Optimal. Leaf size=117 \[ \frac{10}{9} a^2 b^2 x^9 (a B+A b)+\frac{5}{7} a^3 b x^7 (a B+2 A b)+\frac{1}{5} a^4 x^5 (a B+5 A b)+\frac{1}{3} a^5 A x^3+\frac{1}{13} b^4 x^{13} (5 a B+A b)+\frac{5}{11} a b^3 x^{11} (2 a B+A b)+\frac{1}{15} b^5 B x^{15} \]
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Rubi [A] time = 0.0664618, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {448} \[ \frac{10}{9} a^2 b^2 x^9 (a B+A b)+\frac{5}{7} a^3 b x^7 (a B+2 A b)+\frac{1}{5} a^4 x^5 (a B+5 A b)+\frac{1}{3} a^5 A x^3+\frac{1}{13} b^4 x^{13} (5 a B+A b)+\frac{5}{11} a b^3 x^{11} (2 a B+A b)+\frac{1}{15} b^5 B x^{15} \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin{align*} \int x^2 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx &=\int \left (a^5 A x^2+a^4 (5 A b+a B) x^4+5 a^3 b (2 A b+a B) x^6+10 a^2 b^2 (A b+a B) x^8+5 a b^3 (A b+2 a B) x^{10}+b^4 (A b+5 a B) x^{12}+b^5 B x^{14}\right ) \, dx\\ &=\frac{1}{3} a^5 A x^3+\frac{1}{5} a^4 (5 A b+a B) x^5+\frac{5}{7} a^3 b (2 A b+a B) x^7+\frac{10}{9} a^2 b^2 (A b+a B) x^9+\frac{5}{11} a b^3 (A b+2 a B) x^{11}+\frac{1}{13} b^4 (A b+5 a B) x^{13}+\frac{1}{15} b^5 B x^{15}\\ \end{align*}
Mathematica [A] time = 0.0151001, size = 117, normalized size = 1. \[ \frac{10}{9} a^2 b^2 x^9 (a B+A b)+\frac{5}{7} a^3 b x^7 (a B+2 A b)+\frac{1}{5} a^4 x^5 (a B+5 A b)+\frac{1}{3} a^5 A x^3+\frac{1}{13} b^4 x^{13} (5 a B+A b)+\frac{5}{11} a b^3 x^{11} (2 a B+A b)+\frac{1}{15} b^5 B x^{15} \]
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}^{15}}{15}}+{\frac{ \left ({b}^{5}A+5\,a{b}^{4}B \right ){x}^{13}}{13}}+{\frac{ \left ( 5\,a{b}^{4}A+10\,{a}^{2}{b}^{3}B \right ){x}^{11}}{11}}+{\frac{ \left ( 10\,{a}^{2}{b}^{3}A+10\,{a}^{3}{b}^{2}B \right ){x}^{9}}{9}}+{\frac{ \left ( 10\,{a}^{3}{b}^{2}A+5\,{a}^{4}bB \right ){x}^{7}}{7}}+{\frac{ \left ( 5\,{a}^{4}bA+{a}^{5}B \right ){x}^{5}}{5}}+{\frac{{a}^{5}A{x}^{3}}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.985409, size = 161, normalized size = 1.38 \begin{align*} \frac{1}{15} \, B b^{5} x^{15} + \frac{1}{13} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{13} + \frac{5}{11} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{11} + \frac{10}{9} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + \frac{1}{3} \, A a^{5} x^{3} + \frac{5}{7} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{7} + \frac{1}{5} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.31291, size = 304, normalized size = 2.6 \begin{align*} \frac{1}{15} x^{15} b^{5} B + \frac{5}{13} x^{13} b^{4} a B + \frac{1}{13} x^{13} b^{5} A + \frac{10}{11} x^{11} b^{3} a^{2} B + \frac{5}{11} x^{11} b^{4} a A + \frac{10}{9} x^{9} b^{2} a^{3} B + \frac{10}{9} x^{9} b^{3} a^{2} A + \frac{5}{7} x^{7} b a^{4} B + \frac{10}{7} x^{7} b^{2} a^{3} A + \frac{1}{5} x^{5} a^{5} B + x^{5} b a^{4} A + \frac{1}{3} x^{3} a^{5} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.082216, size = 134, normalized size = 1.15 \begin{align*} \frac{A a^{5} x^{3}}{3} + \frac{B b^{5} x^{15}}{15} + x^{13} \left (\frac{A b^{5}}{13} + \frac{5 B a b^{4}}{13}\right ) + x^{11} \left (\frac{5 A a b^{4}}{11} + \frac{10 B a^{2} b^{3}}{11}\right ) + x^{9} \left (\frac{10 A a^{2} b^{3}}{9} + \frac{10 B a^{3} b^{2}}{9}\right ) + x^{7} \left (\frac{10 A a^{3} b^{2}}{7} + \frac{5 B a^{4} b}{7}\right ) + x^{5} \left (A a^{4} b + \frac{B a^{5}}{5}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15428, size = 167, normalized size = 1.43 \begin{align*} \frac{1}{15} \, B b^{5} x^{15} + \frac{5}{13} \, B a b^{4} x^{13} + \frac{1}{13} \, A b^{5} x^{13} + \frac{10}{11} \, B a^{2} b^{3} x^{11} + \frac{5}{11} \, A a b^{4} x^{11} + \frac{10}{9} \, B a^{3} b^{2} x^{9} + \frac{10}{9} \, A a^{2} b^{3} x^{9} + \frac{5}{7} \, B a^{4} b x^{7} + \frac{10}{7} \, A a^{3} b^{2} x^{7} + \frac{1}{5} \, B a^{5} x^{5} + A a^{4} b x^{5} + \frac{1}{3} \, A a^{5} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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