Optimal. Leaf size=31 \[ -\frac{A c+b B}{2 x^2}-\frac{A b}{3 x^3}-\frac{B c}{x} \]
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Rubi [A] time = 0.0166305, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {765} \[ -\frac{A c+b B}{2 x^2}-\frac{A b}{3 x^3}-\frac{B c}{x} \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (b x+c x^2\right )}{x^5} \, dx &=\int \left (\frac{A b}{x^4}+\frac{b B+A c}{x^3}+\frac{B c}{x^2}\right ) \, dx\\ &=-\frac{A b}{3 x^3}-\frac{b B+A c}{2 x^2}-\frac{B c}{x}\\ \end{align*}
Mathematica [A] time = 0.008783, size = 28, normalized size = 0.9 \[ -\frac{A (2 b+3 c x)+3 B x (b+2 c x)}{6 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 28, normalized size = 0.9 \begin{align*} -{\frac{Bc}{x}}-{\frac{Ab}{3\,{x}^{3}}}-{\frac{Ac+bB}{2\,{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02574, size = 36, normalized size = 1.16 \begin{align*} -\frac{6 \, B c x^{2} + 2 \, A b + 3 \,{\left (B b + A c\right )} x}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6412, size = 65, normalized size = 2.1 \begin{align*} -\frac{6 \, B c x^{2} + 2 \, A b + 3 \,{\left (B b + A c\right )} x}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.435433, size = 31, normalized size = 1. \begin{align*} - \frac{2 A b + 6 B c x^{2} + x \left (3 A c + 3 B b\right )}{6 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14418, size = 36, normalized size = 1.16 \begin{align*} -\frac{6 \, B c x^{2} + 3 \, B b x + 3 \, A c x + 2 \, A b}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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