Optimal. Leaf size=53 \[ \frac{a^2 \left (a+b x^2\right )^9}{18 b^3}+\frac{\left (a+b x^2\right )^{11}}{22 b^3}-\frac{a \left (a+b x^2\right )^{10}}{10 b^3} \]
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Rubi [A] time = 0.0839906, 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} \[ \frac{a^2 \left (a+b x^2\right )^9}{18 b^3}+\frac{\left (a+b x^2\right )^{11}}{22 b^3}-\frac{a \left (a+b x^2\right )^{10}}{10 b^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^5 \left (a+b x^2\right )^8 \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^2 (a+b x)^8 \, dx,x,x^2\right )\\ &=\frac{1}{2} \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^2\right )\\ &=\frac{a^2 \left (a+b x^2\right )^9}{18 b^3}-\frac{a \left (a+b x^2\right )^{10}}{10 b^3}+\frac{\left (a+b x^2\right )^{11}}{22 b^3}\\ \end{align*}
Mathematica [A] time = 0.0024927, size = 103, normalized size = 1.94 \[ \frac{14}{9} a^2 b^6 x^{18}+\frac{7}{2} a^3 b^5 x^{16}+5 a^4 b^4 x^{14}+\frac{14}{3} a^5 b^3 x^{12}+\frac{14}{5} a^6 b^2 x^{10}+a^7 b x^8+\frac{a^8 x^6}{6}+\frac{2}{5} a b^7 x^{20}+\frac{b^8 x^{22}}{22} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 90, normalized size = 1.7 \begin{align*}{\frac{{b}^{8}{x}^{22}}{22}}+{\frac{2\,a{b}^{7}{x}^{20}}{5}}+{\frac{14\,{a}^{2}{b}^{6}{x}^{18}}{9}}+{\frac{7\,{a}^{3}{b}^{5}{x}^{16}}{2}}+5\,{a}^{4}{b}^{4}{x}^{14}+{\frac{14\,{a}^{5}{b}^{3}{x}^{12}}{3}}+{\frac{14\,{a}^{6}{b}^{2}{x}^{10}}{5}}+{a}^{7}b{x}^{8}+{\frac{{a}^{8}{x}^{6}}{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.3005, size = 120, normalized size = 2.26 \begin{align*} \frac{1}{22} \, b^{8} x^{22} + \frac{2}{5} \, a b^{7} x^{20} + \frac{14}{9} \, a^{2} b^{6} x^{18} + \frac{7}{2} \, a^{3} b^{5} x^{16} + 5 \, a^{4} b^{4} x^{14} + \frac{14}{3} \, a^{5} b^{3} x^{12} + \frac{14}{5} \, a^{6} b^{2} x^{10} + a^{7} b x^{8} + \frac{1}{6} \, a^{8} x^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.41869, size = 208, normalized size = 3.92 \begin{align*} \frac{1}{22} x^{22} b^{8} + \frac{2}{5} x^{20} b^{7} a + \frac{14}{9} x^{18} b^{6} a^{2} + \frac{7}{2} x^{16} b^{5} a^{3} + 5 x^{14} b^{4} a^{4} + \frac{14}{3} x^{12} b^{3} a^{5} + \frac{14}{5} x^{10} b^{2} a^{6} + x^{8} b a^{7} + \frac{1}{6} x^{6} a^{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.083992, size = 102, normalized size = 1.92 \begin{align*} \frac{a^{8} x^{6}}{6} + a^{7} b x^{8} + \frac{14 a^{6} b^{2} x^{10}}{5} + \frac{14 a^{5} b^{3} x^{12}}{3} + 5 a^{4} b^{4} x^{14} + \frac{7 a^{3} b^{5} x^{16}}{2} + \frac{14 a^{2} b^{6} x^{18}}{9} + \frac{2 a b^{7} x^{20}}{5} + \frac{b^{8} x^{22}}{22} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.43434, size = 120, normalized size = 2.26 \begin{align*} \frac{1}{22} \, b^{8} x^{22} + \frac{2}{5} \, a b^{7} x^{20} + \frac{14}{9} \, a^{2} b^{6} x^{18} + \frac{7}{2} \, a^{3} b^{5} x^{16} + 5 \, a^{4} b^{4} x^{14} + \frac{14}{3} \, a^{5} b^{3} x^{12} + \frac{14}{5} \, a^{6} b^{2} x^{10} + a^{7} b x^{8} + \frac{1}{6} \, a^{8} x^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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