3.563 \(\int a^x b^x x^2 \, dx\)
Optimal. Leaf size=49 \[ \frac {x^2 a^x b^x}{\log (a)+\log (b)}-\frac {2 x a^x b^x}{(\log (a)+\log (b))^2}+\frac {2 a^x b^x}{(\log (a)+\log (b))^3} \]
[Out]
2*a^x*b^x/(ln(a)+ln(b))^3-2*a^x*b^x*x/(ln(a)+ln(b))^2+a^x*b^x*x^2/(ln(a)+ln(b))
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Rubi [A] time = 0.06, antiderivative size = 49, 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, 2176, 2194} \[ \frac {x^2 a^x b^x}{\log (a)+\log (b)}-\frac {2 x a^x b^x}{(\log (a)+\log (b))^2}+\frac {2 a^x b^x}{(\log (a)+\log (b))^3} \]
Antiderivative was successfully verified.
[In]
Int[a^x*b^x*x^2,x]
[Out]
(2*a^x*b^x)/(Log[a] + Log[b])^3 - (2*a^x*b^x*x)/(Log[a] + Log[b])^2 + (a^x*b^x*x^2)/(Log[a] + Log[b])
Rule 2176
Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), 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] && GtQ[m, 0] && IntegerQ[2*m] && !$UseGamma === True
Rule 2194
Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]
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 a^x b^x x^2 \, dx &=\int e^{x (\log (a)+\log (b))} x^2 \, dx\\ &=\frac {a^x b^x x^2}{\log (a)+\log (b)}-\frac {2 \int e^{x (\log (a)+\log (b))} x \, dx}{\log (a)+\log (b)}\\ &=-\frac {2 a^x b^x x}{(\log (a)+\log (b))^2}+\frac {a^x b^x x^2}{\log (a)+\log (b)}+\frac {2 \int e^{x (\log (a)+\log (b))} \, dx}{(\log (a)+\log (b))^2}\\ &=\frac {2 a^x b^x}{(\log (a)+\log (b))^3}-\frac {2 a^x b^x x}{(\log (a)+\log (b))^2}+\frac {a^x b^x x^2}{\log (a)+\log (b)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 35, normalized size = 0.71 \[ \frac {a^x b^x \left (x^2 (\log (a)+\log (b))^2-2 x (\log (a)+\log (b))+2\right )}{(\log (a)+\log (b))^3} \]
Antiderivative was successfully verified.
[In]
Integrate[a^x*b^x*x^2,x]
[Out]
(a^x*b^x*(2 - 2*x*(Log[a] + Log[b]) + x^2*(Log[a] + Log[b])^2))/(Log[a] + Log[b])^3
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fricas [A] time = 0.43, size = 71, normalized size = 1.45 \[ \frac {{\left (x^{2} \log \relax (a)^{2} + x^{2} \log \relax (b)^{2} - 2 \, x \log \relax (a) + 2 \, {\left (x^{2} \log \relax (a) - x\right )} \log \relax (b) + 2\right )} a^{x} b^{x}}{\log \relax (a)^{3} + 3 \, \log \relax (a)^{2} \log \relax (b) + 3 \, \log \relax (a) \log \relax (b)^{2} + \log \relax (b)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(a^x*b^x*x^2,x, algorithm="fricas")
[Out]
(x^2*log(a)^2 + x^2*log(b)^2 - 2*x*log(a) + 2*(x^2*log(a) - x)*log(b) + 2)*a^x*b^x/(log(a)^3 + 3*log(a)^2*log(
b) + 3*log(a)*log(b)^2 + log(b)^3)
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giac [B] time = 0.85, size = 2679, normalized size = 54.67 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(a^x*b^x*x^2,x, algorithm="giac")
