3.82 \(\int \frac{a+b \log (e x)}{x} \, dx\)

Optimal. Leaf size=17 \[ \frac{(a+b \log (e x))^2}{2 b} \]

[Out]

(a + b*Log[e*x])^2/(2*b)

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Rubi [A]  time = 0.0112685, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {2301} \[ \frac{(a+b \log (e x))^2}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[e*x])/x,x]

[Out]

(a + b*Log[e*x])^2/(2*b)

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rubi steps

\begin{align*} \int \frac{a+b \log (e x)}{x} \, dx &=\frac{(a+b \log (e x))^2}{2 b}\\ \end{align*}

Mathematica [A]  time = 0.0013259, size = 16, normalized size = 0.94 \[ a \log (x)+\frac{1}{2} b \log ^2(e x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[e*x])/x,x]

[Out]

a*Log[x] + (b*Log[e*x]^2)/2

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Maple [A]  time = 0.059, size = 17, normalized size = 1. \begin{align*}{\frac{ \left ( \ln \left ( ex \right ) \right ) ^{2}b}{2}}+a\ln \left ( ex \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(e*x))/x,x)

[Out]

1/2*ln(e*x)^2*b+a*ln(e*x)

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Maxima [A]  time = 1.18602, size = 20, normalized size = 1.18 \begin{align*} \frac{{\left (b \log \left (e x\right ) + a\right )}^{2}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(e*x))/x,x, algorithm="maxima")

[Out]

1/2*(b*log(e*x) + a)^2/b

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Fricas [A]  time = 1.92302, size = 42, normalized size = 2.47 \begin{align*} \frac{1}{2} \, b \log \left (e x\right )^{2} + a \log \left (e x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(e*x))/x,x, algorithm="fricas")

[Out]

1/2*b*log(e*x)^2 + a*log(e*x)

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Sympy [A]  time = 0.275289, size = 14, normalized size = 0.82 \begin{align*} a \log{\left (x \right )} + \frac{b \log{\left (e x \right )}^{2}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(e*x))/x,x)

[Out]

a*log(x) + b*log(e*x)**2/2

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Giac [A]  time = 1.20601, size = 24, normalized size = 1.41 \begin{align*} \frac{1}{2} \, b \log \left (x e\right )^{2} + a \log \left (x e\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(e*x))/x,x, algorithm="giac")

[Out]

1/2*b*log(x*e)^2 + a*log(x*e)