Optimal. Leaf size=13 \[ -\frac{c}{e (d+e x)} \]
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Rubi [A] time = 0.0064377, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {24, 21, 32} \[ -\frac{c}{e (d+e x)} \]
Antiderivative was successfully verified.
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Rule 24
Rule 21
Rule 32
Rubi steps
\begin{align*} \int \frac{c d^2+2 c d e x+c e^2 x^2}{(d+e x)^4} \, dx &=\frac{\int \frac{c d e^2+c e^3 x}{(d+e x)^3} \, dx}{e^2}\\ &=c \int \frac{1}{(d+e x)^2} \, dx\\ &=-\frac{c}{e (d+e x)}\\ \end{align*}
Mathematica [A] time = 0.0033505, size = 13, normalized size = 1. \[ -\frac{c}{e (d+e x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 14, normalized size = 1.1 \begin{align*} -{\frac{c}{e \left ( ex+d \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16147, size = 19, normalized size = 1.46 \begin{align*} -\frac{c}{e^{2} x + d e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98543, size = 24, normalized size = 1.85 \begin{align*} -\frac{c}{e^{2} x + d e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.314067, size = 10, normalized size = 0.77 \begin{align*} - \frac{c}{d e + e^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.22292, size = 46, normalized size = 3.54 \begin{align*} -\frac{{\left (c x^{2} e^{4} + 2 \, c d x e^{3} + c d^{2} e^{2}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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