Optimal. Leaf size=72 \[ \frac{a^2 \left (a+b x^3\right )^7}{7 b^4}-\frac{a^3 \left (a+b x^3\right )^6}{18 b^4}+\frac{\left (a+b x^3\right )^9}{27 b^4}-\frac{a \left (a+b x^3\right )^8}{8 b^4} \]
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Rubi [A] time = 0.0908243, 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} \[ \frac{a^2 \left (a+b x^3\right )^7}{7 b^4}-\frac{a^3 \left (a+b x^3\right )^6}{18 b^4}+\frac{\left (a+b x^3\right )^9}{27 b^4}-\frac{a \left (a+b x^3\right )^8}{8 b^4} \]
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 )^5 \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x^3 (a+b x)^5 \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{a^3 (a+b x)^5}{b^3}+\frac{3 a^2 (a+b x)^6}{b^3}-\frac{3 a (a+b x)^7}{b^3}+\frac{(a+b x)^8}{b^3}\right ) \, dx,x,x^3\right )\\ &=-\frac{a^3 \left (a+b x^3\right )^6}{18 b^4}+\frac{a^2 \left (a+b x^3\right )^7}{7 b^4}-\frac{a \left (a+b x^3\right )^8}{8 b^4}+\frac{\left (a+b x^3\right )^9}{27 b^4}\\ \end{align*}
Mathematica [A] time = 0.0022456, size = 69, normalized size = 0.96 \[ \frac{10}{21} a^2 b^3 x^{21}+\frac{5}{9} a^3 b^2 x^{18}+\frac{1}{3} a^4 b x^{15}+\frac{a^5 x^{12}}{12}+\frac{5}{24} a b^4 x^{24}+\frac{b^5 x^{27}}{27} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 58, normalized size = 0.8 \begin{align*}{\frac{{b}^{5}{x}^{27}}{27}}+{\frac{5\,a{b}^{4}{x}^{24}}{24}}+{\frac{10\,{a}^{2}{b}^{3}{x}^{21}}{21}}+{\frac{5\,{a}^{3}{b}^{2}{x}^{18}}{9}}+{\frac{{a}^{4}b{x}^{15}}{3}}+{\frac{{a}^{5}{x}^{12}}{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.981493, size = 77, normalized size = 1.07 \begin{align*} \frac{1}{27} \, b^{5} x^{27} + \frac{5}{24} \, a b^{4} x^{24} + \frac{10}{21} \, a^{2} b^{3} x^{21} + \frac{5}{9} \, a^{3} b^{2} x^{18} + \frac{1}{3} \, a^{4} b x^{15} + \frac{1}{12} \, a^{5} x^{12} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47038, size = 143, normalized size = 1.99 \begin{align*} \frac{1}{27} x^{27} b^{5} + \frac{5}{24} x^{24} b^{4} a + \frac{10}{21} x^{21} b^{3} a^{2} + \frac{5}{9} x^{18} b^{2} a^{3} + \frac{1}{3} x^{15} b a^{4} + \frac{1}{12} x^{12} a^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.094567, size = 65, normalized size = 0.9 \begin{align*} \frac{a^{5} x^{12}}{12} + \frac{a^{4} b x^{15}}{3} + \frac{5 a^{3} b^{2} x^{18}}{9} + \frac{10 a^{2} b^{3} x^{21}}{21} + \frac{5 a b^{4} x^{24}}{24} + \frac{b^{5} x^{27}}{27} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10086, size = 77, normalized size = 1.07 \begin{align*} \frac{1}{27} \, b^{5} x^{27} + \frac{5}{24} \, a b^{4} x^{24} + \frac{10}{21} \, a^{2} b^{3} x^{21} + \frac{5}{9} \, a^{3} b^{2} x^{18} + \frac{1}{3} \, a^{4} b x^{15} + \frac{1}{12} \, a^{5} x^{12} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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