Optimal. Leaf size=77 \[ \frac{-a B e-A b e+2 b B d}{4 e^3 (d+e x)^4}-\frac{(b d-a e) (B d-A e)}{5 e^3 (d+e x)^5}-\frac{b B}{3 e^3 (d+e x)^3} \]
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Rubi [A] time = 0.0489671, antiderivative size = 77, 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} \[ \frac{-a B e-A b e+2 b B d}{4 e^3 (d+e x)^4}-\frac{(b d-a e) (B d-A e)}{5 e^3 (d+e x)^5}-\frac{b B}{3 e^3 (d+e x)^3} \]
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)^6} \, dx &=\int \left (\frac{(-b d+a e) (-B d+A e)}{e^2 (d+e x)^6}+\frac{-2 b B d+A b e+a B e}{e^2 (d+e x)^5}+\frac{b B}{e^2 (d+e x)^4}\right ) \, dx\\ &=-\frac{(b d-a e) (B d-A e)}{5 e^3 (d+e x)^5}+\frac{2 b B d-A b e-a B e}{4 e^3 (d+e x)^4}-\frac{b B}{3 e^3 (d+e x)^3}\\ \end{align*}
Mathematica [A] time = 0.0331288, size = 65, normalized size = 0.84 \[ -\frac{3 a e (4 A e+B (d+5 e x))+b \left (3 A e (d+5 e x)+2 B \left (d^2+5 d e x+10 e^2 x^2\right )\right )}{60 e^3 (d+e x)^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 79, normalized size = 1. \begin{align*} -{\frac{Bb}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{aA{e}^{2}-Adbe-Bdae+bB{d}^{2}}{5\,{e}^{3} \left ( ex+d \right ) ^{5}}}-{\frac{Abe+Bae-2\,Bbd}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08327, size = 158, normalized size = 2.05 \begin{align*} -\frac{20 \, B b e^{2} x^{2} + 2 \, B b d^{2} + 12 \, A a e^{2} + 3 \,{\left (B a + A b\right )} d e + 5 \,{\left (2 \, B b d e + 3 \,{\left (B a + A b\right )} e^{2}\right )} x}{60 \,{\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.855, size = 255, normalized size = 3.31 \begin{align*} -\frac{20 \, B b e^{2} x^{2} + 2 \, B b d^{2} + 12 \, A a e^{2} + 3 \,{\left (B a + A b\right )} d e + 5 \,{\left (2 \, B b d e + 3 \,{\left (B a + A b\right )} e^{2}\right )} x}{60 \,{\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.12953, size = 134, normalized size = 1.74 \begin{align*} - \frac{12 A a e^{2} + 3 A b d e + 3 B a d e + 2 B b d^{2} + 20 B b e^{2} x^{2} + x \left (15 A b e^{2} + 15 B a e^{2} + 10 B b d e\right )}{60 d^{5} e^{3} + 300 d^{4} e^{4} x + 600 d^{3} e^{5} x^{2} + 600 d^{2} e^{6} x^{3} + 300 d e^{7} x^{4} + 60 e^{8} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.05032, size = 96, normalized size = 1.25 \begin{align*} -\frac{{\left (20 \, B b x^{2} e^{2} + 10 \, B b d x e + 2 \, B b d^{2} + 15 \, B a x e^{2} + 15 \, A b x e^{2} + 3 \, B a d e + 3 \, A b d e + 12 \, A a e^{2}\right )} e^{\left (-3\right )}}{60 \,{\left (x e + d\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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