[Out]
((2*(pi*x^2*log(abs(a))*sgn(a) + pi*x^2*log(abs(b))*sgn(a) + pi*x^2*log(abs(a))*sgn(b) + pi*x^2*log(abs(b))*sg
n(b) - 2*pi*x^2*log(abs(a)) - 2*pi*x^2*log(abs(b)) - pi*x*sgn(a) - pi*x*sgn(b) + 2*pi*x)*(3*pi^3*sgn(a)*sgn(b)
- 4*pi^3*sgn(a) + 3*pi*log(abs(a))^2*sgn(a) + 6*pi*log(abs(a))*log(abs(b))*sgn(a) + 3*pi*log(abs(b))^2*sgn(a)
- 4*pi^3*sgn(b) + 3*pi*log(abs(a))^2*sgn(b) + 6*pi*log(abs(a))*log(abs(b))*sgn(b) + 3*pi*log(abs(b))^2*sgn(b)
+ 5*pi^3 - 6*pi*log(abs(a))^2 - 12*pi*log(abs(a))*log(abs(b)) - 6*pi*log(abs(b))^2)/((3*pi^3*sgn(a)*sgn(b) -
4*pi^3*sgn(a) + 3*pi*log(abs(a))^2*sgn(a) + 6*pi*log(abs(a))*log(abs(b))*sgn(a) + 3*pi*log(abs(b))^2*sgn(a) -
4*pi^3*sgn(b) + 3*pi*log(abs(a))^2*sgn(b) + 6*pi*log(abs(a))*log(abs(b))*sgn(b) + 3*pi*log(abs(b))^2*sgn(b) +
5*pi^3 - 6*pi*log(abs(a))^2 - 12*pi*log(abs(a))*log(abs(b)) - 6*pi*log(abs(b))^2)^2 + (3*pi^2*log(abs(a))*sgn(
a)*sgn(b) + 3*pi^2*log(abs(b))*sgn(a)*sgn(b) - 6*pi^2*log(abs(a))*sgn(a) - 6*pi^2*log(abs(b))*sgn(a) - 6*pi^2*
log(abs(a))*sgn(b) - 6*pi^2*log(abs(b))*sgn(b) + 9*pi^2*log(abs(a)) - 2*log(abs(a))^3 + 9*pi^2*log(abs(b)) - 6
*log(abs(a))^2*log(abs(b)) - 6*log(abs(a))*log(abs(b))^2 - 2*log(abs(b))^3)^2) + (pi^2*x^2*sgn(a)*sgn(b) - 2*p
i^2*x^2*sgn(a) - 2*pi^2*x^2*sgn(b) + 3*pi^2*x^2 - 2*x^2*log(abs(a))^2 - 4*x^2*log(abs(a))*log(abs(b)) - 2*x^2*
log(abs(b))^2 + 4*x*log(abs(a)) + 4*x*log(abs(b)) - 4)*(3*pi^2*log(abs(a))*sgn(a)*sgn(b) + 3*pi^2*log(abs(b))*
sgn(a)*sgn(b) - 6*pi^2*log(abs(a))*sgn(a) - 6*pi^2*log(abs(b))*sgn(a) - 6*pi^2*log(abs(a))*sgn(b) - 6*pi^2*log
(abs(b))*sgn(b) + 9*pi^2*log(abs(a)) - 2*log(abs(a))^3 + 9*pi^2*log(abs(b)) - 6*log(abs(a))^2*log(abs(b)) - 6*
log(abs(a))*log(abs(b))^2 - 2*log(abs(b))^3)/((3*pi^3*sgn(a)*sgn(b) - 4*pi^3*sgn(a) + 3*pi*log(abs(a))^2*sgn(a
) + 6*pi*log(abs(a))*log(abs(b))*sgn(a) + 3*pi*log(abs(b))^2*sgn(a) - 4*pi^3*sgn(b) + 3*pi*log(abs(a))^2*sgn(b
) + 6*pi*log(abs(a))*log(abs(b))*sgn(b) + 3*pi*log(abs(b))^2*sgn(b) + 5*pi^3 - 6*pi*log(abs(a))^2 - 12*pi*log(
abs(a))*log(abs(b)) - 6*pi*log(abs(b))^2)^2 + (3*pi^2*log(abs(a))*sgn(a)*sgn(b) + 3*pi^2*log(abs(b))*sgn(a)*sg
n(b) - 6*pi^2*log(abs(a))*sgn(a) - 6*pi^2*log(abs(b))*sgn(a) - 6*pi^2*log(abs(a))*sgn(b) - 6*pi^2*log(abs(b))*
sgn(b) + 9*pi^2*log(abs(a)) - 2*log(abs(a))^3 + 9*pi^2*log(abs(b)) - 6*log(abs(a))^2*log(abs(b)) - 6*log(abs(a
))*log(abs(b))^2 - 2*log(abs(b))^3)^2))*cos(-1/2*pi*x*sgn(a) - 1/2*pi*x*sgn(b) + pi*x) + ((pi^2*x^2*sgn(a)*sgn
(b) - 2*pi^2*x^2*sgn(a) - 2*pi^2*x^2*sgn(b) + 3*pi^2*x^2 - 2*x^2*log(abs(a))^2 - 4*x^2*log(abs(a))*log(abs(b))
- 2*x^2*log(abs(b))^2 + 4*x*log(abs(a)) + 4*x*log(abs(b)) - 4)*(3*pi^3*sgn(a)*sgn(b) - 4*pi^3*sgn(a) + 3*pi*l
