Optimal. Leaf size=23 \[ \frac{\log \left (\sqrt{-a}+c e+d e x\right )}{d e} \]
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Rubi [A] time = 0.0140041, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {33, 31} \[ \frac{\log \left (\sqrt{-a}+c e+d e x\right )}{d e} \]
Antiderivative was successfully verified.
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Rule 33
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-a}+e (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{-a}+e x} \, dx,x,c+d x\right )}{d}\\ &=\frac{\log \left (\sqrt{-a}+c e+d e x\right )}{d e}\\ \end{align*}
Mathematica [A] time = 0.0088897, size = 23, normalized size = 1. \[ \frac{\log \left (\sqrt{-a}+c e+d e x\right )}{d e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 22, normalized size = 1. \begin{align*}{\frac{1}{ed}\ln \left ( ce+dex+\sqrt{-a} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02679, size = 28, normalized size = 1.22 \begin{align*} \frac{\log \left ({\left (d x + c\right )} e + \sqrt{-a}\right )}{d e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76498, size = 47, normalized size = 2.04 \begin{align*} \frac{\log \left (d e x + c e + \sqrt{-a}\right )}{d e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.069685, size = 19, normalized size = 0.83 \begin{align*} \frac{\log{\left (c e + d e x + \sqrt{- a} \right )}}{d e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18278, size = 30, normalized size = 1.3 \begin{align*} \frac{e^{\left (-1\right )} \log \left ({\left |{\left (d x + c\right )} e + \sqrt{-a} \right |}\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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