Optimal. Leaf size=113 \[ -\frac{5 a^2 b^2 (a B+A b)}{2 x^4}-\frac{a^4 (a B+5 A b)}{8 x^8}-\frac{5 a^3 b (a B+2 A b)}{6 x^6}-\frac{a^5 A}{10 x^{10}}-\frac{5 a b^3 (2 a B+A b)}{2 x^2}+b^4 \log (x) (5 a B+A b)+\frac{1}{2} b^5 B x^2 \]
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Rubi [A] time = 0.0893554, antiderivative size = 113, 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} \[ -\frac{5 a^2 b^2 (a B+A b)}{2 x^4}-\frac{a^4 (a B+5 A b)}{8 x^8}-\frac{5 a^3 b (a B+2 A b)}{6 x^6}-\frac{a^5 A}{10 x^{10}}-\frac{5 a b^3 (2 a B+A b)}{2 x^2}+b^4 \log (x) (5 a B+A b)+\frac{1}{2} b^5 B x^2 \]
Antiderivative was successfully verified.
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Rule 446
Rule 76
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{11}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^5 (A+B x)}{x^6} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (b^5 B+\frac{a^5 A}{x^6}+\frac{a^4 (5 A b+a B)}{x^5}+\frac{5 a^3 b (2 A b+a B)}{x^4}+\frac{10 a^2 b^2 (A b+a B)}{x^3}+\frac{5 a b^3 (A b+2 a B)}{x^2}+\frac{b^4 (A b+5 a B)}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^5 A}{10 x^{10}}-\frac{a^4 (5 A b+a B)}{8 x^8}-\frac{5 a^3 b (2 A b+a B)}{6 x^6}-\frac{5 a^2 b^2 (A b+a B)}{2 x^4}-\frac{5 a b^3 (A b+2 a B)}{2 x^2}+\frac{1}{2} b^5 B x^2+b^4 (A b+5 a B) \log (x)\\ \end{align*}
Mathematica [A] time = 0.0569273, size = 116, normalized size = 1.03 \[ b^4 \log (x) (5 a B+A b)-\frac{300 a^2 b^3 x^6 \left (A+2 B x^2\right )+100 a^3 b^2 x^4 \left (2 A+3 B x^2\right )+25 a^4 b x^2 \left (3 A+4 B x^2\right )+3 a^5 \left (4 A+5 B x^2\right )+300 a A b^4 x^8-60 b^5 B x^{12}}{120 x^{10}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 123, normalized size = 1.1 \begin{align*}{\frac{{b}^{5}B{x}^{2}}{2}}+A\ln \left ( x \right ){b}^{5}+5\,B\ln \left ( x \right ) a{b}^{4}-{\frac{5\,{a}^{2}{b}^{3}A}{2\,{x}^{4}}}-{\frac{5\,{a}^{3}{b}^{2}B}{2\,{x}^{4}}}-{\frac{5\,a{b}^{4}A}{2\,{x}^{2}}}-5\,{\frac{{a}^{2}{b}^{3}B}{{x}^{2}}}-{\frac{5\,{a}^{3}{b}^{2}A}{3\,{x}^{6}}}-{\frac{5\,{a}^{4}bB}{6\,{x}^{6}}}-{\frac{A{a}^{5}}{10\,{x}^{10}}}-{\frac{5\,{a}^{4}bA}{8\,{x}^{8}}}-{\frac{{a}^{5}B}{8\,{x}^{8}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0346, size = 166, normalized size = 1.47 \begin{align*} \frac{1}{2} \, B b^{5} x^{2} + \frac{1}{2} \,{\left (5 \, B a b^{4} + A b^{5}\right )} \log \left (x^{2}\right ) - \frac{300 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 300 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 12 \, A a^{5} + 100 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 15 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{120 \, x^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49088, size = 281, normalized size = 2.49 \begin{align*} \frac{60 \, B b^{5} x^{12} + 120 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} \log \left (x\right ) - 300 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} - 300 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} - 12 \, A a^{5} - 100 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} - 15 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{120 \, x^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.39761, size = 122, normalized size = 1.08 \begin{align*} \frac{B b^{5} x^{2}}{2} + b^{4} \left (A b + 5 B a\right ) \log{\left (x \right )} - \frac{12 A a^{5} + x^{8} \left (300 A a b^{4} + 600 B a^{2} b^{3}\right ) + x^{6} \left (300 A a^{2} b^{3} + 300 B a^{3} b^{2}\right ) + x^{4} \left (200 A a^{3} b^{2} + 100 B a^{4} b\right ) + x^{2} \left (75 A a^{4} b + 15 B a^{5}\right )}{120 x^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10586, size = 198, normalized size = 1.75 \begin{align*} \frac{1}{2} \, B b^{5} x^{2} + \frac{1}{2} \,{\left (5 \, B a b^{4} + A b^{5}\right )} \log \left (x^{2}\right ) - \frac{685 \, B a b^{4} x^{10} + 137 \, A b^{5} x^{10} + 600 \, B a^{2} b^{3} x^{8} + 300 \, A a b^{4} x^{8} + 300 \, B a^{3} b^{2} x^{6} + 300 \, A a^{2} b^{3} x^{6} + 100 \, B a^{4} b x^{4} + 200 \, A a^{3} b^{2} x^{4} + 15 \, B a^{5} x^{2} + 75 \, A a^{4} b x^{2} + 12 \, A a^{5}}{120 \, x^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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