, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{15}{4} a^3 b^2 x^{8/3}+\frac{10}{3} a^2 b^3 x^3+\frac{15}{7} a^4 b x^{7/3}+\frac{a^5 x^2}{2}+\frac{3}{2} a b^4 x^{10/3}+\frac{3}{11} b^5 x^{11/3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \left (a+b \sqrt [3]{x}\right )^5 x \, dx &=3 \operatorname{Subst}\left (\int x^5 (a+b x)^5 \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (a^5 x^5+5 a^4 b x^6+10 a^3 b^2 x^7+10 a^2 b^3 x^8+5 a b^4 x^9+b^5 x^{10}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{a^5 x^2}{2}+\frac{15}{7} a^4 b x^{7/3}+\frac{15}{4} a^3 b^2 x^{8/3}+\frac{10}{3} a^2 b^3 x^3+\frac{3}{2} a b^4 x^{10/3}+\frac{3}{11} b^5 x^{11/3}\\ \end{align*}
Mathematica [A] time = 0.0249687, size = 77, normalized size = 1. \[ \frac{15}{4} a^3 b^2 x^{8/3}+\frac{10}{3} a^2 b^3 x^3+\frac{15}{7} a^4 b x^{7/3}+\frac{a^5 x^2}{2}+\frac{3}{2} a b^4 x^{10/3}+\frac{3}{11} b^5 x^{11/3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 58, normalized size = 0.8 \begin{align*}{\frac{{a}^{5}{x}^{2}}{2}}+{\frac{15\,{a}^{4}b}{7}{x}^{{\frac{7}{3}}}}+{\frac{15\,{a}^{3}{b}^{2}}{4}{x}^{{\frac{8}{3}}}}+{\frac{10\,{a}^{2}{b}^{3}{x}^{3}}{3}}+{\frac{3\,a{b}^{4}}{2}{x}^{{\frac{10}{3}}}}+{\frac{3\,{b}^{5}}{11}{x}^{{\frac{11}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.965591, size = 132, normalized size = 1.71 \begin{align*} \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{11}}{11 \, b^{6}} - \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{10} a}{2 \, b^{6}} + \frac{10 \,{\left (b x^{\frac{1}{3}} + a\right )}^{9} a^{2}}{3 \, b^{6}} - \frac{15 \,{\left (b x^{\frac{1}{3}} + a\right )}^{8} a^{3}}{4 \, b^{6}} + \frac{15 \,{\left (b x^{\frac{1}{3}} + a\right )}^{7} a^{4}}{7 \, b^{6}} - \frac{{\left (b x^{\frac{1}{3}} + a\right )}^{6} a^{5}}{2 \, b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43596, size = 162, normalized size = 2.1 \begin{align*} \frac{10}{3} \, a^{2} b^{3} x^{3} + \frac{1}{2} \, a^{5} x^{2} + \frac{3}{44} \,{\left (4 \, b^{5} x^{3} + 55 \, a^{3} b^{2} x^{2}\right )} x^{\frac{2}{3}} + \frac{3}{14} \,{\left (7 \, a b^{4} x^{3} + 10 \, a^{4} b x^{2}\right )} x^{\frac{1}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.76514, size = 75, normalized size = 0.97 \begin{align*} \frac{a^{5} x^{2}}{2} + \frac{15 a^{4} b x^{\frac{7}{3}}}{7} + \frac{15 a^{3} b^{2} x^{\frac{8}{3}}}{4} + \frac{10 a^{2} b^{3} x^{3}}{3} + \frac{3 a b^{4} x^{\frac{10}{3}}}{2} + \frac{3 b^{5} x^{\frac{11}{3}}}{11} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16508, size = 77, normalized size = 1. \begin{align*} \frac{3}{11} \, b^{5} x^{\frac{11}{3}} + \frac{3}{2} \, a b^{4} x^{\frac{10}{3}} + \frac{10}{3} \, a^{2} b^{3} x^{3} + \frac{15}{4} \, a^{3} b^{2} x^{\frac{8}{3}} + \frac{15}{7} \, a^{4} b x^{\frac{7}{3}} + \frac{1}{2} \, a^{5} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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