Optimal. Leaf size=108 \[ \frac{10}{3} a^2 b^2 x^3 (a B+A b)+5 a^3 b x (a B+2 A b)-\frac{a^4 (a B+5 A b)}{x}-\frac{a^5 A}{3 x^3}+\frac{1}{7} b^4 x^7 (5 a B+A b)+a b^3 x^5 (2 a B+A b)+\frac{1}{9} b^5 B x^9 \]
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Rubi [A] time = 0.0610051, antiderivative size = 108, 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} \[ \frac{10}{3} a^2 b^2 x^3 (a B+A b)+5 a^3 b x (a B+2 A b)-\frac{a^4 (a B+5 A b)}{x}-\frac{a^5 A}{3 x^3}+\frac{1}{7} b^4 x^7 (5 a B+A b)+a b^3 x^5 (2 a B+A b)+\frac{1}{9} b^5 B x^9 \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^4} \, dx &=\int \left (5 a^3 b (2 A b+a B)+\frac{a^5 A}{x^4}+\frac{a^4 (5 A b+a B)}{x^2}+10 a^2 b^2 (A b+a B) x^2+5 a b^3 (A b+2 a B) x^4+b^4 (A b+5 a B) x^6+b^5 B x^8\right ) \, dx\\ &=-\frac{a^5 A}{3 x^3}-\frac{a^4 (5 A b+a B)}{x}+5 a^3 b (2 A b+a B) x+\frac{10}{3} a^2 b^2 (A b+a B) x^3+a b^3 (A b+2 a B) x^5+\frac{1}{7} b^4 (A b+5 a B) x^7+\frac{1}{9} b^5 B x^9\\ \end{align*}
Mathematica [A] time = 0.0338894, size = 110, normalized size = 1.02 \[ \frac{10}{3} a^2 b^2 x^3 (a B+A b)+5 a^3 b x (a B+2 A b)+\frac{a^5 (-B)-5 a^4 A b}{x}-\frac{a^5 A}{3 x^3}+\frac{1}{7} b^4 x^7 (5 a B+A b)+a b^3 x^5 (2 a B+A b)+\frac{1}{9} b^5 B x^9 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 118, normalized size = 1.1 \begin{align*}{\frac{{b}^{5}B{x}^{9}}{9}}+{\frac{A{x}^{7}{b}^{5}}{7}}+{\frac{5\,B{x}^{7}a{b}^{4}}{7}}+A{x}^{5}a{b}^{4}+2\,B{x}^{5}{a}^{2}{b}^{3}+{\frac{10\,A{x}^{3}{a}^{2}{b}^{3}}{3}}+{\frac{10\,B{x}^{3}{a}^{3}{b}^{2}}{3}}+10\,{a}^{3}{b}^{2}Ax+5\,{a}^{4}bBx-{\frac{A{a}^{5}}{3\,{x}^{3}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0016, size = 159, normalized size = 1.47 \begin{align*} \frac{1}{9} \, B b^{5} x^{9} + \frac{1}{7} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{7} +{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{5} + \frac{10}{3} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 5 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x - \frac{A a^{5} + 3 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.41363, size = 263, normalized size = 2.44 \begin{align*} \frac{7 \, B b^{5} x^{12} + 9 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 63 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 210 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} - 21 \, A a^{5} + 315 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} - 63 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{63 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.481343, size = 126, normalized size = 1.17 \begin{align*} \frac{B b^{5} x^{9}}{9} + x^{7} \left (\frac{A b^{5}}{7} + \frac{5 B a b^{4}}{7}\right ) + x^{5} \left (A a b^{4} + 2 B a^{2} b^{3}\right ) + x^{3} \left (\frac{10 A a^{2} b^{3}}{3} + \frac{10 B a^{3} b^{2}}{3}\right ) + x \left (10 A a^{3} b^{2} + 5 B a^{4} b\right ) - \frac{A a^{5} + x^{2} \left (15 A a^{4} b + 3 B a^{5}\right )}{3 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.135, size = 165, normalized size = 1.53 \begin{align*} \frac{1}{9} \, B b^{5} x^{9} + \frac{5}{7} \, B a b^{4} x^{7} + \frac{1}{7} \, A b^{5} x^{7} + 2 \, B a^{2} b^{3} x^{5} + A a b^{4} x^{5} + \frac{10}{3} \, B a^{3} b^{2} x^{3} + \frac{10}{3} \, A a^{2} b^{3} x^{3} + 5 \, B a^{4} b x + 10 \, A a^{3} b^{2} x - \frac{3 \, B a^{5} x^{2} + 15 \, A a^{4} b x^{2} + A a^{5}}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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