Optimal. Leaf size=26 \[ -\frac{a A}{x}+a B \log (x)+A c x+\frac{1}{2} B c x^2 \]
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Rubi [A] time = 0.0143074, antiderivative size = 26, 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 = {766} \[ -\frac{a A}{x}+a B \log (x)+A c x+\frac{1}{2} B c x^2 \]
Antiderivative was successfully verified.
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Rule 766
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a+c x^2\right )}{x^2} \, dx &=\int \left (A c+\frac{a A}{x^2}+\frac{a B}{x}+B c x\right ) \, dx\\ &=-\frac{a A}{x}+A c x+\frac{1}{2} B c x^2+a B \log (x)\\ \end{align*}
Mathematica [A] time = 0.00318, size = 26, normalized size = 1. \[ -\frac{a A}{x}+a B \log (x)+A c x+\frac{1}{2} B c x^2 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 25, normalized size = 1. \begin{align*} -{\frac{aA}{x}}+Acx+{\frac{Bc{x}^{2}}{2}}+aB\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03454, size = 32, normalized size = 1.23 \begin{align*} \frac{1}{2} \, B c x^{2} + A c x + B a \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 = 1.55518, size = 73, normalized size = 2.81 \begin{align*} \frac{B c x^{3} + 2 \, A c x^{2} + 2 \, B a 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 = 0.287171, size = 24, normalized size = 0.92 \begin{align*} - \frac{A a}{x} + A c x + B a \log{\left (x \right )} + \frac{B c x^{2}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10117, size = 34, normalized size = 1.31 \begin{align*} \frac{1}{2} \, B c x^{2} + A c x + B a \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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