3.568 \(\int \frac{a^x b^x}{x^3} \, dx\)

Optimal. Leaf size=51 \[ -\frac{a^x b^x}{2 x^2}-\frac{a^x b^x (\log (a)+\log (b))}{2 x}+\frac{1}{2} (\log (a)+\log (b))^2 \text{Ei}(x (\log (a)+\log (b))) \]

[Out]

-(a^x*b^x)/(2*x^2) - (a^x*b^x*(Log[a] + Log[b]))/(2*x) + (ExpIntegralEi[x*(Log[a] + Log[b])]*(Log[a] + Log[b])
^2)/2

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Rubi [A]  time = 0.0853705, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2287, 2177, 2178} \[ -\frac{a^x b^x}{2 x^2}-\frac{a^x b^x (\log (a)+\log (b))}{2 x}+\frac{1}{2} (\log (a)+\log (b))^2 \text{Ei}(x (\log (a)+\log (b))) \]

Antiderivative was successfully verified.

[In]

Int[(a^x*b^x)/x^3,x]

[Out]

-(a^x*b^x)/(2*x^2) - (a^x*b^x*(Log[a] + Log[b]))/(2*x) + (ExpIntegralEi[x*(Log[a] + Log[b])]*(Log[a] + Log[b])
^2)/2

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 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

Rubi steps

\begin{align*} \int \frac{a^x b^x}{x^3} \, dx &=\int \frac{e^{x (\log (a)+\log (b))}}{x^3} \, dx\\ &=-\frac{a^x b^x}{2 x^2}-\frac{1}{2} (-\log (a)-\log (b)) \int \frac{e^{x (\log (a)+\log (b))}}{x^2} \, dx\\ &=-\frac{a^x b^x}{2 x^2}-\frac{a^x b^x (\log (a)+\log (b))}{2 x}+\frac{1}{2} (\log (a)+\log (b))^2 \int \frac{e^{x (\log (a)+\log (b))}}{x} \, dx\\ &=-\frac{a^x b^x}{2 x^2}-\frac{a^x b^x (\log (a)+\log (b))}{2 x}+\frac{1}{2} \text{Ei}(x (\log (a)+\log (b))) (\log (a)+\log (b))^2\\ \end{align*}

Mathematica [F]  time = 0.0573304, size = 0, normalized size = 0. \[ \int \frac{a^x b^x}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a^x*b^x)/x^3,x]

[Out]

Integrate[(a^x*b^x)/x^3, x]

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Maple [C]  time = 0.044, size = 225, normalized size = 4.4 \begin{align*} \left ( \ln \left ( b \right ) \right ) ^{2} \left ( 1+{\frac{\ln \left ( a \right ) }{\ln \left ( b \right ) }} \right ) ^{2} \left ( -{\frac{1}{2\, \left ( \ln \left ( b \right ) \right ) ^{2}{x}^{2}} \left ( 1+{\frac{\ln \left ( a \right ) }{\ln \left ( b \right ) }} \right ) ^{-2}}-{\frac{1}{\ln \left ( b \right ) x} \left ( 1+{\frac{\ln \left ( a \right ) }{\ln \left ( b \right ) }} \right ) ^{-1}}-{\frac{3}{4}}+{\frac{\ln \left ( x \right ) }{2}}+{\frac{i}{2}}\pi +{\frac{\ln \left ( \ln \left ( b \right ) \right ) }{2}}+{\frac{1}{2}\ln \left ( 1+{\frac{\ln \left ( a \right ) }{\ln \left ( b \right ) }} \right ) }+{\frac{1}{12\, \left ( \ln \left ( b \right ) \right ) ^{2}{x}^{2}} \left ( 9\,{x}^{2} \left ( \ln \left ( b \right ) \right ) ^{2} \left ( 1+{\frac{\ln \left ( a \right ) }{\ln \left ( b \right ) }} \right ) ^{2}+12\,x\ln \left ( b \right ) \left ( 1+{\frac{\ln \left ( a \right ) }{\ln \left ( b \right ) }} \right ) +6 \right ) \left ( 1+{\frac{\ln \left ( a \right ) }{\ln \left ( b \right ) }} \right ) ^{-2}}-{\frac{1}{6\, \left ( \ln \left ( b \right ) \right ) ^{2}{x}^{2}} \left ( 3+3\,x\ln \left ( b \right ) \left ( 1+{\frac{\ln \left ( a \right ) }{\ln \left ( b \right ) }} \right ) \right ){{\rm e}^{x\ln \left ( b \right ) \left ( 1+{\frac{\ln \left ( a \right ) }{\ln \left ( b \right ) }} \right ) }} \left ( 1+{\frac{\ln \left ( a \right ) }{\ln \left ( b \right ) }} \right ) ^{-2}}-{\frac{1}{2}\ln \left ( -x\ln \left ( b \right ) \left ( 1+{\frac{\ln \left ( a \right ) }{\ln \left ( b \right ) }} \right ) \right ) }-{\frac{1}{2}{\it Ei} \left ( 1,-x\ln \left ( b \right ) \left ( 1+{\frac{\ln \left ( a \right ) }{\ln \left ( b \right ) }} \right ) \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a^x*b^x/x^3,x)

[Out]

ln(b)^2*(1+ln(a)/ln(b))^2*(-1/2/x^2/ln(b)^2/(1+ln(a)/ln(b))^2-1/x/ln(b)/(1+ln(a)/ln(b))-3/4+1/2*ln(x)+1/2*I*Pi
+1/2*ln(ln(b))+1/2*ln(1+ln(a)/ln(b))+1/12/x^2/ln(b)^2/(1+ln(a)/ln(b))^2*(9*x^2*ln(b)^2*(1+ln(a)/ln(b))^2+12*x*
ln(b)*(1+ln(a)/ln(b))+6)-1/6/x^2/ln(b)^2/(1+ln(a)/ln(b))^2*(3+3*x*ln(b)*(1+ln(a)/ln(b)))*exp(x*ln(b)*(1+ln(a)/
ln(b)))-1/2*ln(-x*ln(b)*(1+ln(a)/ln(b)))-1/2*Ei(1,-x*ln(b)*(1+ln(a)/ln(b))))

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Maxima [A]  time = 1.0967, size = 26, normalized size = 0.51 \begin{align*} -{\left (\log \left (a\right ) + \log \left (b\right )\right )}^{2} \Gamma \left (-2, -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^3,x, algorithm="maxima")

[Out]

-(log(a) + log(b))^2*gamma(-2, -x*(log(a) + log(b)))

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Fricas [A]  time = 1.33229, size = 167, normalized size = 3.27 \begin{align*} -\frac{{\left (x \log \left (a\right ) + x \log \left (b\right ) + 1\right )} a^{x} b^{x} -{\left (x^{2} \log \left (a\right )^{2} + 2 \, x^{2} \log \left (a\right ) \log \left (b\right ) + x^{2} \log \left (b\right )^{2}\right )}{\rm Ei}\left (x \log \left (a\right ) + x \log \left (b\right )\right )}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a^x*b^x/x^3,x, algorithm="fricas")

[Out]

-1/2*((x*log(a) + x*log(b) + 1)*a^x*b^x - (x^2*log(a)^2 + 2*x^2*log(a)*log(b) + x^2*log(b)^2)*Ei(x*log(a) + x*
log(b)))/x^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a^{x} b^{x}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a**x*b**x/x**3,x)

[Out]

Integral(a**x*b**x/x**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a^{x} b^{x}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a^x*b^x/x^3,x, algorithm="giac")

[Out]

integrate(a^x*b^x/x^3, x)