Optimal. Leaf size=45 \[ -\frac{a B+A b}{3 x^3}-\frac{a A}{4 x^4}-\frac{A c+b B}{2 x^2}-\frac{B c}{x} \]
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Rubi [A] time = 0.0235477, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {765} \[ -\frac{a B+A b}{3 x^3}-\frac{a A}{4 x^4}-\frac{A c+b B}{2 x^2}-\frac{B c}{x} \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a+b x+c x^2\right )}{x^5} \, dx &=\int \left (\frac{a A}{x^5}+\frac{A b+a B}{x^4}+\frac{b B+A c}{x^3}+\frac{B c}{x^2}\right ) \, dx\\ &=-\frac{a A}{4 x^4}-\frac{A b+a B}{3 x^3}-\frac{b B+A c}{2 x^2}-\frac{B c}{x}\\ \end{align*}
Mathematica [A] time = 0.0155992, size = 42, normalized size = 0.93 \[ -\frac{a (3 A+4 B x)+2 x (A (2 b+3 c x)+3 B x (b+2 c x))}{12 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 40, normalized size = 0.9 \begin{align*} -{\frac{Ab+aB}{3\,{x}^{3}}}-{\frac{Ac+bB}{2\,{x}^{2}}}-{\frac{Bc}{x}}-{\frac{aA}{4\,{x}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04347, size = 53, normalized size = 1.18 \begin{align*} -\frac{12 \, B c x^{3} + 6 \,{\left (B b + A c\right )} x^{2} + 3 \, A a + 4 \,{\left (B a + A b\right )} x}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.29218, size = 95, normalized size = 2.11 \begin{align*} -\frac{12 \, B c x^{3} + 6 \,{\left (B b + A c\right )} x^{2} + 3 \, A a + 4 \,{\left (B a + A b\right )} x}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.69182, size = 44, normalized size = 0.98 \begin{align*} - \frac{3 A a + 12 B c x^{3} + x^{2} \left (6 A c + 6 B b\right ) + x \left (4 A b + 4 B a\right )}{12 x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30148, size = 55, normalized size = 1.22 \begin{align*} -\frac{12 \, B c x^{3} + 6 \, B b x^{2} + 6 \, A c x^{2} + 4 \, B a x + 4 \, A b x + 3 \, A a}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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