Optimal. Leaf size=53 \[ \frac{a^2 \left (a+b x^3\right )^6}{18 b^3}+\frac{\left (a+b x^3\right )^8}{24 b^3}-\frac{2 a \left (a+b x^3\right )^7}{21 b^3} \]
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Rubi [A] time = 0.0637328, 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} \[ \frac{a^2 \left (a+b x^3\right )^6}{18 b^3}+\frac{\left (a+b x^3\right )^8}{24 b^3}-\frac{2 a \left (a+b x^3\right )^7}{21 b^3} \]
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 )^5 \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x^2 (a+b x)^5 \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{a^2 (a+b x)^5}{b^2}-\frac{2 a (a+b x)^6}{b^2}+\frac{(a+b x)^7}{b^2}\right ) \, dx,x,x^3\right )\\ &=\frac{a^2 \left (a+b x^3\right )^6}{18 b^3}-\frac{2 a \left (a+b x^3\right )^7}{21 b^3}+\frac{\left (a+b x^3\right )^8}{24 b^3}\\ \end{align*}
Mathematica [A] time = 0.0021005, size = 69, normalized size = 1.3 \[ \frac{5}{9} a^2 b^3 x^{18}+\frac{2}{3} a^3 b^2 x^{15}+\frac{5}{12} a^4 b x^{12}+\frac{a^5 x^9}{9}+\frac{5}{21} a b^4 x^{21}+\frac{b^5 x^{24}}{24} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 58, normalized size = 1.1 \begin{align*}{\frac{{b}^{5}{x}^{24}}{24}}+{\frac{5\,a{b}^{4}{x}^{21}}{21}}+{\frac{5\,{a}^{2}{b}^{3}{x}^{18}}{9}}+{\frac{2\,{a}^{3}{b}^{2}{x}^{15}}{3}}+{\frac{5\,{a}^{4}b{x}^{12}}{12}}+{\frac{{a}^{5}{x}^{9}}{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.971419, size = 77, normalized size = 1.45 \begin{align*} \frac{1}{24} \, b^{5} x^{24} + \frac{5}{21} \, a b^{4} x^{21} + \frac{5}{9} \, a^{2} b^{3} x^{18} + \frac{2}{3} \, a^{3} b^{2} x^{15} + \frac{5}{12} \, a^{4} b x^{12} + \frac{1}{9} \, a^{5} x^{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45869, size = 139, normalized size = 2.62 \begin{align*} \frac{1}{24} x^{24} b^{5} + \frac{5}{21} x^{21} b^{4} a + \frac{5}{9} x^{18} b^{3} a^{2} + \frac{2}{3} x^{15} b^{2} a^{3} + \frac{5}{12} x^{12} b a^{4} + \frac{1}{9} x^{9} a^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.102751, size = 66, normalized size = 1.25 \begin{align*} \frac{a^{5} x^{9}}{9} + \frac{5 a^{4} b x^{12}}{12} + \frac{2 a^{3} b^{2} x^{15}}{3} + \frac{5 a^{2} b^{3} x^{18}}{9} + \frac{5 a b^{4} x^{21}}{21} + \frac{b^{5} x^{24}}{24} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12573, size = 77, normalized size = 1.45 \begin{align*} \frac{1}{24} \, b^{5} x^{24} + \frac{5}{21} \, a b^{4} x^{21} + \frac{5}{9} \, a^{2} b^{3} x^{18} + \frac{2}{3} \, a^{3} b^{2} x^{15} + \frac{5}{12} \, a^{4} b x^{12} + \frac{1}{9} \, a^{5} x^{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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