Optimal. Leaf size=48 \[ -\frac{a^2 A}{5 x^5}-\frac{a (a B+2 A b)}{3 x^3}-\frac{b (2 a B+A b)}{x}+b^2 B x \]
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Rubi [A] time = 0.0280537, antiderivative size = 48, 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} \[ -\frac{a^2 A}{5 x^5}-\frac{a (a B+2 A b)}{3 x^3}-\frac{b (2 a B+A b)}{x}+b^2 B x \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^6} \, dx &=\int \left (b^2 B+\frac{a^2 A}{x^6}+\frac{a (2 A b+a B)}{x^4}+\frac{b (A b+2 a B)}{x^2}\right ) \, dx\\ &=-\frac{a^2 A}{5 x^5}-\frac{a (2 A b+a B)}{3 x^3}-\frac{b (A b+2 a B)}{x}+b^2 B x\\ \end{align*}
Mathematica [A] time = 0.0192372, size = 48, normalized size = 1. \[ -\frac{a^2 A}{5 x^5}-\frac{a (a B+2 A b)}{3 x^3}-\frac{b (2 a B+A b)}{x}+b^2 B x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 45, normalized size = 0.9 \begin{align*} -{\frac{A{a}^{2}}{5\,{x}^{5}}}-{\frac{a \left ( 2\,Ab+Ba \right ) }{3\,{x}^{3}}}-{\frac{b \left ( Ab+2\,Ba \right ) }{x}}+{b}^{2}Bx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.985484, size = 69, normalized size = 1.44 \begin{align*} B b^{2} x - \frac{15 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} + 3 \, A a^{2} + 5 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}}{15 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.39845, size = 119, normalized size = 2.48 \begin{align*} \frac{15 \, B b^{2} x^{6} - 15 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} - 3 \, A a^{2} - 5 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}}{15 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.70299, size = 51, normalized size = 1.06 \begin{align*} B b^{2} x - \frac{3 A a^{2} + x^{4} \left (15 A b^{2} + 30 B a b\right ) + x^{2} \left (10 A a b + 5 B a^{2}\right )}{15 x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13707, size = 72, normalized size = 1.5 \begin{align*} B b^{2} x - \frac{30 \, B a b x^{4} + 15 \, A b^{2} x^{4} + 5 \, B a^{2} x^{2} + 10 \, A a b x^{2} + 3 \, A a^{2}}{15 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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