Optimal. Leaf size=11 \[ \frac{c \log (d+e x)}{e} \]
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Rubi [A] time = 0.0061493, antiderivative size = 11, 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, 31} \[ \frac{c \log (d+e x)}{e} \]
Antiderivative was successfully verified.
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Rule 24
Rule 21
Rule 31
Rubi steps
\begin{align*} \int \frac{c d^2+2 c d e x+c e^2 x^2}{(d+e x)^3} \, dx &=\frac{\int \frac{c d e^2+c e^3 x}{(d+e x)^2} \, dx}{e^2}\\ &=c \int \frac{1}{d+e x} \, dx\\ &=\frac{c \log (d+e x)}{e}\\ \end{align*}
Mathematica [A] time = 0.0013314, size = 11, normalized size = 1. \[ \frac{c \log (d+e x)}{e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 12, normalized size = 1.1 \begin{align*}{\frac{c\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 = 1.15981, size = 15, normalized size = 1.36 \begin{align*} \frac{c \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 = 2.05157, size = 24, normalized size = 2.18 \begin{align*} \frac{c \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 = 0.083459, size = 8, normalized size = 0.73 \begin{align*} \frac{c \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 = 1.20382, size = 16, normalized size = 1.45 \begin{align*} c e^{\left (-1\right )} \log \left ({\left | x e + d \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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