Optimal. Leaf size=41 \[ -\frac {a B+A b}{2 x^2}-\frac {a A}{3 x^3}-\frac {A c+b B}{x}+B c \log (x) \]
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Rubi [A] time = 0.03, antiderivative size = 41, 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}{2 x^2}-\frac {a A}{3 x^3}-\frac {A c+b B}{x}+B c \log (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^4} \, dx &=\int \left (\frac {a A}{x^4}+\frac {A b+a B}{x^3}+\frac {b B+A c}{x^2}+\frac {B c}{x}\right ) \, dx\\ &=-\frac {a A}{3 x^3}-\frac {A b+a B}{2 x^2}-\frac {b B+A c}{x}+B c \log (x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 41, normalized size = 1.00 \begin {gather*} B c \log (x)-\frac {a (2 A+3 B x)+3 x (A b+2 A c x+2 b B x)}{6 x^3} \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^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 41, normalized size = 1.00 \begin {gather*} \frac {6 \, B c x^{3} \log \relax (x) - 6 \, {\left (B b + A c\right )} x^{2} - 2 \, A a - 3 \, {\left (B a + A b\right )} x}{6 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 39, normalized size = 0.95 \begin {gather*} B c \log \left ({\left | x \right |}\right ) - \frac {6 \, {\left (B b + A c\right )} x^{2} + 2 \, A a + 3 \, {\left (B a + A b\right )} x}{6 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 42, normalized size = 1.02 \begin {gather*} B c \ln \relax (x )-\frac {A c}{x}-\frac {B b}{x}-\frac {A b}{2 x^{2}}-\frac {B a}{2 x^{2}}-\frac {A a}{3 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 38, normalized size = 0.93 \begin {gather*} B c \log \relax (x) - \frac {6 \, {\left (B b + A c\right )} x^{2} + 2 \, A a + 3 \, {\left (B a + A b\right )} x}{6 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 38, normalized size = 0.93 \begin {gather*} B\,c\,\ln \relax (x)-\frac {\left (A\,c+B\,b\right )\,x^2+\left (\frac {A\,b}{2}+\frac {B\,a}{2}\right )\,x+\frac {A\,a}{3}}{x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.58, size = 44, normalized size = 1.07 \begin {gather*} B c \log {\relax (x )} + \frac {- 2 A a + x^{2} \left (- 6 A c - 6 B b\right ) + x \left (- 3 A b - 3 B a\right )}{6 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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