Optimal. Leaf size=59 \[ \frac{(A b-a B) (b d-a e) \log (a+b x)}{b^3}+\frac{B x (b d-a e)}{b^2}+\frac{e (A+B x)^2}{2 b B} \]
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Rubi [A] time = 0.0410191, antiderivative size = 59, 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} \[ \frac{(A b-a B) (b d-a e) \log (a+b x)}{b^3}+\frac{B x (b d-a e)}{b^2}+\frac{e (A+B x)^2}{2 b B} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{(A+B x) (d+e x)}{a+b x} \, dx &=\int \left (\frac{B (b d-a e)}{b^2}+\frac{(A b-a B) (b d-a e)}{b^2 (a+b x)}+\frac{e (A+B x)}{b}\right ) \, dx\\ &=\frac{B (b d-a e) x}{b^2}+\frac{e (A+B x)^2}{2 b B}+\frac{(A b-a B) (b d-a e) \log (a+b x)}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0228887, size = 56, normalized size = 0.95 \[ \frac{b x (b (2 A e+2 B d+B e x)-2 a B e)+2 (A b-a B) (b d-a e) \log (a+b x)}{2 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 90, normalized size = 1.5 \begin{align*}{\frac{B{x}^{2}e}{2\,b}}+{\frac{Aex}{b}}-{\frac{Baex}{{b}^{2}}}+{\frac{Bdx}{b}}-{\frac{\ln \left ( bx+a \right ) Aae}{{b}^{2}}}+{\frac{\ln \left ( bx+a \right ) Ad}{b}}+{\frac{\ln \left ( bx+a \right ) B{a}^{2}e}{{b}^{3}}}-{\frac{\ln \left ( bx+a \right ) Bad}{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08111, size = 97, normalized size = 1.64 \begin{align*} \frac{B b e x^{2} + 2 \,{\left (B b d -{\left (B a - A b\right )} e\right )} x}{2 \, b^{2}} - \frac{{\left ({\left (B a b - A b^{2}\right )} d -{\left (B a^{2} - A a b\right )} e\right )} \log \left (b x + a\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4929, size = 157, normalized size = 2.66 \begin{align*} \frac{B b^{2} e x^{2} + 2 \,{\left (B b^{2} d -{\left (B a b - A b^{2}\right )} e\right )} x - 2 \,{\left ({\left (B a b - A b^{2}\right )} d -{\left (B a^{2} - A a b\right )} e\right )} \log \left (b x + a\right )}{2 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.43304, size = 53, normalized size = 0.9 \begin{align*} \frac{B e x^{2}}{2 b} - \frac{x \left (- A b e + B a e - B b d\right )}{b^{2}} + \frac{\left (- A b + B a\right ) \left (a e - b d\right ) \log{\left (a + b x \right )}}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 3.62051, size = 100, normalized size = 1.69 \begin{align*} \frac{B b x^{2} e + 2 \, B b d x - 2 \, B a x e + 2 \, A b x e}{2 \, b^{2}} - \frac{{\left (B a b d - A b^{2} d - B a^{2} e + A a b e\right )} \log \left ({\left | b x + a \right |}\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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