Optimal. Leaf size=13 \[ \frac{c^2 \log (d+e x)}{e} \]
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Rubi [A] time = 0.0044983, antiderivative size = 13, 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, 31} \[ \frac{c^2 \log (d+e x)}{e} \]
Antiderivative was successfully verified.
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Rule 27
Rule 12
Rule 31
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)^5} \, dx &=\int \frac{c^2}{d+e x} \, dx\\ &=c^2 \int \frac{1}{d+e x} \, dx\\ &=\frac{c^2 \log (d+e x)}{e}\\ \end{align*}
Mathematica [A] time = 0.0013471, size = 13, normalized size = 1. \[ \frac{c^2 \log (d+e x)}{e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 14, normalized size = 1.1 \begin{align*}{\frac{{c}^{2}\ln \left ( ex+d \right ) }{e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17349, size = 18, normalized size = 1.38 \begin{align*} \frac{c^{2} \log \left (e x + d\right )}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98157, size = 27, normalized size = 2.08 \begin{align*} \frac{c^{2} \log \left (e x + d\right )}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.138259, size = 10, normalized size = 0.77 \begin{align*} \frac{c^{2} \log{\left (d + e x \right )}}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19822, size = 35, normalized size = 2.69 \begin{align*} -c^{2} e^{\left (-1\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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