Optimal. Leaf size=55 \[ -\frac{a^2 A}{5 x^5}-\frac{a (a B+2 A b)}{4 x^4}-\frac{b (2 a B+A b)}{3 x^3}-\frac{b^2 B}{2 x^2} \]
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Rubi [A] time = 0.0235665, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {27, 76} \[ -\frac{a^2 A}{5 x^5}-\frac{a (a B+2 A b)}{4 x^4}-\frac{b (2 a B+A b)}{3 x^3}-\frac{b^2 B}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{x^6} \, dx &=\int \frac{(a+b x)^2 (A+B x)}{x^6} \, dx\\ &=\int \left (\frac{a^2 A}{x^6}+\frac{a (2 A b+a B)}{x^5}+\frac{b (A b+2 a B)}{x^4}+\frac{b^2 B}{x^3}\right ) \, dx\\ &=-\frac{a^2 A}{5 x^5}-\frac{a (2 A b+a B)}{4 x^4}-\frac{b (A b+2 a B)}{3 x^3}-\frac{b^2 B}{2 x^2}\\ \end{align*}
Mathematica [A] time = 0.0154738, size = 50, normalized size = 0.91 \[ -\frac{3 a^2 (4 A+5 B x)+10 a b x (3 A+4 B x)+10 b^2 x^2 (2 A+3 B x)}{60 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 48, normalized size = 0.9 \begin{align*} -{\frac{A{a}^{2}}{5\,{x}^{5}}}-{\frac{a \left ( 2\,Ab+aB \right ) }{4\,{x}^{4}}}-{\frac{b \left ( Ab+2\,aB \right ) }{3\,{x}^{3}}}-{\frac{{b}^{2}B}{2\,{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.995954, size = 69, normalized size = 1.25 \begin{align*} -\frac{30 \, B b^{2} x^{3} + 12 \, A a^{2} + 20 \,{\left (2 \, B a b + A b^{2}\right )} x^{2} + 15 \,{\left (B a^{2} + 2 \, A a b\right )} x}{60 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.2413, size = 120, normalized size = 2.18 \begin{align*} -\frac{30 \, B b^{2} x^{3} + 12 \, A a^{2} + 20 \,{\left (2 \, B a b + A b^{2}\right )} x^{2} + 15 \,{\left (B a^{2} + 2 \, A a b\right )} x}{60 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.10722, size = 54, normalized size = 0.98 \begin{align*} - \frac{12 A a^{2} + 30 B b^{2} x^{3} + x^{2} \left (20 A b^{2} + 40 B a b\right ) + x \left (30 A a b + 15 B a^{2}\right )}{60 x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10557, size = 69, normalized size = 1.25 \begin{align*} -\frac{30 \, B b^{2} x^{3} + 40 \, B a b x^{2} + 20 \, A b^{2} x^{2} + 15 \, B a^{2} x + 30 \, A a b x + 12 \, A a^{2}}{60 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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