Optimal. Leaf size=67 \[ -\frac{a e^2-b d e+c d^2}{3 e^3 (d+e x)^3}+\frac{2 c d-b e}{2 e^3 (d+e x)^2}-\frac{c}{e^3 (d+e x)} \]
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Rubi [A] time = 0.0443593, antiderivative size = 67, 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 = {698} \[ -\frac{a e^2-b d e+c d^2}{3 e^3 (d+e x)^3}+\frac{2 c d-b e}{2 e^3 (d+e x)^2}-\frac{c}{e^3 (d+e x)} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int \frac{a+b x+c x^2}{(d+e x)^4} \, dx &=\int \left (\frac{c d^2-b d e+a e^2}{e^2 (d+e x)^4}+\frac{-2 c d+b e}{e^2 (d+e x)^3}+\frac{c}{e^2 (d+e x)^2}\right ) \, dx\\ &=-\frac{c d^2-b d e+a e^2}{3 e^3 (d+e x)^3}+\frac{2 c d-b e}{2 e^3 (d+e x)^2}-\frac{c}{e^3 (d+e x)}\\ \end{align*}
Mathematica [A] time = 0.0194428, size = 50, normalized size = 0.75 \[ -\frac{e (2 a e+b (d+3 e x))+2 c \left (d^2+3 d e x+3 e^2 x^2\right )}{6 e^3 (d+e x)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 63, normalized size = 0.9 \begin{align*} -{\frac{be-2\,cd}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{c}{{e}^{3} \left ( ex+d \right ) }}-{\frac{a{e}^{2}-bde+c{d}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.992108, size = 104, normalized size = 1.55 \begin{align*} -\frac{6 \, c e^{2} x^{2} + 2 \, c d^{2} + b d e + 2 \, a e^{2} + 3 \,{\left (2 \, c d e + b 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 = 2.029, size = 162, normalized size = 2.42 \begin{align*} -\frac{6 \, c e^{2} x^{2} + 2 \, c d^{2} + b d e + 2 \, a e^{2} + 3 \,{\left (2 \, c d e + b 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 = 1.02867, size = 82, normalized size = 1.22 \begin{align*} - \frac{2 a e^{2} + b d e + 2 c d^{2} + 6 c e^{2} x^{2} + x \left (3 b e^{2} + 6 c 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 = 1.13463, size = 68, normalized size = 1.01 \begin{align*} -\frac{{\left (6 \, c x^{2} e^{2} + 6 \, c d x e + 2 \, c d^{2} + 3 \, b x e^{2} + b d e + 2 \, 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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