Optimal. Leaf size=44 \[ -\frac{A b^2}{2 x^2}-\frac{b (2 A c+b B)}{x}+c \log (x) (A c+2 b B)+B c^2 x \]
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Rubi [A] time = 0.0299489, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {765} \[ -\frac{A b^2}{2 x^2}-\frac{b (2 A c+b B)}{x}+c \log (x) (A c+2 b B)+B c^2 x \]
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 )^2}{x^5} \, dx &=\int \left (B c^2+\frac{A b^2}{x^3}+\frac{b (b B+2 A c)}{x^2}+\frac{c (2 b B+A c)}{x}\right ) \, dx\\ &=-\frac{A b^2}{2 x^2}-\frac{b (b B+2 A c)}{x}+B c^2 x+c (2 b B+A c) \log (x)\\ \end{align*}
Mathematica [A] time = 0.0252772, size = 44, normalized size = 1. \[ -\frac{A b^2}{2 x^2}-\frac{b (2 A c+b B)}{x}+c \log (x) (A c+2 b B)+B c^2 x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 48, normalized size = 1.1 \begin{align*} B{c}^{2}x+A\ln \left ( x \right ){c}^{2}+2\,B\ln \left ( x \right ) bc-2\,{\frac{Abc}{x}}-{\frac{{b}^{2}B}{x}}-{\frac{A{b}^{2}}{2\,{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16344, size = 62, normalized size = 1.41 \begin{align*} B c^{2} x +{\left (2 \, B b c + A c^{2}\right )} \log \left (x\right ) - \frac{A b^{2} + 2 \,{\left (B b^{2} + 2 \, A b c\right )} x}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76216, size = 119, normalized size = 2.7 \begin{align*} \frac{2 \, B c^{2} x^{3} + 2 \,{\left (2 \, B b c + A c^{2}\right )} x^{2} \log \left (x\right ) - A b^{2} - 2 \,{\left (B b^{2} + 2 \, A b c\right )} x}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.565436, size = 44, normalized size = 1. \begin{align*} B c^{2} x + c \left (A c + 2 B b\right ) \log{\left (x \right )} - \frac{A b^{2} + x \left (4 A b c + 2 B b^{2}\right )}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15981, size = 63, normalized size = 1.43 \begin{align*} B c^{2} x +{\left (2 \, B b c + A c^{2}\right )} \log \left ({\left | x \right |}\right ) - \frac{A b^{2} + 2 \,{\left (B b^{2} + 2 \, A b c\right )} x}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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