Optimal. Leaf size=69 \[ \frac{1}{2} a^2 b^3 x^{20}+\frac{5}{9} a^3 b^2 x^{18}+\frac{5}{16} a^4 b x^{16}+\frac{a^5 x^{14}}{14}+\frac{5}{22} a b^4 x^{22}+\frac{b^5 x^{24}}{24} \]
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Rubi [A] time = 0.0488556, antiderivative size = 69, 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} \[ \frac{1}{2} a^2 b^3 x^{20}+\frac{5}{9} a^3 b^2 x^{18}+\frac{5}{16} a^4 b x^{16}+\frac{a^5 x^{14}}{14}+\frac{5}{22} a b^4 x^{22}+\frac{b^5 x^{24}}{24} \]
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 )^5 \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^6 (a+b x)^5 \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (a^5 x^6+5 a^4 b x^7+10 a^3 b^2 x^8+10 a^2 b^3 x^9+5 a b^4 x^{10}+b^5 x^{11}\right ) \, dx,x,x^2\right )\\ &=\frac{a^5 x^{14}}{14}+\frac{5}{16} a^4 b x^{16}+\frac{5}{9} a^3 b^2 x^{18}+\frac{1}{2} a^2 b^3 x^{20}+\frac{5}{22} a b^4 x^{22}+\frac{b^5 x^{24}}{24}\\ \end{align*}
Mathematica [A] time = 0.002849, size = 69, normalized size = 1. \[ \frac{1}{2} a^2 b^3 x^{20}+\frac{5}{9} a^3 b^2 x^{18}+\frac{5}{16} a^4 b x^{16}+\frac{a^5 x^{14}}{14}+\frac{5}{22} a b^4 x^{22}+\frac{b^5 x^{24}}{24} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 58, normalized size = 0.8 \begin{align*}{\frac{{a}^{5}{x}^{14}}{14}}+{\frac{5\,{a}^{4}b{x}^{16}}{16}}+{\frac{5\,{a}^{3}{b}^{2}{x}^{18}}{9}}+{\frac{{a}^{2}{b}^{3}{x}^{20}}{2}}+{\frac{5\,a{b}^{4}{x}^{22}}{22}}+{\frac{{b}^{5}{x}^{24}}{24}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.1205, size = 77, normalized size = 1.12 \begin{align*} \frac{1}{24} \, b^{5} x^{24} + \frac{5}{22} \, a b^{4} x^{22} + \frac{1}{2} \, a^{2} b^{3} x^{20} + \frac{5}{9} \, a^{3} b^{2} x^{18} + \frac{5}{16} \, a^{4} b x^{16} + \frac{1}{14} \, a^{5} x^{14} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.11462, size = 142, normalized size = 2.06 \begin{align*} \frac{1}{24} x^{24} b^{5} + \frac{5}{22} x^{22} b^{4} a + \frac{1}{2} x^{20} b^{3} a^{2} + \frac{5}{9} x^{18} b^{2} a^{3} + \frac{5}{16} x^{16} b a^{4} + \frac{1}{14} x^{14} a^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.072784, size = 65, normalized size = 0.94 \begin{align*} \frac{a^{5} x^{14}}{14} + \frac{5 a^{4} b x^{16}}{16} + \frac{5 a^{3} b^{2} x^{18}}{9} + \frac{a^{2} b^{3} x^{20}}{2} + \frac{5 a b^{4} x^{22}}{22} + \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 = 2.58715, size = 77, normalized size = 1.12 \begin{align*} \frac{1}{24} \, b^{5} x^{24} + \frac{5}{22} \, a b^{4} x^{22} + \frac{1}{2} \, a^{2} b^{3} x^{20} + \frac{5}{9} \, a^{3} b^{2} x^{18} + \frac{5}{16} \, a^{4} b x^{16} + \frac{1}{14} \, a^{5} x^{14} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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