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 = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.00, size = 26, normalized size = 1.00 \[ -\frac {a A}{x}+a B \log (x)+A c x+\frac {1}{2} B c x^2 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 30, normalized size = 1.15 \[ \frac {B c x^{3} + 2 \, A c x^{2} + 2 \, B a x \log \relax (x) - 2 \, A a}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 25, normalized size = 0.96 \[ \frac {1}{2} \, B c x^{2} + A c x + B a \log \left ({\left | x \right |}\right ) - \frac {A a}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 25, normalized size = 0.96 \[ \frac {B c \,x^{2}}{2}+A c x +B a \ln \relax (x )-\frac {A a}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 24, normalized size = 0.92 \[ \frac {1}{2} \, B c x^{2} + A c x + B a \log \relax (x) - \frac {A a}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 24, normalized size = 0.92 \[ A\,c\,x-\frac {A\,a}{x}+\frac {B\,c\,x^2}{2}+B\,a\,\ln \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 24, normalized size = 0.92 \[ - \frac {A a}{x} + A c x + B a \log {\relax (x )} + \frac {B c x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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