Optimal. Leaf size=134 \[ \frac{A e (2 c d-b e)-B \left (3 c d^2-e (2 b d-a e)\right )}{5 e^4 (d+e x)^5}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )}{6 e^4 (d+e x)^6}+\frac{-A c e-b B e+3 B c d}{4 e^4 (d+e x)^4}-\frac{B c}{3 e^4 (d+e x)^3} \]
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Rubi [A] time = 0.102488, antiderivative size = 133, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {771} \[ -\frac{-B e (2 b d-a e)-A e (2 c d-b e)+3 B c d^2}{5 e^4 (d+e x)^5}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )}{6 e^4 (d+e x)^6}+\frac{-A c e-b B e+3 B c d}{4 e^4 (d+e x)^4}-\frac{B c}{3 e^4 (d+e x)^3} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a+b x+c x^2\right )}{(d+e x)^7} \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2-b d e+a e^2\right )}{e^3 (d+e x)^7}+\frac{3 B c d^2-B e (2 b d-a e)-A e (2 c d-b e)}{e^3 (d+e x)^6}+\frac{-3 B c d+b B e+A c e}{e^3 (d+e x)^5}+\frac{B c}{e^3 (d+e x)^4}\right ) \, dx\\ &=\frac{(B d-A e) \left (c d^2-b d e+a e^2\right )}{6 e^4 (d+e x)^6}-\frac{3 B c d^2-B e (2 b d-a e)-A e (2 c d-b e)}{5 e^4 (d+e x)^5}+\frac{3 B c d-b B e-A c e}{4 e^4 (d+e x)^4}-\frac{B c}{3 e^4 (d+e x)^3}\\ \end{align*}
Mathematica [A] time = 0.0673116, size = 119, normalized size = 0.89 \[ -\frac{A e \left (2 e (5 a e+b d+6 b e x)+c \left (d^2+6 d e x+15 e^2 x^2\right )\right )+B \left (e \left (2 a e (d+6 e x)+b \left (d^2+6 d e x+15 e^2 x^2\right )\right )+c \left (6 d^2 e x+d^3+15 d e^2 x^2+20 e^3 x^3\right )\right )}{60 e^4 (d+e x)^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 142, normalized size = 1.1 \begin{align*} -{\frac{aA{e}^{3}-Abd{e}^{2}+Ac{d}^{2}e-aBd{e}^{2}+B{d}^{2}be-Bc{d}^{3}}{6\,{e}^{4} \left ( ex+d \right ) ^{6}}}-{\frac{Ab{e}^{2}-2\,Acde+aB{e}^{2}-2\,Bbde+3\,Bc{d}^{2}}{5\,{e}^{4} \left ( ex+d \right ) ^{5}}}-{\frac{Bc}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{Ace+bBe-3\,Bcd}{4\,{e}^{4} \left ( ex+d \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05701, size = 240, normalized size = 1.79 \begin{align*} -\frac{20 \, B c e^{3} x^{3} + B c d^{3} + 10 \, A a e^{3} +{\left (B b + A c\right )} d^{2} e + 2 \,{\left (B a + A b\right )} d e^{2} + 15 \,{\left (B c d e^{2} +{\left (B b + A c\right )} e^{3}\right )} x^{2} + 6 \,{\left (B c d^{2} e +{\left (B b + A c\right )} d e^{2} + 2 \,{\left (B a + A b\right )} e^{3}\right )} x}{60 \,{\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.2487, size = 386, normalized size = 2.88 \begin{align*} -\frac{20 \, B c e^{3} x^{3} + B c d^{3} + 10 \, A a e^{3} +{\left (B b + A c\right )} d^{2} e + 2 \,{\left (B a + A b\right )} d e^{2} + 15 \,{\left (B c d e^{2} +{\left (B b + A c\right )} e^{3}\right )} x^{2} + 6 \,{\left (B c d^{2} e +{\left (B b + A c\right )} d e^{2} + 2 \,{\left (B a + A b\right )} e^{3}\right )} x}{60 \,{\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11216, size = 178, normalized size = 1.33 \begin{align*} -\frac{{\left (20 \, B c x^{3} e^{3} + 15 \, B c d x^{2} e^{2} + 6 \, B c d^{2} x e + B c d^{3} + 15 \, B b x^{2} e^{3} + 15 \, A c x^{2} e^{3} + 6 \, B b d x e^{2} + 6 \, A c d x e^{2} + B b d^{2} e + A c d^{2} e + 12 \, B a x e^{3} + 12 \, A b x e^{3} + 2 \, B a d e^{2} + 2 \, A b d e^{2} + 10 \, A a e^{3}\right )} e^{\left (-4\right )}}{60 \,{\left (x e + d\right )}^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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