Optimal. Leaf size=39 \[ \frac{(d+e x) \log (d+e x)}{e \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]
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Rubi [A] time = 0.0093111, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {608, 31} \[ \frac{(d+e x) \log (d+e x)}{e \sqrt{c d^2+2 c d e x+c e^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 608
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{c d^2+2 c d e x+c e^2 x^2}} \, dx &=\frac{\left (c d e+c e^2 x\right ) \int \frac{1}{c d e+c e^2 x} \, dx}{\sqrt{c d^2+2 c d e x+c e^2 x^2}}\\ &=\frac{(d+e x) \log (d+e x)}{e \sqrt{c d^2+2 c d e x+c e^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.006548, size = 28, normalized size = 0.72 \[ \frac{(d+e x) \log (d+e x)}{e \sqrt{c (d+e x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 38, normalized size = 1. \begin{align*}{\frac{ \left ( ex+d \right ) \ln \left ( ex+d \right ) }{e}{\frac{1}{\sqrt{c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14099, size = 24, normalized size = 0.62 \begin{align*} \sqrt{\frac{1}{c e^{2}}} \log \left (x + \frac{d}{e}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.33178, size = 92, normalized size = 2.36 \begin{align*} \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} \log \left (e x + d\right )}{c e^{2} x + c d e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3566, size = 73, normalized size = 1.87 \begin{align*} -\frac{e^{\left (-1\right )} \log \left ({\left | -\sqrt{c} d e^{2} -{\left (\sqrt{c} x e - \sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}}\right )} e^{2} \right |}\right )}{\sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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