Optimal. Leaf size=15 \[ -\frac{c^2}{e (d+e x)} \]
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Rubi [A] time = 0.0046743, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {27, 12, 32} \[ -\frac{c^2}{e (d+e x)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 12
Rule 32
Rubi steps
\begin{align*} \int \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^6} \, dx &=\int \frac{c^2}{(d+e x)^2} \, dx\\ &=c^2 \int \frac{1}{(d+e x)^2} \, dx\\ &=-\frac{c^2}{e (d+e x)}\\ \end{align*}
Mathematica [A] time = 0.0017567, size = 15, normalized size = 1. \[ -\frac{c^2}{e (d+e x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 16, normalized size = 1.1 \begin{align*} -{\frac{{c}^{2}}{e \left ( ex+d \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17471, size = 22, normalized size = 1.47 \begin{align*} -\frac{c^{2}}{e^{2} x + d e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03494, size = 27, normalized size = 1.8 \begin{align*} -\frac{c^{2}}{e^{2} x + d e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.314408, size = 12, normalized size = 0.8 \begin{align*} - \frac{c^{2}}{d e + e^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.20877, size = 89, normalized size = 5.93 \begin{align*} -\frac{{\left (c^{2} x^{4} e^{8} + 4 \, c^{2} d x^{3} e^{7} + 6 \, c^{2} d^{2} x^{2} e^{6} + 4 \, c^{2} d^{3} x e^{5} + c^{2} d^{4} e^{4}\right )} e^{\left (-5\right )}}{{\left (x e + d\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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