\(\int \frac {a^x b^x}{x} \, dx\) [566]

Optimal result

Integrand size = 10, antiderivative size = 8 \[ \int \frac {a^x b^x}{x} \, dx=\operatorname {ExpIntegralEi}(x (\log (a)+\log (b))) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2325, 2209} \[ \int \frac {a^x b^x}{x} \, dx=\operatorname {ExpIntegralEi}(x (\log (a)+\log (b))) \]

[In]

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

[Out]

ExpIntegralEi[x*(Log[a] + Log[b])]

Rule 2209

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

Rule 2325

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.25 \[ \int \frac {a^x b^x}{x} \, dx=\operatorname {ExpIntegralEi}(x \log (a)+x \log (b)) \]

[In]

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

[Out]

ExpIntegralEi[x*Log[a] + x*Log[b]]

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 7.00

method	result	size
meijerg	\(\ln \left (x \right )+i \pi +\ln \left (\ln \left (b \right )\right )+\ln \left (1+\frac {\ln \left (a \right )}{\ln \left (b \right )}\right )-\ln \left (-x \ln \left (b \right ) \left (1+\frac {\ln \left (a \right )}{\ln \left (b \right )}\right )\right )-\operatorname {Ei}_{1}\left (-x \ln \left (b \right ) \left (1+\frac {\ln \left (a \right )}{\ln \left (b \right )}\right )\right )\)	\(56\)

[In]

int(a^x*b^x/x,x,method=_RETURNVERBOSE)

[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)))

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.25 \[ \int \frac {a^x b^x}{x} \, dx={\rm Ei}\left (x \log \left (a\right ) + x \log \left (b\right )\right ) \]

[In]

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

[Out]

Ei(x*log(a) + x*log(b))

Sympy [F]

\[ \int \frac {a^x b^x}{x} \, dx=\int \frac {a^{x} b^{x}}{x}\, dx \]

[In]

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

[Out]

Integral(a**x*b**x/x, x)

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {a^x b^x}{x} \, dx={\rm Ei}\left (x {\left (\log \left (a\right ) + \log \left (b\right )\right )}\right ) \]

[In]

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

[Out]

Ei(x*(log(a) + log(b)))

Giac [F]

\[ \int \frac {a^x b^x}{x} \, dx=\int { \frac {a^{x} b^{x}}{x} \,d x } \]

[In]

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

[Out]

integrate(a^x*b^x/x, x)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {a^x b^x}{x} \, dx=\mathrm {ei}\left (x\,\left (\ln \left (a\right )+\ln \left (b\right )\right )\right ) \]

[In]

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

[Out]

ei(x*(log(a) + log(b)))