Optimal. Leaf size=55 \[ -\frac{a^2 A}{7 x^7}-\frac{a (a B+2 A b)}{6 x^6}-\frac{b (2 a B+A b)}{5 x^5}-\frac{b^2 B}{4 x^4} \]
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Rubi [A] time = 0.0221769, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {76} \[ -\frac{a^2 A}{7 x^7}-\frac{a (a B+2 A b)}{6 x^6}-\frac{b (2 a B+A b)}{5 x^5}-\frac{b^2 B}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin{align*} \int \frac{(a+b x)^2 (A+B x)}{x^8} \, dx &=\int \left (\frac{a^2 A}{x^8}+\frac{a (2 A b+a B)}{x^7}+\frac{b (A b+2 a B)}{x^6}+\frac{b^2 B}{x^5}\right ) \, dx\\ &=-\frac{a^2 A}{7 x^7}-\frac{a (2 A b+a B)}{6 x^6}-\frac{b (A b+2 a B)}{5 x^5}-\frac{b^2 B}{4 x^4}\\ \end{align*}
Mathematica [A] time = 0.0156682, size = 50, normalized size = 0.91 \[ -\frac{10 a^2 (6 A+7 B x)+28 a b x (5 A+6 B x)+21 b^2 x^2 (4 A+5 B x)}{420 x^7} \]
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 ) }{6\,{x}^{6}}}-{\frac{b \left ( Ab+2\,Ba \right ) }{5\,{x}^{5}}}-{\frac{B{b}^{2}}{4\,{x}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01831, size = 69, normalized size = 1.25 \begin{align*} -\frac{105 \, B b^{2} x^{3} + 60 \, A a^{2} + 84 \,{\left (2 \, B a b + A b^{2}\right )} x^{2} + 70 \,{\left (B a^{2} + 2 \, A a b\right )} x}{420 \, x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85972, size = 123, normalized size = 2.24 \begin{align*} -\frac{105 \, B b^{2} x^{3} + 60 \, A a^{2} + 84 \,{\left (2 \, B a b + A b^{2}\right )} x^{2} + 70 \,{\left (B a^{2} + 2 \, A a b\right )} x}{420 \, x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.51608, size = 54, normalized size = 0.98 \begin{align*} - \frac{60 A a^{2} + 105 B b^{2} x^{3} + x^{2} \left (84 A b^{2} + 168 B a b\right ) + x \left (140 A a b + 70 B a^{2}\right )}{420 x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24168, size = 69, normalized size = 1.25 \begin{align*} -\frac{105 \, B b^{2} x^{3} + 168 \, B a b x^{2} + 84 \, A b^{2} x^{2} + 70 \, B a^{2} x + 140 \, A a b x + 60 \, A a^{2}}{420 \, x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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