Optimal. Leaf size=36 \[ -\frac{a B+A b}{x}-\frac{a A}{2 x^2}+\log (x) (A c+b B)+B c x \]
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Rubi [A] time = 0.0237494, antiderivative size = 36, 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} \[ -\frac{a B+A b}{x}-\frac{a A}{2 x^2}+\log (x) (A c+b B)+B c x \]
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^3} \, dx &=\int \left (B c+\frac{a A}{x^3}+\frac{A b+a B}{x^2}+\frac{b B+A c}{x}\right ) \, dx\\ &=-\frac{a A}{2 x^2}-\frac{A b+a B}{x}+B c x+(b B+A c) \log (x)\\ \end{align*}
Mathematica [A] time = 0.0197833, size = 37, normalized size = 1.03 \[ \frac{-a B-A b}{x}-\frac{a A}{2 x^2}+\log (x) (A c+b B)+B c x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 37, normalized size = 1. \begin{align*} Bcx+Ac\ln \left ( x \right ) +B\ln \left ( x \right ) b-{\frac{aA}{2\,{x}^{2}}}-{\frac{Ab}{x}}-{\frac{aB}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0563, size = 46, normalized size = 1.28 \begin{align*} B c x +{\left (B b + A c\right )} \log \left (x\right ) - \frac{A a + 2 \,{\left (B a + A b\right )} x}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.27495, size = 97, normalized size = 2.69 \begin{align*} \frac{2 \, B c x^{3} + 2 \,{\left (B b + A c\right )} x^{2} \log \left (x\right ) - A a - 2 \,{\left (B a + A b\right )} x}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.784262, size = 34, normalized size = 0.94 \begin{align*} B c x + \left (A c + B b\right ) \log{\left (x \right )} - \frac{A a + x \left (2 A b + 2 B a\right )}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17604, size = 47, normalized size = 1.31 \begin{align*} B c x +{\left (B b + A c\right )} \log \left ({\left | x \right |}\right ) - \frac{A a + 2 \,{\left (B a + A b\right )} x}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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