Optimal. Leaf size=26 \[ -\frac {a A}{2 x^2}-\frac {a B}{x}+A c \log (x)+B c x \]
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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} \begin {gather*} -\frac {a A}{2 x^2}-\frac {a B}{x}+A c \log (x)+B c x \end {gather*}
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^3} \, dx &=\int \left (B c+\frac {a A}{x^3}+\frac {a B}{x^2}+\frac {A c}{x}\right ) \, dx\\ &=-\frac {a A}{2 x^2}-\frac {a B}{x}+B c x+A c \log (x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 26, normalized size = 1.00 \begin {gather*} -\frac {a A}{2 x^2}-\frac {a B}{x}+A c \log (x)+B c x \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a+c x^2\right )}{x^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 31, normalized size = 1.19 \begin {gather*} \frac {2 \, B c x^{3} + 2 \, A c x^{2} \log \relax (x) - 2 \, B a x - A a}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 25, normalized size = 0.96 \begin {gather*} B c x + A c \log \left ({\left | x \right |}\right ) - \frac {2 \, B a x + A a}{2 \, x^{2}} \end {gather*}
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 \begin {gather*} A c \ln \relax (x )+B c x -\frac {B a}{x}-\frac {A a}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 24, normalized size = 0.92 \begin {gather*} B c x + A c \log \relax (x) - \frac {2 \, B a x + A a}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.03, size = 24, normalized size = 0.92 \begin {gather*} B\,c\,x-\frac {\frac {A\,a}{2}+B\,a\,x}{x^2}+A\,c\,\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 27, normalized size = 1.04 \begin {gather*} A c \log {\relax (x )} + B c x + \frac {- A a - 2 B a x}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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