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]

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

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Rubi [A]  time = 0.09, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 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^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*}

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Mathematica [F]  time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {a^x b^x}{x^3} \, dx \]

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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fricas [A]  time = 0.42, size = 61, normalized size = 1.20 \[ -\frac {{\left (x \log \relax (a) + x \log \relax (b) + 1\right )} a^{x} b^{x} - {\left (x^{2} \log \relax (a)^{2} + 2 \, x^{2} \log \relax (a) \log \relax (b) + x^{2} \log \relax (b)^{2}\right )} {\rm Ei}\left (x \log \relax (a) + x \log \relax (b)\right )}{2 \, x^{2}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a^{x} b^{x}}{x^{3}}\,{d x} \]

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)

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maple [C]  time = 0.07, size = 225, normalized size = 4.41 \[ \left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right )^{2} \left (-\frac {\Ei \left (1, -\left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right ) x \ln \relax (b )\right )}{2}+\frac {\ln \relax (x )}{2}-\frac {\ln \left (-\left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right ) x \ln \relax (b )\right )}{2}+\frac {\ln \left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right )}{2}+\frac {\ln \left (\ln \relax (b )\right )}{2}-\frac {1}{\left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right ) x \ln \relax (b )}-\frac {\left (3 \left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right ) x \ln \relax (b )+3\right ) {\mathrm e}^{\left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right ) x \ln \relax (b )}}{6 \left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right )^{2} x^{2} \ln \relax (b )^{2}}+\frac {9 \left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right )^{2} x^{2} \ln \relax (b )^{2}+12 \left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right ) x \ln \relax (b )+6}{12 \left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right )^{2} x^{2} \ln \relax (b )^{2}}-\frac {1}{2 \left (\frac {\ln \relax (a )}{\ln \relax (b )}+1\right )^{2} x^{2} \ln \relax (b )^{2}}-\frac {3}{4}+\frac {i \pi }{2}\right ) \ln \relax (b )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.33, size = 19, normalized size = 0.37 \[ -{\left (\log \relax (a) + \log \relax (b)\right )}^{2} \Gamma \left (-2, -x {\left (\log \relax (a) + \log \relax (b)\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.05, size = 59, normalized size = 1.16 \[ -\frac {\mathrm {expint}\left (-x\,\left (\ln \relax (a)+\ln \relax (b)\right )\right )\,{\left (\ln \relax (a)+\ln \relax (b)\right )}^2}{2}-a^x\,b^x\,\left (\frac {1}{2\,x\,\left (\ln \relax (a)+\ln \relax (b)\right )}+\frac {1}{2\,x^2\,{\left (\ln \relax (a)+\ln \relax (b)\right )}^2}\right )\,{\left (\ln \relax (a)+\ln \relax (b)\right )}^2 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a^{x} b^{x}}{x^{3}}\, dx \]

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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