Optimal. Leaf size=129 \[ \frac{15 a^2 \left (a+b x^2\right )^{13}}{26 b^7}-\frac{5 a^3 \left (a+b x^2\right )^{12}}{6 b^7}+\frac{15 a^4 \left (a+b x^2\right )^{11}}{22 b^7}-\frac{3 a^5 \left (a+b x^2\right )^{10}}{10 b^7}+\frac{a^6 \left (a+b x^2\right )^9}{18 b^7}+\frac{\left (a+b x^2\right )^{15}}{30 b^7}-\frac{3 a \left (a+b x^2\right )^{14}}{14 b^7} \]
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Rubi [A] time = 0.210454, antiderivative size = 129, 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} \[ \frac{15 a^2 \left (a+b x^2\right )^{13}}{26 b^7}-\frac{5 a^3 \left (a+b x^2\right )^{12}}{6 b^7}+\frac{15 a^4 \left (a+b x^2\right )^{11}}{22 b^7}-\frac{3 a^5 \left (a+b x^2\right )^{10}}{10 b^7}+\frac{a^6 \left (a+b x^2\right )^9}{18 b^7}+\frac{\left (a+b x^2\right )^{15}}{30 b^7}-\frac{3 a \left (a+b x^2\right )^{14}}{14 b^7} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{13} \left (a+b x^2\right )^8 \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^6 (a+b x)^8 \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^6 (a+b x)^8}{b^6}-\frac{6 a^5 (a+b x)^9}{b^6}+\frac{15 a^4 (a+b x)^{10}}{b^6}-\frac{20 a^3 (a+b x)^{11}}{b^6}+\frac{15 a^2 (a+b x)^{12}}{b^6}-\frac{6 a (a+b x)^{13}}{b^6}+\frac{(a+b x)^{14}}{b^6}\right ) \, dx,x,x^2\right )\\ &=\frac{a^6 \left (a+b x^2\right )^9}{18 b^7}-\frac{3 a^5 \left (a+b x^2\right )^{10}}{10 b^7}+\frac{15 a^4 \left (a+b x^2\right )^{11}}{22 b^7}-\frac{5 a^3 \left (a+b x^2\right )^{12}}{6 b^7}+\frac{15 a^2 \left (a+b x^2\right )^{13}}{26 b^7}-\frac{3 a \left (a+b x^2\right )^{14}}{14 b^7}+\frac{\left (a+b x^2\right )^{15}}{30 b^7}\\ \end{align*}
Mathematica [A] time = 0.0028276, size = 108, normalized size = 0.84 \[ \frac{14}{13} a^2 b^6 x^{26}+\frac{7}{3} a^3 b^5 x^{24}+\frac{35}{11} a^4 b^4 x^{22}+\frac{14}{5} a^5 b^3 x^{20}+\frac{14}{9} a^6 b^2 x^{18}+\frac{1}{2} a^7 b x^{16}+\frac{a^8 x^{14}}{14}+\frac{2}{7} a b^7 x^{28}+\frac{b^8 x^{30}}{30} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 91, normalized size = 0.7 \begin{align*}{\frac{{b}^{8}{x}^{30}}{30}}+{\frac{2\,a{b}^{7}{x}^{28}}{7}}+{\frac{14\,{b}^{6}{a}^{2}{x}^{26}}{13}}+{\frac{7\,{a}^{3}{b}^{5}{x}^{24}}{3}}+{\frac{35\,{a}^{4}{b}^{4}{x}^{22}}{11}}+{\frac{14\,{a}^{5}{b}^{3}{x}^{20}}{5}}+{\frac{14\,{a}^{6}{b}^{2}{x}^{18}}{9}}+{\frac{{a}^{7}b{x}^{16}}{2}}+{\frac{{a}^{8}{x}^{14}}{14}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.46955, size = 122, normalized size = 0.95 \begin{align*} \frac{1}{30} \, b^{8} x^{30} + \frac{2}{7} \, a b^{7} x^{28} + \frac{14}{13} \, a^{2} b^{6} x^{26} + \frac{7}{3} \, a^{3} b^{5} x^{24} + \frac{35}{11} \, a^{4} b^{4} x^{22} + \frac{14}{5} \, a^{5} b^{3} x^{20} + \frac{14}{9} \, a^{6} b^{2} x^{18} + \frac{1}{2} \, a^{7} b x^{16} + \frac{1}{14} \, a^{8} x^{14} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.03541, size = 224, normalized size = 1.74 \begin{align*} \frac{1}{30} x^{30} b^{8} + \frac{2}{7} x^{28} b^{7} a + \frac{14}{13} x^{26} b^{6} a^{2} + \frac{7}{3} x^{24} b^{5} a^{3} + \frac{35}{11} x^{22} b^{4} a^{4} + \frac{14}{5} x^{20} b^{3} a^{5} + \frac{14}{9} x^{18} b^{2} a^{6} + \frac{1}{2} x^{16} b a^{7} + \frac{1}{14} x^{14} a^{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.082409, size = 105, normalized size = 0.81 \begin{align*} \frac{a^{8} x^{14}}{14} + \frac{a^{7} b x^{16}}{2} + \frac{14 a^{6} b^{2} x^{18}}{9} + \frac{14 a^{5} b^{3} x^{20}}{5} + \frac{35 a^{4} b^{4} x^{22}}{11} + \frac{7 a^{3} b^{5} x^{24}}{3} + \frac{14 a^{2} b^{6} x^{26}}{13} + \frac{2 a b^{7} x^{28}}{7} + \frac{b^{8} x^{30}}{30} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.27867, size = 122, normalized size = 0.95 \begin{align*} \frac{1}{30} \, b^{8} x^{30} + \frac{2}{7} \, a b^{7} x^{28} + \frac{14}{13} \, a^{2} b^{6} x^{26} + \frac{7}{3} \, a^{3} b^{5} x^{24} + \frac{35}{11} \, a^{4} b^{4} x^{22} + \frac{14}{5} \, a^{5} b^{3} x^{20} + \frac{14}{9} \, a^{6} b^{2} x^{18} + \frac{1}{2} \, a^{7} b x^{16} + \frac{1}{14} \, a^{8} x^{14} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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