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

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

[Out]

Ei(x*(ln(a)+ln(b)))

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Rubi [A]  time = 0.04, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 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

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]

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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.40, size = 10, normalized size = 1.25 \[ {\rm Ei}\left (x \log \relax (a) + x \log \relax (b)\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a^{x} b^{x}}{x}\,{d x} \]

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)

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maple [C]  time = 0.06, size = 56, normalized size = 7.00 \[ -\Ei \left (1, -\left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right ) x \ln \relax (b )\right )+\ln \relax (x )-\ln \left (-\left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right ) x \ln \relax (b )\right )+\ln \left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right )+\ln \left (\ln \relax (b )\right )+i \pi \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.34, size = 8, normalized size = 1.00 \[ {\rm Ei}\left (x {\left (\log \relax (a) + \log \relax (b)\right )}\right ) \]

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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mupad [B]  time = 0.03, size = 8, normalized size = 1.00 \[ \mathrm {ei}\left (x\,\left (\ln \relax (a)+\ln \relax (b)\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ei(x*(log(a) + log(b)))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a^{x} b^{x}}{x}\, dx \]

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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