Optimal. Leaf size=75 \[ \frac{-a B e-A b e+2 b B d}{2 e^3 (d+e x)^2}-\frac{(b d-a e) (B d-A e)}{3 e^3 (d+e x)^3}-\frac{b B}{e^3 (d+e x)} \]
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Rubi [A] time = 0.0490479, antiderivative size = 75, 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} \[ \frac{-a B e-A b e+2 b B d}{2 e^3 (d+e x)^2}-\frac{(b d-a e) (B d-A e)}{3 e^3 (d+e x)^3}-\frac{b B}{e^3 (d+e x)} \]
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)^4} \, dx &=\int \left (\frac{(-b d+a e) (-B d+A e)}{e^2 (d+e x)^4}+\frac{-2 b B d+A b e+a B e}{e^2 (d+e x)^3}+\frac{b B}{e^2 (d+e x)^2}\right ) \, dx\\ &=-\frac{(b d-a e) (B d-A e)}{3 e^3 (d+e x)^3}+\frac{2 b B d-A b e-a B e}{2 e^3 (d+e x)^2}-\frac{b B}{e^3 (d+e x)}\\ \end{align*}
Mathematica [A] time = 0.0312522, size = 63, normalized size = 0.84 \[ -\frac{a e (2 A e+B (d+3 e x))+b \left (A e (d+3 e x)+2 B \left (d^2+3 d e x+3 e^2 x^2\right )\right )}{6 e^3 (d+e x)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 79, normalized size = 1.1 \begin{align*} -{\frac{aA{e}^{2}-Adbe-Bdae+bB{d}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{Abe+Bae-2\,Bbd}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{Bb}{{e}^{3} \left ( ex+d \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.20904, size = 126, normalized size = 1.68 \begin{align*} -\frac{6 \, B b e^{2} x^{2} + 2 \, B b d^{2} + 2 \, A a e^{2} +{\left (B a + A b\right )} d e + 3 \,{\left (2 \, B b d e +{\left (B a + A b\right )} e^{2}\right )} x}{6 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81589, size = 200, normalized size = 2.67 \begin{align*} -\frac{6 \, B b e^{2} x^{2} + 2 \, B b d^{2} + 2 \, A a e^{2} +{\left (B a + A b\right )} d e + 3 \,{\left (2 \, B b d e +{\left (B a + A b\right )} e^{2}\right )} x}{6 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.15033, size = 107, normalized size = 1.43 \begin{align*} - \frac{2 A a e^{2} + A b d e + B a d e + 2 B b d^{2} + 6 B b e^{2} x^{2} + x \left (3 A b e^{2} + 3 B a e^{2} + 6 B b d e\right )}{6 d^{3} e^{3} + 18 d^{2} e^{4} x + 18 d e^{5} x^{2} + 6 e^{6} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.9704, size = 93, normalized size = 1.24 \begin{align*} -\frac{{\left (6 \, B b x^{2} e^{2} + 6 \, B b d x e + 2 \, B b d^{2} + 3 \, B a x e^{2} + 3 \, A b x e^{2} + B a d e + A b d e + 2 \, A a e^{2}\right )} e^{\left (-3\right )}}{6 \,{\left (x e + d\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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