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 = 0.0253187, antiderivative size = 41, normalized size of antiderivative = 1., 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} \[ -\frac{a B+A b}{2 x^2}-\frac{a A}{3 x^3}-\frac{A c+b B}{x}+B c \log (x) \]
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*}
Mathematica [A] time = 0.0272206, size = 41, normalized size = 1. \[ B c \log (x)-\frac{a (2 A+3 B x)+3 x (A b+2 A c x+2 b B x)}{6 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 42, normalized size = 1. \begin{align*} Bc\ln \left ( x \right ) -{\frac{aA}{3\,{x}^{3}}}-{\frac{Ab}{2\,{x}^{2}}}-{\frac{aB}{2\,{x}^{2}}}-{\frac{Ac}{x}}-{\frac{bB}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02508, size = 51, normalized size = 1.24 \begin{align*} B c \log \left (x\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.25606, size = 100, normalized size = 2.44 \begin{align*} \frac{6 \, B c x^{3} \log \left (x\right ) - 6 \,{\left (B b + A c\right )} x^{2} - 2 \, A a - 3 \,{\left (B a + A b\right )} x}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.42573, size = 41, normalized size = 1. \begin{align*} B c \log{\left (x \right )} - \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2402, size = 53, normalized size = 1.29 \begin{align*} 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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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