Optimal. Leaf size=60 \[ \frac{(b d-a e) (B d-A e) \log (d+e x)}{e^3}+\frac{B (a+b x)^2}{2 b e}-\frac{b x (B d-A e)}{e^2} \]
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Rubi [A] time = 0.0422003, antiderivative size = 60, 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 = {77} \[ \frac{(b d-a e) (B d-A e) \log (d+e x)}{e^3}+\frac{B (a+b x)^2}{2 b e}-\frac{b x (B d-A e)}{e^2} \]
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} \, dx &=\int \left (\frac{b (-B d+A e)}{e^2}+\frac{B (a+b x)}{e}+\frac{(-b d+a e) (-B d+A e)}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac{b (B d-A e) x}{e^2}+\frac{B (a+b x)^2}{2 b e}+\frac{(b d-a e) (B d-A e) \log (d+e x)}{e^3}\\ \end{align*}
Mathematica [A] time = 0.0265425, size = 56, normalized size = 0.93 \[ \frac{e x (2 a B e+b (2 A e-2 B d+B e x))+2 (b d-a e) (B d-A e) \log (d+e x)}{2 e^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 90, normalized size = 1.5 \begin{align*}{\frac{bB{x}^{2}}{2\,e}}+{\frac{Abx}{e}}+{\frac{Bax}{e}}-{\frac{Bbdx}{{e}^{2}}}+{\frac{\ln \left ( ex+d \right ) aA}{e}}-{\frac{\ln \left ( ex+d \right ) Abd}{{e}^{2}}}-{\frac{\ln \left ( ex+d \right ) Bad}{{e}^{2}}}+{\frac{\ln \left ( ex+d \right ) bB{d}^{2}}{{e}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.122, size = 89, normalized size = 1.48 \begin{align*} \frac{B b e x^{2} - 2 \,{\left (B b d -{\left (B a + A b\right )} e\right )} x}{2 \, e^{2}} + \frac{{\left (B b d^{2} + A a e^{2} -{\left (B a + A b\right )} d e\right )} \log \left (e x + d\right )}{e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67598, size = 151, normalized size = 2.52 \begin{align*} \frac{B b e^{2} x^{2} - 2 \,{\left (B b d e -{\left (B a + A b\right )} e^{2}\right )} x + 2 \,{\left (B b d^{2} + A a e^{2} -{\left (B a + A b\right )} d e\right )} \log \left (e x + d\right )}{2 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.47639, size = 53, normalized size = 0.88 \begin{align*} \frac{B b x^{2}}{2 e} + \frac{x \left (A b e + B a e - B b d\right )}{e^{2}} - \frac{\left (- A e + B d\right ) \left (a e - b d\right ) \log{\left (d + e x \right )}}{e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.62367, size = 96, normalized size = 1.6 \begin{align*}{\left (B b d^{2} - B a d e - A b d e + A a e^{2}\right )} e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (B b x^{2} e - 2 \, B b d x + 2 \, B a x e + 2 \, A b x e\right )} e^{\left (-2\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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