Optimal. Leaf size=63 \[ -\frac {(b d-a e) (B d-A e)}{e^3 (d+e x)}-\frac {\log (d+e x) (-a B e-A b e+2 b B d)}{e^3}+\frac {b B x}{e^2} \]
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Rubi [A] time = 0.05, antiderivative size = 63, normalized size of antiderivative = 1.00, 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 = {77} \[ -\frac {(b d-a e) (B d-A e)}{e^3 (d+e x)}-\frac {\log (d+e x) (-a B e-A b e+2 b B d)}{e^3}+\frac {b B x}{e^2} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {(a+b x) (A+B x)}{(d+e x)^2} \, dx &=\int \left (\frac {b B}{e^2}+\frac {(-b d+a e) (-B d+A e)}{e^2 (d+e x)^2}+\frac {-2 b B d+A b e+a B e}{e^2 (d+e x)}\right ) \, dx\\ &=\frac {b B x}{e^2}-\frac {(b d-a e) (B d-A e)}{e^3 (d+e x)}-\frac {(2 b B d-A b e-a B e) \log (d+e x)}{e^3}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 56, normalized size = 0.89 \[ \frac {-\frac {(b d-a e) (B d-A e)}{d+e x}+\log (d+e x) (a B e+A b e-2 b B d)+b B e x}{e^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 102, normalized size = 1.62 \[ \frac {B b e^{2} x^{2} + B b d e x - B b d^{2} - A a e^{2} + {\left (B a + A b\right )} d e - {\left (2 \, B b d^{2} - {\left (B a + A b\right )} d e + {\left (2 \, B b d e - {\left (B a + A b\right )} e^{2}\right )} x\right )} \log \left (e x + d\right )}{e^{4} x + d e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.19, size = 116, normalized size = 1.84 \[ {\left (x e + d\right )} B b e^{\left (-3\right )} + {\left (2 \, B b d - B a e - A b e\right )} e^{\left (-3\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - {\left (\frac {B b d^{2} e}{x e + d} - \frac {B a d e^{2}}{x e + d} - \frac {A b d e^{2}}{x e + d} + \frac {A a e^{3}}{x e + d}\right )} e^{\left (-4\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 106, normalized size = 1.68 \[ -\frac {A a}{\left (e x +d \right ) e}+\frac {A b d}{\left (e x +d \right ) e^{2}}+\frac {A b \ln \left (e x +d \right )}{e^{2}}+\frac {B a d}{\left (e x +d \right ) e^{2}}+\frac {B a \ln \left (e x +d \right )}{e^{2}}-\frac {B b \,d^{2}}{\left (e x +d \right ) e^{3}}-\frac {2 B b d \ln \left (e x +d \right )}{e^{3}}+\frac {B b x}{e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 74, normalized size = 1.17 \[ \frac {B b x}{e^{2}} - \frac {B b d^{2} + A a e^{2} - {\left (B a + A b\right )} d e}{e^{4} x + d e^{3}} - \frac {{\left (2 \, B b d - {\left (B a + A b\right )} e\right )} \log \left (e x + d\right )}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.09, size = 75, normalized size = 1.19 \[ \frac {\ln \left (d+e\,x\right )\,\left (A\,b\,e+B\,a\,e-2\,B\,b\,d\right )}{e^3}-\frac {A\,a\,e^2+B\,b\,d^2-A\,b\,d\,e-B\,a\,d\,e}{e\,\left (x\,e^3+d\,e^2\right )}+\frac {B\,b\,x}{e^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.46, size = 71, normalized size = 1.13 \[ \frac {B b x}{e^{2}} + \frac {- A a e^{2} + A b d e + B a d e - B b d^{2}}{d e^{3} + e^{4} x} + \frac {\left (A b e + B a e - 2 B b d\right ) \log {\left (d + e x \right )}}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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