Optimal. Leaf size=72 \[ \frac{a^2 \left (a+b x^3\right )^{10}}{10 b^4}-\frac{a^3 \left (a+b x^3\right )^9}{27 b^4}+\frac{\left (a+b x^3\right )^{12}}{36 b^4}-\frac{a \left (a+b x^3\right )^{11}}{11 b^4} \]
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Rubi [A] time = 0.109987, antiderivative size = 72, 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} \[ \frac{a^2 \left (a+b x^3\right )^{10}}{10 b^4}-\frac{a^3 \left (a+b x^3\right )^9}{27 b^4}+\frac{\left (a+b x^3\right )^{12}}{36 b^4}-\frac{a \left (a+b x^3\right )^{11}}{11 b^4} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{11} \left (a+b x^3\right )^8 \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x^3 (a+b x)^8 \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{a^3 (a+b x)^8}{b^3}+\frac{3 a^2 (a+b x)^9}{b^3}-\frac{3 a (a+b x)^{10}}{b^3}+\frac{(a+b x)^{11}}{b^3}\right ) \, dx,x,x^3\right )\\ &=-\frac{a^3 \left (a+b x^3\right )^9}{27 b^4}+\frac{a^2 \left (a+b x^3\right )^{10}}{10 b^4}-\frac{a \left (a+b x^3\right )^{11}}{11 b^4}+\frac{\left (a+b x^3\right )^{12}}{36 b^4}\\ \end{align*}
Mathematica [A] time = 0.0028274, size = 108, normalized size = 1.5 \[ \frac{14}{15} a^2 b^6 x^{30}+\frac{56}{27} a^3 b^5 x^{27}+\frac{35}{12} a^4 b^4 x^{24}+\frac{8}{3} a^5 b^3 x^{21}+\frac{14}{9} a^6 b^2 x^{18}+\frac{8}{15} a^7 b x^{15}+\frac{a^8 x^{12}}{12}+\frac{8}{33} a b^7 x^{33}+\frac{b^8 x^{36}}{36} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 91, normalized size = 1.3 \begin{align*}{\frac{{b}^{8}{x}^{36}}{36}}+{\frac{8\,a{b}^{7}{x}^{33}}{33}}+{\frac{14\,{b}^{6}{a}^{2}{x}^{30}}{15}}+{\frac{56\,{a}^{3}{b}^{5}{x}^{27}}{27}}+{\frac{35\,{a}^{4}{b}^{4}{x}^{24}}{12}}+{\frac{8\,{a}^{5}{b}^{3}{x}^{21}}{3}}+{\frac{14\,{a}^{6}{b}^{2}{x}^{18}}{9}}+{\frac{8\,{a}^{7}b{x}^{15}}{15}}+{\frac{{a}^{8}{x}^{12}}{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.962593, size = 122, normalized size = 1.69 \begin{align*} \frac{1}{36} \, b^{8} x^{36} + \frac{8}{33} \, a b^{7} x^{33} + \frac{14}{15} \, a^{2} b^{6} x^{30} + \frac{56}{27} \, a^{3} b^{5} x^{27} + \frac{35}{12} \, a^{4} b^{4} x^{24} + \frac{8}{3} \, a^{5} b^{3} x^{21} + \frac{14}{9} \, a^{6} b^{2} x^{18} + \frac{8}{15} \, a^{7} b x^{15} + \frac{1}{12} \, a^{8} x^{12} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43355, size = 228, normalized size = 3.17 \begin{align*} \frac{1}{36} x^{36} b^{8} + \frac{8}{33} x^{33} b^{7} a + \frac{14}{15} x^{30} b^{6} a^{2} + \frac{56}{27} x^{27} b^{5} a^{3} + \frac{35}{12} x^{24} b^{4} a^{4} + \frac{8}{3} x^{21} b^{3} a^{5} + \frac{14}{9} x^{18} b^{2} a^{6} + \frac{8}{15} x^{15} b a^{7} + \frac{1}{12} x^{12} a^{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.113184, size = 107, normalized size = 1.49 \begin{align*} \frac{a^{8} x^{12}}{12} + \frac{8 a^{7} b x^{15}}{15} + \frac{14 a^{6} b^{2} x^{18}}{9} + \frac{8 a^{5} b^{3} x^{21}}{3} + \frac{35 a^{4} b^{4} x^{24}}{12} + \frac{56 a^{3} b^{5} x^{27}}{27} + \frac{14 a^{2} b^{6} x^{30}}{15} + \frac{8 a b^{7} x^{33}}{33} + \frac{b^{8} x^{36}}{36} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09656, size = 122, normalized size = 1.69 \begin{align*} \frac{1}{36} \, b^{8} x^{36} + \frac{8}{33} \, a b^{7} x^{33} + \frac{14}{15} \, a^{2} b^{6} x^{30} + \frac{56}{27} \, a^{3} b^{5} x^{27} + \frac{35}{12} \, a^{4} b^{4} x^{24} + \frac{8}{3} \, a^{5} b^{3} x^{21} + \frac{14}{9} \, a^{6} b^{2} x^{18} + \frac{8}{15} \, a^{7} b x^{15} + \frac{1}{12} \, a^{8} x^{12} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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