Optimal. Leaf size=117 \[ \frac{5}{8} a^2 b^2 x^{16} (a B+A b)+\frac{5}{14} a^3 b x^{14} (a B+2 A b)+\frac{1}{12} a^4 x^{12} (a B+5 A b)+\frac{1}{10} a^5 A x^{10}+\frac{1}{20} b^4 x^{20} (5 a B+A b)+\frac{5}{18} a b^3 x^{18} (2 a B+A b)+\frac{1}{22} b^5 B x^{22} \]
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Rubi [A] time = 0.152547, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {446, 76} \[ \frac{5}{8} a^2 b^2 x^{16} (a B+A b)+\frac{5}{14} a^3 b x^{14} (a B+2 A b)+\frac{1}{12} a^4 x^{12} (a B+5 A b)+\frac{1}{10} a^5 A x^{10}+\frac{1}{20} b^4 x^{20} (5 a B+A b)+\frac{5}{18} a b^3 x^{18} (2 a B+A b)+\frac{1}{22} b^5 B x^{22} \]
Antiderivative was successfully verified.
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Rule 446
Rule 76
Rubi steps
\begin{align*} \int x^9 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^4 (a+b x)^5 (A+B x) \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (a^5 A x^4+a^4 (5 A b+a B) x^5+5 a^3 b (2 A b+a B) x^6+10 a^2 b^2 (A b+a B) x^7+5 a b^3 (A b+2 a B) x^8+b^4 (A b+5 a B) x^9+b^5 B x^{10}\right ) \, dx,x,x^2\right )\\ &=\frac{1}{10} a^5 A x^{10}+\frac{1}{12} a^4 (5 A b+a B) x^{12}+\frac{5}{14} a^3 b (2 A b+a B) x^{14}+\frac{5}{8} a^2 b^2 (A b+a B) x^{16}+\frac{5}{18} a b^3 (A b+2 a B) x^{18}+\frac{1}{20} b^4 (A b+5 a B) x^{20}+\frac{1}{22} b^5 B x^{22}\\ \end{align*}
Mathematica [A] time = 0.0171334, size = 117, normalized size = 1. \[ \frac{5}{8} a^2 b^2 x^{16} (a B+A b)+\frac{5}{14} a^3 b x^{14} (a B+2 A b)+\frac{1}{12} a^4 x^{12} (a B+5 A b)+\frac{1}{10} a^5 A x^{10}+\frac{1}{20} b^4 x^{20} (5 a B+A b)+\frac{5}{18} a b^3 x^{18} (2 a B+A b)+\frac{1}{22} b^5 B x^{22} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 124, normalized size = 1.1 \begin{align*}{\frac{{b}^{5}B{x}^{22}}{22}}+{\frac{ \left ({b}^{5}A+5\,a{b}^{4}B \right ){x}^{20}}{20}}+{\frac{ \left ( 5\,a{b}^{4}A+10\,{a}^{2}{b}^{3}B \right ){x}^{18}}{18}}+{\frac{ \left ( 10\,{a}^{2}{b}^{3}A+10\,{a}^{3}{b}^{2}B \right ){x}^{16}}{16}}+{\frac{ \left ( 10\,{a}^{3}{b}^{2}A+5\,{a}^{4}bB \right ){x}^{14}}{14}}+{\frac{ \left ( 5\,{a}^{4}bA+{a}^{5}B \right ){x}^{12}}{12}}+{\frac{{a}^{5}A{x}^{10}}{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07332, size = 161, normalized size = 1.38 \begin{align*} \frac{1}{22} \, B b^{5} x^{22} + \frac{1}{20} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{20} + \frac{5}{18} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{18} + \frac{5}{8} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{16} + \frac{1}{10} \, A a^{5} x^{10} + \frac{5}{14} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{14} + \frac{1}{12} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{12} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.23201, size = 316, normalized size = 2.7 \begin{align*} \frac{1}{22} x^{22} b^{5} B + \frac{1}{4} x^{20} b^{4} a B + \frac{1}{20} x^{20} b^{5} A + \frac{5}{9} x^{18} b^{3} a^{2} B + \frac{5}{18} x^{18} b^{4} a A + \frac{5}{8} x^{16} b^{2} a^{3} B + \frac{5}{8} x^{16} b^{3} a^{2} A + \frac{5}{14} x^{14} b a^{4} B + \frac{5}{7} x^{14} b^{2} a^{3} A + \frac{1}{12} x^{12} a^{5} B + \frac{5}{12} x^{12} b a^{4} A + \frac{1}{10} x^{10} a^{5} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.083946, size = 136, normalized size = 1.16 \begin{align*} \frac{A a^{5} x^{10}}{10} + \frac{B b^{5} x^{22}}{22} + x^{20} \left (\frac{A b^{5}}{20} + \frac{B a b^{4}}{4}\right ) + x^{18} \left (\frac{5 A a b^{4}}{18} + \frac{5 B a^{2} b^{3}}{9}\right ) + x^{16} \left (\frac{5 A a^{2} b^{3}}{8} + \frac{5 B a^{3} b^{2}}{8}\right ) + x^{14} \left (\frac{5 A a^{3} b^{2}}{7} + \frac{5 B a^{4} b}{14}\right ) + x^{12} \left (\frac{5 A a^{4} b}{12} + \frac{B a^{5}}{12}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33724, size = 169, normalized size = 1.44 \begin{align*} \frac{1}{22} \, B b^{5} x^{22} + \frac{1}{4} \, B a b^{4} x^{20} + \frac{1}{20} \, A b^{5} x^{20} + \frac{5}{9} \, B a^{2} b^{3} x^{18} + \frac{5}{18} \, A a b^{4} x^{18} + \frac{5}{8} \, B a^{3} b^{2} x^{16} + \frac{5}{8} \, A a^{2} b^{3} x^{16} + \frac{5}{14} \, B a^{4} b x^{14} + \frac{5}{7} \, A a^{3} b^{2} x^{14} + \frac{1}{12} \, B a^{5} x^{12} + \frac{5}{12} \, A a^{4} b x^{12} + \frac{1}{10} \, A a^{5} x^{10} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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