Optimal. Leaf size=99 \[ \frac {d (B d-A e) (c d-b e)}{e^4 (d+e x)}+\frac {\log (d+e x) (B d (3 c d-2 b e)-A e (2 c d-b e))}{e^4}-\frac {x (-A c e-b B e+2 B c d)}{e^3}+\frac {B c x^2}{2 e^2} \]
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Rubi [A] time = 0.11, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {771} \[ -\frac {x (-A c e-b B e+2 B c d)}{e^3}+\frac {d (B d-A e) (c d-b e)}{e^4 (d+e x)}+\frac {\log (d+e x) (B d (3 c d-2 b e)-A e (2 c d-b e))}{e^4}+\frac {B c x^2}{2 e^2} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )}{(d+e x)^2} \, dx &=\int \left (\frac {-2 B c d+b B e+A c e}{e^3}+\frac {B c x}{e^2}-\frac {d (B d-A e) (c d-b e)}{e^3 (d+e x)^2}+\frac {B d (3 c d-2 b e)-A e (2 c d-b e)}{e^3 (d+e x)}\right ) \, dx\\ &=-\frac {(2 B c d-b B e-A c e) x}{e^3}+\frac {B c x^2}{2 e^2}+\frac {d (B d-A e) (c d-b e)}{e^4 (d+e x)}+\frac {(B d (3 c d-2 b e)-A e (2 c d-b e)) \log (d+e x)}{e^4}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 93, normalized size = 0.94 \[ \frac {2 e x (A c e+b B e-2 B c d)+\frac {2 d (B d-A e) (c d-b e)}{d+e x}+2 \log (d+e x) (A e (b e-2 c d)+B d (3 c d-2 b e))+B c e^2 x^2}{2 e^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 170, normalized size = 1.72 \[ \frac {B c e^{3} x^{3} + 2 \, B c d^{3} + 2 \, A b d e^{2} - 2 \, {\left (B b + A c\right )} d^{2} e - {\left (3 \, B c d e^{2} - 2 \, {\left (B b + A c\right )} e^{3}\right )} x^{2} - 2 \, {\left (2 \, B c d^{2} e - {\left (B b + A c\right )} d e^{2}\right )} x + 2 \, {\left (3 \, B c d^{3} + A b d e^{2} - 2 \, {\left (B b + A c\right )} d^{2} e + {\left (3 \, B c d^{2} e + A b e^{3} - 2 \, {\left (B b + A c\right )} d e^{2}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (e^{5} x + d e^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 167, normalized size = 1.69 \[ \frac {1}{2} \, {\left (B c - \frac {2 \, {\left (3 \, B c d e - B b e^{2} - A c e^{2}\right )} e^{\left (-1\right )}}{x e + d}\right )} {\left (x e + d\right )}^{2} e^{\left (-4\right )} - {\left (3 \, B c d^{2} - 2 \, B b d e - 2 \, A c d e + A b e^{2}\right )} e^{\left (-4\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + {\left (\frac {B c d^{3} e^{2}}{x e + d} - \frac {B b d^{2} e^{3}}{x e + d} - \frac {A c d^{2} e^{3}}{x e + d} + \frac {A b d e^{4}}{x e + d}\right )} e^{\left (-6\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 155, normalized size = 1.57 \[ \frac {B c \,x^{2}}{2 e^{2}}+\frac {A b d}{\left (e x +d \right ) e^{2}}+\frac {A b \ln \left (e x +d \right )}{e^{2}}-\frac {A c \,d^{2}}{\left (e x +d \right ) e^{3}}-\frac {2 A c d \ln \left (e x +d \right )}{e^{3}}+\frac {A c x}{e^{2}}-\frac {B b \,d^{2}}{\left (e x +d \right ) e^{3}}-\frac {2 B b d \ln \left (e x +d \right )}{e^{3}}+\frac {B b x}{e^{2}}+\frac {B c \,d^{3}}{\left (e x +d \right ) e^{4}}+\frac {3 B c \,d^{2} \ln \left (e x +d \right )}{e^{4}}-\frac {2 B c d x}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 109, normalized size = 1.10 \[ \frac {B c d^{3} + A b d e^{2} - {\left (B b + A c\right )} d^{2} e}{e^{5} x + d e^{4}} + \frac {B c e x^{2} - 2 \, {\left (2 \, B c d - {\left (B b + A c\right )} e\right )} x}{2 \, e^{3}} + \frac {{\left (3 \, B c d^{2} + A b e^{2} - 2 \, {\left (B b + A c\right )} d e\right )} \log \left (e x + d\right )}{e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.39, size = 116, normalized size = 1.17 \[ x\,\left (\frac {A\,c+B\,b}{e^2}-\frac {2\,B\,c\,d}{e^3}\right )+\frac {\ln \left (d+e\,x\right )\,\left (A\,b\,e^2+3\,B\,c\,d^2-2\,A\,c\,d\,e-2\,B\,b\,d\,e\right )}{e^4}+\frac {B\,c\,d^3+A\,b\,d\,e^2-A\,c\,d^2\,e-B\,b\,d^2\,e}{e\,\left (x\,e^4+d\,e^3\right )}+\frac {B\,c\,x^2}{2\,e^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.66, size = 121, normalized size = 1.22 \[ \frac {B c x^{2}}{2 e^{2}} + x \left (\frac {A c}{e^{2}} + \frac {B b}{e^{2}} - \frac {2 B c d}{e^{3}}\right ) + \frac {A b d e^{2} - A c d^{2} e - B b d^{2} e + B c d^{3}}{d e^{4} + e^{5} x} - \frac {\left (- A b e^{2} + 2 A c d e + 2 B b d e - 3 B c d^{2}\right ) \log {\left (d + e x \right )}}{e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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