3.567 \(\int \frac {a^x b^x}{x^2} \, dx\)

Optimal. Leaf size=26 \[ (\log (a)+\log (b)) \text {Ei}(x (\log (a)+\log (b)))-\frac {a^x b^x}{x} \]

[Out]

-a^x*b^x/x+Ei(x*(ln(a)+ln(b)))*(ln(a)+ln(b))

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Rubi [A]  time = 0.06, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2287, 2177, 2178} \[ (\log (a)+\log (b)) \text {Ei}(x (\log (a)+\log (b)))-\frac {a^x b^x}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a^x*b^x)/x^2,x]

[Out]

-((a^x*b^x)/x) + ExpIntegralEi[x*(Log[a] + Log[b])]*(Log[a] + Log[b])

Rule 2177

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), Int[(c + d*x)^(m + 1)*(b*F^(g*
(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] &&  !$UseGamma ===
True

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

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Mathematica [F]  time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {a^x b^x}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a^x*b^x)/x^2,x]

[Out]

Integrate[(a^x*b^x)/x^2, x]

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fricas [A]  time = 0.40, size = 34, normalized size = 1.31 \[ -\frac {a^{x} b^{x} - {\left (x \log \relax (a) + x \log \relax (b)\right )} {\rm Ei}\left (x \log \relax (a) + x \log \relax (b)\right )}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a^x*b^x/x^2,x, algorithm="fricas")

[Out]

-(a^x*b^x - (x*log(a) + x*log(b))*Ei(x*log(a) + x*log(b)))/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a^{x} b^{x}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a^x*b^x/x^2,x, algorithm="giac")

[Out]

integrate(a^x*b^x/x^2, x)

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maple [C]  time = 0.07, size = 160, normalized size = 6.15 \[ -\left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right ) \left (\Ei \left (1, -\left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right ) x \ln \relax (b )\right )-\ln \relax (x )+\ln \left (-\left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right ) x \ln \relax (b )\right )-\ln \left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right )-\ln \left (\ln \relax (b )\right )+\frac {{\mathrm e}^{\left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right ) x \ln \relax (b )}}{\left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right ) x \ln \relax (b )}-\frac {2 \left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right ) x \ln \relax (b )+2}{2 \left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right ) x \ln \relax (b )}+\frac {1}{\left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right ) x \ln \relax (b )}+1-i \pi \right ) \ln \relax (b ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a^x*b^x/x^2,x)

[Out]

-ln(b)*(ln(a)/ln(b)+1)*(-1/2/(ln(a)/ln(b)+1)/x/ln(b)*(2+2*(ln(a)/ln(b)+1)*x*ln(b))+1/(ln(a)/ln(b)+1)/x/ln(b)*e
xp((ln(a)/ln(b)+1)*x*ln(b))+ln(-(ln(a)/ln(b)+1)*x*ln(b))+Ei(1,-(ln(a)/ln(b)+1)*x*ln(b))+1-ln(x)-I*Pi-ln(ln(b))
-ln(ln(a)/ln(b)+1)+1/x/ln(b)/(ln(a)/ln(b)+1))

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maxima [A]  time = 1.30, size = 16, normalized size = 0.62 \[ {\left (\log \relax (a) + \log \relax (b)\right )} \Gamma \left (-1, -x {\left (\log \relax (a) + \log \relax (b)\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a^x*b^x/x^2,x, algorithm="maxima")

[Out]

(log(a) + log(b))*gamma(-1, -x*(log(a) + log(b)))

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mupad [B]  time = 3.50, size = 28, normalized size = 1.08 \[ -\mathrm {expint}\left (-x\,\left (\ln \relax (a)+\ln \relax (b)\right )\right )\,\left (\ln \relax (a)+\ln \relax (b)\right )-\frac {a^x\,b^x}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^x*b^x)/x^2,x)

[Out]

- expint(-x*(log(a) + log(b)))*(log(a) + log(b)) - (a^x*b^x)/x

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a^{x} b^{x}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a**x*b**x/x**2,x)

[Out]

Integral(a**x*b**x/x**2, x)

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