Optimal. Leaf size=69 \[ -\frac{(A b-a B) (b d-a e)}{2 b^3 (a+b x)^2}-\frac{-2 a B e+A b e+b B d}{b^3 (a+b x)}+\frac{B e \log (a+b x)}{b^3} \]
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Rubi [A] time = 0.0550578, antiderivative size = 69, 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} \[ -\frac{(A b-a B) (b d-a e)}{2 b^3 (a+b x)^2}-\frac{-2 a B e+A b e+b B d}{b^3 (a+b x)}+\frac{B e \log (a+b x)}{b^3} \]
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)^3} \, dx &=\int \left (\frac{(A b-a B) (b d-a e)}{b^2 (a+b x)^3}+\frac{b B d+A b e-2 a B e}{b^2 (a+b x)^2}+\frac{B e}{b^2 (a+b x)}\right ) \, dx\\ &=-\frac{(A b-a B) (b d-a e)}{2 b^3 (a+b x)^2}-\frac{b B d+A b e-2 a B e}{b^3 (a+b x)}+\frac{B e \log (a+b x)}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0331609, size = 75, normalized size = 1.09 \[ \frac{B \left (3 a^2 e-a b d+4 a b e x-2 b^2 d x\right )-A b (a e+b d+2 b e x)+2 B e (a+b x)^2 \log (a+b x)}{2 b^3 (a+b x)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 118, normalized size = 1.7 \begin{align*}{\frac{Be\ln \left ( bx+a \right ) }{{b}^{3}}}-{\frac{Ae}{{b}^{2} \left ( bx+a \right ) }}+2\,{\frac{Bae}{{b}^{3} \left ( bx+a \right ) }}-{\frac{Bd}{{b}^{2} \left ( bx+a \right ) }}+{\frac{Aae}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}}-{\frac{Ad}{2\,b \left ( bx+a \right ) ^{2}}}-{\frac{B{a}^{2}e}{2\,{b}^{3} \left ( bx+a \right ) ^{2}}}+{\frac{Bad}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.19524, size = 124, normalized size = 1.8 \begin{align*} -\frac{{\left (B a b + A b^{2}\right )} d -{\left (3 \, B a^{2} - A a b\right )} e + 2 \,{\left (B b^{2} d -{\left (2 \, B a b - A b^{2}\right )} e\right )} x}{2 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} + \frac{B e \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 = 1.53219, size = 234, normalized size = 3.39 \begin{align*} -\frac{{\left (B a b + A b^{2}\right )} d -{\left (3 \, B a^{2} - A a b\right )} e + 2 \,{\left (B b^{2} d -{\left (2 \, B a b - A b^{2}\right )} e\right )} x - 2 \,{\left (B b^{2} e x^{2} + 2 \, B a b e x + B a^{2} e\right )} \log \left (b x + a\right )}{2 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.05323, size = 94, normalized size = 1.36 \begin{align*} \frac{B e \log{\left (a + b x \right )}}{b^{3}} + \frac{- A a b e - A b^{2} d + 3 B a^{2} e - B a b d + x \left (- 2 A b^{2} e + 4 B a b e - 2 B b^{2} d\right )}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.7912, size = 104, normalized size = 1.51 \begin{align*} \frac{B e \log \left ({\left | b x + a \right |}\right )}{b^{3}} - \frac{2 \,{\left (B b d - 2 \, B a e + A b e\right )} x + \frac{B a b d + A b^{2} d - 3 \, B a^{2} e + A a b e}{b}}{2 \,{\left (b x + a\right )}^{2} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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