3.566 \(\int \frac{a^x b^x}{x} \, dx\)

Optimal. Leaf size=8 \[ \text{Ei}(x (\log (a)+\log (b))) \]

[Out]

ExpIntegralEi[x*(Log[a] + Log[b])]

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Rubi [A]  time = 0.0416269, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2287, 2178} \[ \text{Ei}(x (\log (a)+\log (b))) \]

Antiderivative was successfully verified.

[In]

Int[(a^x*b^x)/x,x]

[Out]

ExpIntegralEi[x*(Log[a] + Log[b])]

Rule 2287

Int[(u_.)*(F_)^(v_)*(G_)^(w_), x_Symbol] :> With[{z = v*Log[F] + w*Log[G]}, Int[u*NormalizeIntegrand[E^z, x],
x] /; BinomialQ[z, x] || (PolynomialQ[z, x] && LeQ[Exponent[z, x], 2])] /; FreeQ[{F, G}, x]

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rubi steps

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

Mathematica [A]  time = 0.0109685, size = 10, normalized size = 1.25 \[ \text{Ei}(x \log (a)+x \log (b)) \]

Antiderivative was successfully verified.

[In]

Integrate[(a^x*b^x)/x,x]

[Out]

ExpIntegralEi[x*Log[a] + x*Log[b]]

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Maple [C]  time = 0.034, size = 56, normalized size = 7. \begin{align*} \ln \left ( x \right ) +i\pi +\ln \left ( \ln \left ( b \right ) \right ) +\ln \left ( 1+{\frac{\ln \left ( a \right ) }{\ln \left ( b \right ) }} \right ) -\ln \left ( -x\ln \left ( b \right ) \left ( 1+{\frac{\ln \left ( a \right ) }{\ln \left ( b \right ) }} \right ) \right ) -{\it Ei} \left ( 1,-x\ln \left ( b \right ) \left ( 1+{\frac{\ln \left ( a \right ) }{\ln \left ( b \right ) }} \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a^x*b^x/x,x)

[Out]

ln(x)+I*Pi+ln(ln(b))+ln(1+ln(a)/ln(b))-ln(-x*ln(b)*(1+ln(a)/ln(b)))-Ei(1,-x*ln(b)*(1+ln(a)/ln(b)))

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Maxima [A]  time = 1.09311, size = 11, normalized size = 1.38 \begin{align*}{\rm Ei}\left (x{\left (\log \left (a\right ) + \log \left (b\right )\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a^x*b^x/x,x, algorithm="maxima")

[Out]

Ei(x*(log(a) + log(b)))

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Fricas [A]  time = 1.22634, size = 34, normalized size = 4.25 \begin{align*}{\rm Ei}\left (x \log \left (a\right ) + x \log \left (b\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a^x*b^x/x,x, algorithm="fricas")

[Out]

Ei(x*log(a) + x*log(b))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a^{x} b^{x}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a**x*b**x/x,x)

[Out]

Integral(a**x*b**x/x, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a^{x} b^{x}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a^x*b^x/x,x, algorithm="giac")

[Out]

integrate(a^x*b^x/x, x)