Optimal. Leaf size=53 \[ \frac{a^2 \left (a+b x^3\right )^9}{27 b^3}+\frac{\left (a+b x^3\right )^{11}}{33 b^3}-\frac{a \left (a+b x^3\right )^{10}}{15 b^3} \]
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Rubi [A] time = 0.0794962, antiderivative size = 53, 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} \[ \frac{a^2 \left (a+b x^3\right )^9}{27 b^3}+\frac{\left (a+b x^3\right )^{11}}{33 b^3}-\frac{a \left (a+b x^3\right )^{10}}{15 b^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^8 \left (a+b x^3\right )^8 \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x^2 (a+b x)^8 \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{a^2 (a+b x)^8}{b^2}-\frac{2 a (a+b x)^9}{b^2}+\frac{(a+b x)^{10}}{b^2}\right ) \, dx,x,x^3\right )\\ &=\frac{a^2 \left (a+b x^3\right )^9}{27 b^3}-\frac{a \left (a+b x^3\right )^{10}}{15 b^3}+\frac{\left (a+b x^3\right )^{11}}{33 b^3}\\ \end{align*}
Mathematica [B] time = 0.0029469, size = 108, normalized size = 2.04 \[ \frac{28}{27} a^2 b^6 x^{27}+\frac{7}{3} a^3 b^5 x^{24}+\frac{10}{3} a^4 b^4 x^{21}+\frac{28}{9} a^5 b^3 x^{18}+\frac{28}{15} a^6 b^2 x^{15}+\frac{2}{3} a^7 b x^{12}+\frac{a^8 x^9}{9}+\frac{4}{15} a b^7 x^{30}+\frac{b^8 x^{33}}{33} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 91, normalized size = 1.7 \begin{align*}{\frac{{b}^{8}{x}^{33}}{33}}+{\frac{4\,a{b}^{7}{x}^{30}}{15}}+{\frac{28\,{b}^{6}{a}^{2}{x}^{27}}{27}}+{\frac{7\,{a}^{3}{b}^{5}{x}^{24}}{3}}+{\frac{10\,{a}^{4}{b}^{4}{x}^{21}}{3}}+{\frac{28\,{a}^{5}{b}^{3}{x}^{18}}{9}}+{\frac{28\,{a}^{6}{b}^{2}{x}^{15}}{15}}+{\frac{2\,{a}^{7}b{x}^{12}}{3}}+{\frac{{a}^{8}{x}^{9}}{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.949077, size = 122, normalized size = 2.3 \begin{align*} \frac{1}{33} \, b^{8} x^{33} + \frac{4}{15} \, a b^{7} x^{30} + \frac{28}{27} \, a^{2} b^{6} x^{27} + \frac{7}{3} \, a^{3} b^{5} x^{24} + \frac{10}{3} \, a^{4} b^{4} x^{21} + \frac{28}{9} \, a^{5} b^{3} x^{18} + \frac{28}{15} \, a^{6} b^{2} x^{15} + \frac{2}{3} \, a^{7} b x^{12} + \frac{1}{9} \, a^{8} x^{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44819, size = 223, normalized size = 4.21 \begin{align*} \frac{1}{33} x^{33} b^{8} + \frac{4}{15} x^{30} b^{7} a + \frac{28}{27} x^{27} b^{6} a^{2} + \frac{7}{3} x^{24} b^{5} a^{3} + \frac{10}{3} x^{21} b^{4} a^{4} + \frac{28}{9} x^{18} b^{3} a^{5} + \frac{28}{15} x^{15} b^{2} a^{6} + \frac{2}{3} x^{12} b a^{7} + \frac{1}{9} x^{9} a^{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.087228, size = 107, normalized size = 2.02 \begin{align*} \frac{a^{8} x^{9}}{9} + \frac{2 a^{7} b x^{12}}{3} + \frac{28 a^{6} b^{2} x^{15}}{15} + \frac{28 a^{5} b^{3} x^{18}}{9} + \frac{10 a^{4} b^{4} x^{21}}{3} + \frac{7 a^{3} b^{5} x^{24}}{3} + \frac{28 a^{2} b^{6} x^{27}}{27} + \frac{4 a b^{7} x^{30}}{15} + \frac{b^{8} x^{33}}{33} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15574, size = 122, normalized size = 2.3 \begin{align*} \frac{1}{33} \, b^{8} x^{33} + \frac{4}{15} \, a b^{7} x^{30} + \frac{28}{27} \, a^{2} b^{6} x^{27} + \frac{7}{3} \, a^{3} b^{5} x^{24} + \frac{10}{3} \, a^{4} b^{4} x^{21} + \frac{28}{9} \, a^{5} b^{3} x^{18} + \frac{28}{15} \, a^{6} b^{2} x^{15} + \frac{2}{3} \, a^{7} b x^{12} + \frac{1}{9} \, a^{8} x^{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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