Optimal. Leaf size=70 \[ -\frac{(b d-a e) (B d-A e)}{2 e^3 (d+e x)^2}+\frac{-a B e-A b e+2 b B d}{e^3 (d+e x)}+\frac{b B \log (d+e x)}{e^3} \]
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Rubi [A] time = 0.0499388, antiderivative size = 70, 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} \[ -\frac{(b d-a e) (B d-A e)}{2 e^3 (d+e x)^2}+\frac{-a B e-A b e+2 b B d}{e^3 (d+e x)}+\frac{b B \log (d+e x)}{e^3} \]
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)^3} \, dx &=\int \left (\frac{(-b d+a e) (-B d+A e)}{e^2 (d+e x)^3}+\frac{-2 b B d+A b e+a B e}{e^2 (d+e x)^2}+\frac{b B}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac{(b d-a e) (B d-A e)}{2 e^3 (d+e x)^2}+\frac{2 b B d-A b e-a B e}{e^3 (d+e x)}+\frac{b B \log (d+e x)}{e^3}\\ \end{align*}
Mathematica [A] time = 0.0362283, size = 72, normalized size = 1.03 \[ \frac{-a e (A e+B (d+2 e x))+b (B d (3 d+4 e x)-A e (d+2 e x))+2 b B (d+e x)^2 \log (d+e x)}{2 e^3 (d+e x)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 118, normalized size = 1.7 \begin{align*} -{\frac{Aa}{2\,e \left ( ex+d \right ) ^{2}}}+{\frac{Adb}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}+{\frac{Bda}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{bB{d}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{Ab}{{e}^{2} \left ( ex+d \right ) }}-{\frac{Ba}{{e}^{2} \left ( ex+d \right ) }}+2\,{\frac{Bbd}{{e}^{3} \left ( ex+d \right ) }}+{\frac{Bb\ln \left ( ex+d \right ) }{{e}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01484, size = 117, normalized size = 1.67 \begin{align*} \frac{3 \, B b d^{2} - A a e^{2} -{\left (B a + A b\right )} d e + 2 \,{\left (2 \, B b d e -{\left (B a + A b\right )} e^{2}\right )} x}{2 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} + \frac{B b \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 = 1.85916, size = 227, normalized size = 3.24 \begin{align*} \frac{3 \, B b d^{2} - A a e^{2} -{\left (B a + A b\right )} d e + 2 \,{\left (2 \, B b d e -{\left (B a + A b\right )} e^{2}\right )} x + 2 \,{\left (B b e^{2} x^{2} + 2 \, B b d e x + B b d^{2}\right )} \log \left (e x + d\right )}{2 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.36353, size = 94, normalized size = 1.34 \begin{align*} \frac{B b \log{\left (d + e x \right )}}{e^{3}} - \frac{A a e^{2} + A b d e + B a d e - 3 B b d^{2} + x \left (2 A b e^{2} + 2 B a e^{2} - 4 B b d e\right )}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.26373, size = 107, normalized size = 1.53 \begin{align*} B b e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{{\left (2 \,{\left (2 \, B b d - B a e - A b e\right )} x +{\left (3 \, B b d^{2} - B a d e - A b d e - A a e^{2}\right )} e^{\left (-1\right )}\right )} e^{\left (-2\right )}}{2 \,{\left (x e + d\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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