Optimal. Leaf size=75 \[ 2 a^2 b^3 x^5+\frac{15}{7} a^3 b^2 x^{14/3}+\frac{15}{13} a^4 b x^{13/3}+\frac{a^5 x^4}{4}+\frac{15}{16} a b^4 x^{16/3}+\frac{3}{17} b^5 x^{17/3} \]
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Rubi [A] time = 0.0453484, antiderivative size = 75, 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} \[ 2 a^2 b^3 x^5+\frac{15}{7} a^3 b^2 x^{14/3}+\frac{15}{13} a^4 b x^{13/3}+\frac{a^5 x^4}{4}+\frac{15}{16} a b^4 x^{16/3}+\frac{3}{17} b^5 x^{17/3} \]
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^3 \, dx &=3 \operatorname{Subst}\left (\int x^{11} (a+b x)^5 \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (a^5 x^{11}+5 a^4 b x^{12}+10 a^3 b^2 x^{13}+10 a^2 b^3 x^{14}+5 a b^4 x^{15}+b^5 x^{16}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{a^5 x^4}{4}+\frac{15}{13} a^4 b x^{13/3}+\frac{15}{7} a^3 b^2 x^{14/3}+2 a^2 b^3 x^5+\frac{15}{16} a b^4 x^{16/3}+\frac{3}{17} b^5 x^{17/3}\\ \end{align*}
Mathematica [A] time = 0.0320669, size = 75, normalized size = 1. \[ 2 a^2 b^3 x^5+\frac{15}{7} a^3 b^2 x^{14/3}+\frac{15}{13} a^4 b x^{13/3}+\frac{a^5 x^4}{4}+\frac{15}{16} a b^4 x^{16/3}+\frac{3}{17} b^5 x^{17/3} \]
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}^{4}}{4}}+{\frac{15\,{a}^{4}b}{13}{x}^{{\frac{13}{3}}}}+{\frac{15\,{a}^{3}{b}^{2}}{7}{x}^{{\frac{14}{3}}}}+2\,{a}^{2}{b}^{3}{x}^{5}+{\frac{15\,a{b}^{4}}{16}{x}^{{\frac{16}{3}}}}+{\frac{3\,{b}^{5}}{17}{x}^{{\frac{17}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.966164, size = 270, normalized size = 3.6 \begin{align*} \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{17}}{17 \, b^{12}} - \frac{33 \,{\left (b x^{\frac{1}{3}} + a\right )}^{16} a}{16 \, b^{12}} + \frac{11 \,{\left (b x^{\frac{1}{3}} + a\right )}^{15} a^{2}}{b^{12}} - \frac{495 \,{\left (b x^{\frac{1}{3}} + a\right )}^{14} a^{3}}{14 \, b^{12}} + \frac{990 \,{\left (b x^{\frac{1}{3}} + a\right )}^{13} a^{4}}{13 \, b^{12}} - \frac{231 \,{\left (b x^{\frac{1}{3}} + a\right )}^{12} a^{5}}{2 \, b^{12}} + \frac{126 \,{\left (b x^{\frac{1}{3}} + a\right )}^{11} a^{6}}{b^{12}} - \frac{99 \,{\left (b x^{\frac{1}{3}} + a\right )}^{10} a^{7}}{b^{12}} + \frac{55 \,{\left (b x^{\frac{1}{3}} + a\right )}^{9} a^{8}}{b^{12}} - \frac{165 \,{\left (b x^{\frac{1}{3}} + a\right )}^{8} a^{9}}{8 \, b^{12}} + \frac{33 \,{\left (b x^{\frac{1}{3}} + a\right )}^{7} a^{10}}{7 \, b^{12}} - \frac{{\left (b x^{\frac{1}{3}} + a\right )}^{6} a^{11}}{2 \, b^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43538, size = 163, normalized size = 2.17 \begin{align*} 2 \, a^{2} b^{3} x^{5} + \frac{1}{4} \, a^{5} x^{4} + \frac{3}{119} \,{\left (7 \, b^{5} x^{5} + 85 \, a^{3} b^{2} x^{4}\right )} x^{\frac{2}{3}} + \frac{15}{208} \,{\left (13 \, a b^{4} x^{5} + 16 \, a^{4} b x^{4}\right )} x^{\frac{1}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.58423, size = 73, normalized size = 0.97 \begin{align*} \frac{a^{5} x^{4}}{4} + \frac{15 a^{4} b x^{\frac{13}{3}}}{13} + \frac{15 a^{3} b^{2} x^{\frac{14}{3}}}{7} + 2 a^{2} b^{3} x^{5} + \frac{15 a b^{4} x^{\frac{16}{3}}}{16} + \frac{3 b^{5} x^{\frac{17}{3}}}{17} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18349, size = 77, normalized size = 1.03 \begin{align*} \frac{3}{17} \, b^{5} x^{\frac{17}{3}} + \frac{15}{16} \, a b^{4} x^{\frac{16}{3}} + 2 \, a^{2} b^{3} x^{5} + \frac{15}{7} \, a^{3} b^{2} x^{\frac{14}{3}} + \frac{15}{13} \, a^{4} b x^{\frac{13}{3}} + \frac{1}{4} \, a^{5} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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