Optimal. Leaf size=53 \[ \frac{a^2 \left (a+b x^2\right )^6}{12 b^3}+\frac{\left (a+b x^2\right )^8}{16 b^3}-\frac{a \left (a+b x^2\right )^7}{7 b^3} \]
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Rubi [A] time = 0.0647074, 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} \[ \frac{a^2 \left (a+b x^2\right )^6}{12 b^3}+\frac{\left (a+b x^2\right )^8}{16 b^3}-\frac{a \left (a+b x^2\right )^7}{7 b^3} \]
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 )^5 \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^2 (a+b x)^5 \, dx,x,x^2\right )\\ &=\frac{1}{2} \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^2\right )\\ &=\frac{a^2 \left (a+b x^2\right )^6}{12 b^3}-\frac{a \left (a+b x^2\right )^7}{7 b^3}+\frac{\left (a+b x^2\right )^8}{16 b^3}\\ \end{align*}
Mathematica [A] time = 0.0020264, size = 66, normalized size = 1.25 \[ \frac{5}{6} a^2 b^3 x^{12}+a^3 b^2 x^{10}+\frac{5}{8} a^4 b x^8+\frac{a^5 x^6}{6}+\frac{5}{14} a b^4 x^{14}+\frac{b^5 x^{16}}{16} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 57, normalized size = 1.1 \begin{align*}{\frac{{b}^{5}{x}^{16}}{16}}+{\frac{5\,a{b}^{4}{x}^{14}}{14}}+{\frac{5\,{a}^{2}{b}^{3}{x}^{12}}{6}}+{a}^{3}{b}^{2}{x}^{10}+{\frac{5\,{a}^{4}b{x}^{8}}{8}}+{\frac{{a}^{5}{x}^{6}}{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.87543, size = 76, normalized size = 1.43 \begin{align*} \frac{1}{16} \, b^{5} x^{16} + \frac{5}{14} \, a b^{4} x^{14} + \frac{5}{6} \, a^{2} b^{3} x^{12} + a^{3} b^{2} x^{10} + \frac{5}{8} \, a^{4} b x^{8} + \frac{1}{6} \, a^{5} x^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.984943, size = 131, normalized size = 2.47 \begin{align*} \frac{1}{16} x^{16} b^{5} + \frac{5}{14} x^{14} b^{4} a + \frac{5}{6} x^{12} b^{3} a^{2} + x^{10} b^{2} a^{3} + \frac{5}{8} x^{8} b a^{4} + \frac{1}{6} x^{6} a^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.069944, size = 63, normalized size = 1.19 \begin{align*} \frac{a^{5} x^{6}}{6} + \frac{5 a^{4} b x^{8}}{8} + a^{3} b^{2} x^{10} + \frac{5 a^{2} b^{3} x^{12}}{6} + \frac{5 a b^{4} x^{14}}{14} + \frac{b^{5} x^{16}}{16} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.01114, size = 76, normalized size = 1.43 \begin{align*} \frac{1}{16} \, b^{5} x^{16} + \frac{5}{14} \, a b^{4} x^{14} + \frac{5}{6} \, a^{2} b^{3} x^{12} + a^{3} b^{2} x^{10} + \frac{5}{8} \, a^{4} b x^{8} + \frac{1}{6} \, a^{5} x^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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