Optimal. Leaf size=36 \[ \log (x) (a B+A b)-\frac{a A}{x}+x (A c+b B)+\frac{1}{2} B c x^2 \]
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Rubi [A] time = 0.0252918, 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} \[ \log (x) (a B+A b)-\frac{a A}{x}+x (A c+b B)+\frac{1}{2} B c x^2 \]
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^2} \, dx &=\int \left (b B \left (1+\frac{A c}{b B}\right )+\frac{a A}{x^2}+\frac{A b+a B}{x}+B c x\right ) \, dx\\ &=-\frac{a A}{x}+(b B+A c) x+\frac{1}{2} B c x^2+(A b+a B) \log (x)\\ \end{align*}
Mathematica [A] time = 0.016727, size = 36, normalized size = 1. \[ \log (x) (a B+A b)-\frac{a A}{x}+x (A c+b B)+\frac{1}{2} B c x^2 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 34, normalized size = 0.9 \begin{align*}{\frac{Bc{x}^{2}}{2}}+Acx+bBx+Ab\ln \left ( x \right ) +aB\ln \left ( x \right ) -{\frac{aA}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07078, size = 46, normalized size = 1.28 \begin{align*} \frac{1}{2} \, B c x^{2} +{\left (B b + A c\right )} x +{\left (B a + A b\right )} \log \left (x\right ) - \frac{A a}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.24768, size = 95, normalized size = 2.64 \begin{align*} \frac{B c x^{3} + 2 \,{\left (B b + A c\right )} x^{2} + 2 \,{\left (B a + A b\right )} x \log \left (x\right ) - 2 \, A a}{2 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.376721, size = 31, normalized size = 0.86 \begin{align*} - \frac{A a}{x} + \frac{B c x^{2}}{2} + x \left (A c + B b\right ) + \left (A b + B a\right ) \log{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17388, size = 46, normalized size = 1.28 \begin{align*} \frac{1}{2} \, B c x^{2} + B b x + A c x +{\left (B a + A b\right )} \log \left ({\left | x \right |}\right ) - \frac{A a}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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