Optimal. Leaf size=53 \[ -\frac{a^2 A}{7 x^7}-\frac{a (a B+2 A b)}{5 x^5}-\frac{b (2 a B+A b)}{3 x^3}-\frac{b^2 B}{x} \]
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Rubi [A] time = 0.0274398, antiderivative size = 53, 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} \[ -\frac{a^2 A}{7 x^7}-\frac{a (a B+2 A b)}{5 x^5}-\frac{b (2 a B+A b)}{3 x^3}-\frac{b^2 B}{x} \]
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^8} \, dx &=\int \left (\frac{a^2 A}{x^8}+\frac{a (2 A b+a B)}{x^6}+\frac{b (A b+2 a B)}{x^4}+\frac{b^2 B}{x^2}\right ) \, dx\\ &=-\frac{a^2 A}{7 x^7}-\frac{a (2 A b+a B)}{5 x^5}-\frac{b (A b+2 a B)}{3 x^3}-\frac{b^2 B}{x}\\ \end{align*}
Mathematica [A] time = 0.0161532, size = 56, normalized size = 1.06 \[ -\frac{3 a^2 \left (5 A+7 B x^2\right )+14 a b x^2 \left (3 A+5 B x^2\right )+35 b^2 x^4 \left (A+3 B x^2\right )}{105 x^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 48, normalized size = 0.9 \begin{align*} -{\frac{A{a}^{2}}{7\,{x}^{7}}}-{\frac{a \left ( 2\,Ab+Ba \right ) }{5\,{x}^{5}}}-{\frac{b \left ( Ab+2\,Ba \right ) }{3\,{x}^{3}}}-{\frac{B{b}^{2}}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00985, size = 72, normalized size = 1.36 \begin{align*} -\frac{105 \, B b^{2} x^{6} + 35 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} + 15 \, A a^{2} + 21 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}}{105 \, x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.38145, size = 126, normalized size = 2.38 \begin{align*} -\frac{105 \, B b^{2} x^{6} + 35 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} + 15 \, A a^{2} + 21 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}}{105 \, x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.13365, size = 56, normalized size = 1.06 \begin{align*} - \frac{15 A a^{2} + 105 B b^{2} x^{6} + x^{4} \left (35 A b^{2} + 70 B a b\right ) + x^{2} \left (42 A a b + 21 B a^{2}\right )}{105 x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16392, size = 74, normalized size = 1.4 \begin{align*} -\frac{105 \, B b^{2} x^{6} + 70 \, B a b x^{4} + 35 \, A b^{2} x^{4} + 21 \, B a^{2} x^{2} + 42 \, A a b x^{2} + 15 \, A a^{2}}{105 \, x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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