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 = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {a B+A b}{x}-\frac {a A}{2 x^2}+\log (x) (A c+b B)+B c x \end {gather*}
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*}
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Mathematica [A] time = 0.02, size = 37, normalized size = 1.03 \begin {gather*} \frac {-a B-A b}{x}-\frac {a A}{2 x^2}+\log (x) (A c+b B)+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+b x+c x^2\right )}{x^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 41, normalized size = 1.14 \begin {gather*} \frac {2 \, B c x^{3} + 2 \, {\left (B b + A c\right )} x^{2} \log \relax (x) - A a - 2 \, {\left (B a + A b\right )} x}{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 = 35, normalized size = 0.97 \begin {gather*} 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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 37, normalized size = 1.03 \begin {gather*} A c \ln \relax (x )+B b \ln \relax (x )+B c x -\frac {A b}{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 = 0.50, size = 34, normalized size = 0.94 \begin {gather*} B c x + {\left (B b + A c\right )} \log \relax (x) - \frac {A a + 2 \, {\left (B a + A b\right )} x}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.17, size = 34, normalized size = 0.94 \begin {gather*} \ln \relax (x)\,\left (A\,c+B\,b\right )-\frac {\frac {A\,a}{2}+x\,\left (A\,b+B\,a\right )}{x^2}+B\,c\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.31, size = 36, normalized size = 1.00 \begin {gather*} B c x + \left (A c + B b\right ) \log {\relax (x )} + \frac {- A a + x \left (- 2 A b - 2 B a\right )}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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