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 = 0.03, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, 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*}
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Mathematica [A] time = 0.03, size = 44, normalized size = 1.00 \[ -\frac {A b^2}{2 x^2}-\frac {b (2 A c+b B)}{x}+c \log (x) (A c+2 b B)+B c^2 x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 53, normalized size = 1.20 \[ \frac {2 \, B c^{2} x^{3} + 2 \, {\left (2 \, B b c + A c^{2}\right )} x^{2} \log \relax (x) - A b^{2} - 2 \, {\left (B b^{2} + 2 \, A b c\right )} x}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 47, normalized size = 1.07 \[ B c^{2} x + {\left (2 \, B b c + A c^{2}\right )} \log \left ({\left | x \right |}\right ) - \frac {A b^{2} + 2 \, {\left (B b^{2} + 2 \, A b c\right )} x}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 48, normalized size = 1.09 \[ A \,c^{2} \ln \relax (x )+2 B b c \ln \relax (x )+B \,c^{2} x -\frac {2 A b c}{x}-\frac {B \,b^{2}}{x}-\frac {A \,b^{2}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.93, size = 46, normalized size = 1.05 \[ B c^{2} x + {\left (2 \, B b c + A c^{2}\right )} \log \relax (x) - \frac {A b^{2} + 2 \, {\left (B b^{2} + 2 \, A b c\right )} x}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.04, size = 46, normalized size = 1.05 \[ \ln \relax (x)\,\left (A\,c^2+2\,B\,b\,c\right )-\frac {\frac {A\,b^2}{2}+x\,\left (B\,b^2+2\,A\,c\,b\right )}{x^2}+B\,c^2\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.33, size = 46, normalized size = 1.05 \[ B c^{2} x + c \left (A c + 2 B b\right ) \log {\relax (x )} + \frac {- A b^{2} + x \left (- 4 A b c - 2 B b^{2}\right )}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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