Optimal. Leaf size=95 \[ \frac{a^2 \left (a+b x^2\right )^6 (A b-a B)}{12 b^4}+\frac{\left (a+b x^2\right )^8 (A b-3 a B)}{16 b^4}-\frac{a \left (a+b x^2\right )^7 (2 A b-3 a B)}{14 b^4}+\frac{B \left (a+b x^2\right )^9}{18 b^4} \]
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Rubi [A] time = 0.213331, antiderivative size = 95, 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} \[ \frac{a^2 \left (a+b x^2\right )^6 (A b-a B)}{12 b^4}+\frac{\left (a+b x^2\right )^8 (A b-3 a B)}{16 b^4}-\frac{a \left (a+b x^2\right )^7 (2 A b-3 a B)}{14 b^4}+\frac{B \left (a+b x^2\right )^9}{18 b^4} \]
Antiderivative was successfully verified.
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Rule 446
Rule 76
Rubi steps
\begin{align*} \int x^5 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^2 (a+b x)^5 (A+B x) \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a^2 (-A b+a B) (a+b x)^5}{b^3}+\frac{a (-2 A b+3 a B) (a+b x)^6}{b^3}+\frac{(A b-3 a B) (a+b x)^7}{b^3}+\frac{B (a+b x)^8}{b^3}\right ) \, dx,x,x^2\right )\\ &=\frac{a^2 (A b-a B) \left (a+b x^2\right )^6}{12 b^4}-\frac{a (2 A b-3 a B) \left (a+b x^2\right )^7}{14 b^4}+\frac{(A b-3 a B) \left (a+b x^2\right )^8}{16 b^4}+\frac{B \left (a+b x^2\right )^9}{18 b^4}\\ \end{align*}
Mathematica [A] time = 0.0236781, size = 107, normalized size = 1.13 \[ \frac{x^6 \left (840 a^2 b^2 x^6 (a B+A b)+504 a^3 b x^4 (a B+2 A b)+126 a^4 x^2 (a B+5 A b)+168 a^5 A+63 b^4 x^{10} (5 a B+A b)+360 a b^3 x^8 (2 a B+A b)+56 b^5 B x^{12}\right )}{1008} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 124, normalized size = 1.3 \begin{align*}{\frac{{b}^{5}B{x}^{18}}{18}}+{\frac{ \left ({b}^{5}A+5\,a{b}^{4}B \right ){x}^{16}}{16}}+{\frac{ \left ( 5\,a{b}^{4}A+10\,{a}^{2}{b}^{3}B \right ){x}^{14}}{14}}+{\frac{ \left ( 10\,{a}^{2}{b}^{3}A+10\,{a}^{3}{b}^{2}B \right ){x}^{12}}{12}}+{\frac{ \left ( 10\,{a}^{3}{b}^{2}A+5\,{a}^{4}bB \right ){x}^{10}}{10}}+{\frac{ \left ( 5\,{a}^{4}bA+{a}^{5}B \right ){x}^{8}}{8}}+{\frac{{a}^{5}A{x}^{6}}{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.99256, size = 161, normalized size = 1.69 \begin{align*} \frac{1}{18} \, B b^{5} x^{18} + \frac{1}{16} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{16} + \frac{5}{14} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{14} + \frac{5}{6} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{12} + \frac{1}{6} \, A a^{5} x^{6} + \frac{1}{2} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{10} + \frac{1}{8} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.27962, size = 302, normalized size = 3.18 \begin{align*} \frac{1}{18} x^{18} b^{5} B + \frac{5}{16} x^{16} b^{4} a B + \frac{1}{16} x^{16} b^{5} A + \frac{5}{7} x^{14} b^{3} a^{2} B + \frac{5}{14} x^{14} b^{4} a A + \frac{5}{6} x^{12} b^{2} a^{3} B + \frac{5}{6} x^{12} b^{3} a^{2} A + \frac{1}{2} x^{10} b a^{4} B + x^{10} b^{2} a^{3} A + \frac{1}{8} x^{8} a^{5} B + \frac{5}{8} x^{8} b a^{4} A + \frac{1}{6} x^{6} a^{5} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.083754, size = 133, normalized size = 1.4 \begin{align*} \frac{A a^{5} x^{6}}{6} + \frac{B b^{5} x^{18}}{18} + x^{16} \left (\frac{A b^{5}}{16} + \frac{5 B a b^{4}}{16}\right ) + x^{14} \left (\frac{5 A a b^{4}}{14} + \frac{5 B a^{2} b^{3}}{7}\right ) + x^{12} \left (\frac{5 A a^{2} b^{3}}{6} + \frac{5 B a^{3} b^{2}}{6}\right ) + x^{10} \left (A a^{3} b^{2} + \frac{B a^{4} b}{2}\right ) + x^{8} \left (\frac{5 A a^{4} b}{8} + \frac{B a^{5}}{8}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12368, size = 167, normalized size = 1.76 \begin{align*} \frac{1}{18} \, B b^{5} x^{18} + \frac{5}{16} \, B a b^{4} x^{16} + \frac{1}{16} \, A b^{5} x^{16} + \frac{5}{7} \, B a^{2} b^{3} x^{14} + \frac{5}{14} \, A a b^{4} x^{14} + \frac{5}{6} \, B a^{3} b^{2} x^{12} + \frac{5}{6} \, A a^{2} b^{3} x^{12} + \frac{1}{2} \, B a^{4} b x^{10} + A a^{3} b^{2} x^{10} + \frac{1}{8} \, B a^{5} x^{8} + \frac{5}{8} \, A a^{4} b x^{8} + \frac{1}{6} \, A a^{5} x^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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