Optimal. Leaf size=101 \[ -\frac{3 c d^2-e (2 b d-a e)}{3 e^4 (d+e x)^3}+\frac{d \left (a e^2-b d e+c d^2\right )}{4 e^4 (d+e x)^4}+\frac{3 c d-b e}{2 e^4 (d+e x)^2}-\frac{c}{e^4 (d+e x)} \]
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Rubi [A] time = 0.0733396, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {771} \[ -\frac{3 c d^2-e (2 b d-a e)}{3 e^4 (d+e x)^3}+\frac{d \left (a e^2-b d e+c d^2\right )}{4 e^4 (d+e x)^4}+\frac{3 c d-b e}{2 e^4 (d+e x)^2}-\frac{c}{e^4 (d+e x)} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int \frac{x \left (a+b x+c x^2\right )}{(d+e x)^5} \, dx &=\int \left (-\frac{d \left (c d^2-b d e+a e^2\right )}{e^3 (d+e x)^5}+\frac{3 c d^2-e (2 b d-a e)}{e^3 (d+e x)^4}+\frac{-3 c d+b e}{e^3 (d+e x)^3}+\frac{c}{e^3 (d+e x)^2}\right ) \, dx\\ &=\frac{d \left (c d^2-b d e+a e^2\right )}{4 e^4 (d+e x)^4}-\frac{3 c d^2-e (2 b d-a e)}{3 e^4 (d+e x)^3}+\frac{3 c d-b e}{2 e^4 (d+e x)^2}-\frac{c}{e^4 (d+e x)}\\ \end{align*}
Mathematica [A] time = 0.0308944, size = 77, normalized size = 0.76 \[ -\frac{e \left (a e (d+4 e x)+b \left (d^2+4 d e x+6 e^2 x^2\right )\right )+3 c \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )}{12 e^4 (d+e x)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 93, normalized size = 0.9 \begin{align*} -{\frac{a{e}^{2}-2\,bde+3\,c{d}^{2}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{be-3\,cd}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{c}{{e}^{4} \left ( ex+d \right ) }}+{\frac{d \left ( a{e}^{2}-bde+c{d}^{2} \right ) }{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 = 1.06958, size = 157, normalized size = 1.55 \begin{align*} -\frac{12 \, c e^{3} x^{3} + 3 \, c d^{3} + b d^{2} e + a d e^{2} + 6 \,{\left (3 \, c d e^{2} + b e^{3}\right )} x^{2} + 4 \,{\left (3 \, c d^{2} e + b d e^{2} + a e^{3}\right )} x}{12 \,{\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.3139, size = 243, normalized size = 2.41 \begin{align*} -\frac{12 \, c e^{3} x^{3} + 3 \, c d^{3} + b d^{2} e + a d e^{2} + 6 \,{\left (3 \, c d e^{2} + b e^{3}\right )} x^{2} + 4 \,{\left (3 \, c d^{2} e + b d e^{2} + a e^{3}\right )} x}{12 \,{\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.57528, size = 124, normalized size = 1.23 \begin{align*} - \frac{a d e^{2} + b d^{2} e + 3 c d^{3} + 12 c e^{3} x^{3} + x^{2} \left (6 b e^{3} + 18 c d e^{2}\right ) + x \left (4 a e^{3} + 4 b d e^{2} + 12 c d^{2} e\right )}{12 d^{4} e^{4} + 48 d^{3} e^{5} x + 72 d^{2} e^{6} x^{2} + 48 d e^{7} x^{3} + 12 e^{8} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10768, size = 173, normalized size = 1.71 \begin{align*} -\frac{1}{12} \,{\left (\frac{12 \, c e^{\left (-1\right )}}{x e + d} - \frac{18 \, c d e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}} + \frac{12 \, c d^{2} e^{\left (-1\right )}}{{\left (x e + d\right )}^{3}} - \frac{3 \, c d^{3} e^{\left (-1\right )}}{{\left (x e + d\right )}^{4}} + \frac{6 \, b}{{\left (x e + d\right )}^{2}} - \frac{8 \, b d}{{\left (x e + d\right )}^{3}} + \frac{3 \, b d^{2}}{{\left (x e + d\right )}^{4}} + \frac{4 \, a e}{{\left (x e + d\right )}^{3}} - \frac{3 \, a d e}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-3\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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