Optimal. Leaf size=179 \[ \frac{84 a^2 \left (a+b \sqrt [3]{x}\right )^{17}}{17 b^9}-\frac{21 a^3 \left (a+b \sqrt [3]{x}\right )^{16}}{2 b^9}+\frac{14 a^4 \left (a+b \sqrt [3]{x}\right )^{15}}{b^9}-\frac{12 a^5 \left (a+b \sqrt [3]{x}\right )^{14}}{b^9}+\frac{84 a^6 \left (a+b \sqrt [3]{x}\right )^{13}}{13 b^9}-\frac{2 a^7 \left (a+b \sqrt [3]{x}\right )^{12}}{b^9}+\frac{3 a^8 \left (a+b \sqrt [3]{x}\right )^{11}}{11 b^9}+\frac{3 \left (a+b \sqrt [3]{x}\right )^{19}}{19 b^9}-\frac{4 a \left (a+b \sqrt [3]{x}\right )^{18}}{3 b^9} \]
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Rubi [A] time = 0.0919547, antiderivative size = 179, 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} \[ \frac{84 a^2 \left (a+b \sqrt [3]{x}\right )^{17}}{17 b^9}-\frac{21 a^3 \left (a+b \sqrt [3]{x}\right )^{16}}{2 b^9}+\frac{14 a^4 \left (a+b \sqrt [3]{x}\right )^{15}}{b^9}-\frac{12 a^5 \left (a+b \sqrt [3]{x}\right )^{14}}{b^9}+\frac{84 a^6 \left (a+b \sqrt [3]{x}\right )^{13}}{13 b^9}-\frac{2 a^7 \left (a+b \sqrt [3]{x}\right )^{12}}{b^9}+\frac{3 a^8 \left (a+b \sqrt [3]{x}\right )^{11}}{11 b^9}+\frac{3 \left (a+b \sqrt [3]{x}\right )^{19}}{19 b^9}-\frac{4 a \left (a+b \sqrt [3]{x}\right )^{18}}{3 b^9} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \left (a+b \sqrt [3]{x}\right )^{10} x^2 \, dx &=3 \operatorname{Subst}\left (\int x^8 (a+b x)^{10} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{a^8 (a+b x)^{10}}{b^8}-\frac{8 a^7 (a+b x)^{11}}{b^8}+\frac{28 a^6 (a+b x)^{12}}{b^8}-\frac{56 a^5 (a+b x)^{13}}{b^8}+\frac{70 a^4 (a+b x)^{14}}{b^8}-\frac{56 a^3 (a+b x)^{15}}{b^8}+\frac{28 a^2 (a+b x)^{16}}{b^8}-\frac{8 a (a+b x)^{17}}{b^8}+\frac{(a+b x)^{18}}{b^8}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3 a^8 \left (a+b \sqrt [3]{x}\right )^{11}}{11 b^9}-\frac{2 a^7 \left (a+b \sqrt [3]{x}\right )^{12}}{b^9}+\frac{84 a^6 \left (a+b \sqrt [3]{x}\right )^{13}}{13 b^9}-\frac{12 a^5 \left (a+b \sqrt [3]{x}\right )^{14}}{b^9}+\frac{14 a^4 \left (a+b \sqrt [3]{x}\right )^{15}}{b^9}-\frac{21 a^3 \left (a+b \sqrt [3]{x}\right )^{16}}{2 b^9}+\frac{84 a^2 \left (a+b \sqrt [3]{x}\right )^{17}}{17 b^9}-\frac{4 a \left (a+b \sqrt [3]{x}\right )^{18}}{3 b^9}+\frac{3 \left (a+b \sqrt [3]{x}\right )^{19}}{19 b^9}\\ \end{align*}
Mathematica [A] time = 0.051242, size = 140, normalized size = 0.78 \[ \frac{135}{11} a^8 b^2 x^{11/3}+30 a^7 b^3 x^4+\frac{630}{13} a^6 b^4 x^{13/3}+54 a^5 b^5 x^{14/3}+42 a^4 b^6 x^5+\frac{45}{2} a^3 b^7 x^{16/3}+\frac{135}{17} a^2 b^8 x^{17/3}+3 a^9 b x^{10/3}+\frac{a^{10} x^3}{3}+\frac{5}{3} a b^9 x^6+\frac{3}{19} b^{10} x^{19/3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 113, normalized size = 0.6 \begin{align*}{\frac{3\,{b}^{10}}{19}{x}^{{\frac{19}{3}}}}+{\frac{5\,a{b}^{9}{x}^{6}}{3}}+{\frac{135\,{a}^{2}{b}^{8}}{17}{x}^{{\frac{17}{3}}}}+{\frac{45\,{a}^{3}{b}^{7}}{2}{x}^{{\frac{16}{3}}}}+42\,{a}^{4}{b}^{6}{x}^{5}+54\,{a}^{5}{b}^{5}{x}^{14/3}+{\frac{630\,{a}^{6}{b}^{4}}{13}{x}^{{\frac{13}{3}}}}+30\,{a}^{7}{b}^{3}{x}^{4}+{\frac{135\,{a}^{8}{b}^{2}}{11}{x}^{{\frac{11}{3}}}}+3\,{a}^{9}b{x}^{10/3}+{\frac{{x}^{3}{a}^{10}}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.970359, size = 201, normalized size = 1.12 \begin{align*} \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{19}}{19 \, b^{9}} - \frac{4 \,{\left (b x^{\frac{1}{3}} + a\right )}^{18} a}{3 \, b^{9}} + \frac{84 \,{\left (b x^{\frac{1}{3}} + a\right )}^{17} a^{2}}{17 \, b^{9}} - \frac{21 \,{\left (b x^{\frac{1}{3}} + a\right )}^{16} a^{3}}{2 \, b^{9}} + \frac{14 \,{\left (b x^{\frac{1}{3}} + a\right )}^{15} a^{4}}{b^{9}} - \frac{12 \,{\left (b x^{\frac{1}{3}} + a\right )}^{14} a^{5}}{b^{9}} + \frac{84 \,{\left (b x^{\frac{1}{3}} + a\right )}^{13} a^{6}}{13 \, b^{9}} - \frac{2 \,{\left (b x^{\frac{1}{3}} + a\right )}^{12} a^{7}}{b^{9}} + \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{11} a^{8}}{11 \, b^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42361, size = 293, normalized size = 1.64 \begin{align*} \frac{5}{3} \, a b^{9} x^{6} + 42 \, a^{4} b^{6} x^{5} + 30 \, a^{7} b^{3} x^{4} + \frac{1}{3} \, a^{10} x^{3} + \frac{27}{187} \,{\left (55 \, a^{2} b^{8} x^{5} + 374 \, a^{5} b^{5} x^{4} + 85 \, a^{8} b^{2} x^{3}\right )} x^{\frac{2}{3}} + \frac{3}{494} \,{\left (26 \, b^{10} x^{6} + 3705 \, a^{3} b^{7} x^{5} + 7980 \, a^{6} b^{4} x^{4} + 494 \, a^{9} b x^{3}\right )} x^{\frac{1}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.9175, size = 141, normalized size = 0.79 \begin{align*} \frac{a^{10} x^{3}}{3} + 3 a^{9} b x^{\frac{10}{3}} + \frac{135 a^{8} b^{2} x^{\frac{11}{3}}}{11} + 30 a^{7} b^{3} x^{4} + \frac{630 a^{6} b^{4} x^{\frac{13}{3}}}{13} + 54 a^{5} b^{5} x^{\frac{14}{3}} + 42 a^{4} b^{6} x^{5} + \frac{45 a^{3} b^{7} x^{\frac{16}{3}}}{2} + \frac{135 a^{2} b^{8} x^{\frac{17}{3}}}{17} + \frac{5 a b^{9} x^{6}}{3} + \frac{3 b^{10} x^{\frac{19}{3}}}{19} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21116, size = 151, normalized size = 0.84 \begin{align*} \frac{3}{19} \, b^{10} x^{\frac{19}{3}} + \frac{5}{3} \, a b^{9} x^{6} + \frac{135}{17} \, a^{2} b^{8} x^{\frac{17}{3}} + \frac{45}{2} \, a^{3} b^{7} x^{\frac{16}{3}} + 42 \, a^{4} b^{6} x^{5} + 54 \, a^{5} b^{5} x^{\frac{14}{3}} + \frac{630}{13} \, a^{6} b^{4} x^{\frac{13}{3}} + 30 \, a^{7} b^{3} x^{4} + \frac{135}{11} \, a^{8} b^{2} x^{\frac{11}{3}} + 3 \, a^{9} b x^{\frac{10}{3}} + \frac{1}{3} \, a^{10} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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