og(abs(a))^2*sgn(a) + 6*pi*log(abs(a))*log(abs(b))*sgn(a) + 3*pi*log(abs(b))^2*sgn(a) - 4*pi^3*sgn(b) + 3*pi*l
og(abs(a))^2*sgn(b) + 6*pi*log(abs(a))*log(abs(b))*sgn(b) + 3*pi*log(abs(b))^2*sgn(b) + 5*pi^3 - 6*pi*log(abs(
a))^2 - 12*pi*log(abs(a))*log(abs(b)) - 6*pi*log(abs(b))^2)/((3*pi^3*sgn(a)*sgn(b) - 4*pi^3*sgn(a) + 3*pi*log(
abs(a))^2*sgn(a) + 6*pi*log(abs(a))*log(abs(b))*sgn(a) + 3*pi*log(abs(b))^2*sgn(a) - 4*pi^3*sgn(b) + 3*pi*log(
abs(a))^2*sgn(b) + 6*pi*log(abs(a))*log(abs(b))*sgn(b) + 3*pi*log(abs(b))^2*sgn(b) + 5*pi^3 - 6*pi*log(abs(a))
^2 - 12*pi*log(abs(a))*log(abs(b)) - 6*pi*log(abs(b))^2)^2 + (3*pi^2*log(abs(a))*sgn(a)*sgn(b) + 3*pi^2*log(ab
s(b))*sgn(a)*sgn(b) - 6*pi^2*log(abs(a))*sgn(a) - 6*pi^2*log(abs(b))*sgn(a) - 6*pi^2*log(abs(a))*sgn(b) - 6*pi
^2*log(abs(b))*sgn(b) + 9*pi^2*log(abs(a)) - 2*log(abs(a))^3 + 9*pi^2*log(abs(b)) - 6*log(abs(a))^2*log(abs(b)
) - 6*log(abs(a))*log(abs(b))^2 - 2*log(abs(b))^3)^2) - 2*(pi*x^2*log(abs(a))*sgn(a) + pi*x^2*log(abs(b))*sgn(
a) + pi*x^2*log(abs(a))*sgn(b) + pi*x^2*log(abs(b))*sgn(b) - 2*pi*x^2*log(abs(a)) - 2*pi*x^2*log(abs(b)) - pi*
x*sgn(a) - pi*x*sgn(b) + 2*pi*x)*(3*pi^2*log(abs(a))*sgn(a)*sgn(b) + 3*pi^2*log(abs(b))*sgn(a)*sgn(b) - 6*pi^2
*log(abs(a))*sgn(a) - 6*pi^2*log(abs(b))*sgn(a) - 6*pi^2*log(abs(a))*sgn(b) - 6*pi^2*log(abs(b))*sgn(b) + 9*pi
^2*log(abs(a)) - 2*log(abs(a))^3 + 9*pi^2*log(abs(b)) - 6*log(abs(a))^2*log(abs(b)) - 6*log(abs(a))*log(abs(b)
)^2 - 2*log(abs(b))^3)/((3*pi^3*sgn(a)*sgn(b) - 4*pi^3*sgn(a) + 3*pi*log(abs(a))^2*sgn(a) + 6*pi*log(abs(a))*l
og(abs(b))*sgn(a) + 3*pi*log(abs(b))^2*sgn(a) - 4*pi^3*sgn(b) + 3*pi*log(abs(a))^2*sgn(b) + 6*pi*log(abs(a))*l
og(abs(b))*sgn(b) + 3*pi*log(abs(b))^2*sgn(b) + 5*pi^3 - 6*pi*log(abs(a))^2 - 12*pi*log(abs(a))*log(abs(b)) -
6*pi*log(abs(b))^2)^2 + (3*pi^2*log(abs(a))*sgn(a)*sgn(b) + 3*pi^2*log(abs(b))*sgn(a)*sgn(b) - 6*pi^2*log(abs(
a))*sgn(a) - 6*pi^2*log(abs(b))*sgn(a) - 6*pi^2*log(abs(a))*sgn(b) - 6*pi^2*log(abs(b))*sgn(b) + 9*pi^2*log(ab
s(a)) - 2*log(abs(a))^3 + 9*pi^2*log(abs(b)) - 6*log(abs(a))^2*log(abs(b)) - 6*log(abs(a))*log(abs(b))^2 - 2*l
og(abs(b))^3)^2))*sin(-1/2*pi*x*sgn(a) - 1/2*pi*x*sgn(b) + pi*x))*e^(x*(log(abs(a)) + log(abs(b)))) + 1/2*((pi
^2*i*x^2*sgn(a)*sgn(b) - 2*pi^2*i*x^2*sgn(a) - 2*pi^2*i*x^2*sgn(b) + 3*pi^2*i*x^2 - 2*i*x^2*log(abs(a))^2 - 4*
i*x^2*log(abs(a))*log(abs(b)) - 2*i*x^2*log(abs(b))^2 + 2*pi*x^2*log(abs(a))*sgn(a) + 2*pi*x^2*log(abs(b))*sgn
(a) + 2*pi*x^2*log(abs(a))*sgn(b) + 2*pi*x^2*log(abs(b))*sgn(b) - 4*pi*x^2*log(abs(a)) - 4*pi*x^2*log(abs(b))
+ 4*i*x*log(abs(a)) + 4*i*x*log(abs(b)) - 2*pi*x*sgn(a) - 2*pi*x*sgn(b) + 4*pi*x - 4*i)*e^(1/2*(pi*(sgn(a) - 1
) + pi*(sgn(b) - 1))*i*x)/(3*pi^3*i*sgn(a)*sgn(b) - 4*pi^3*i*sgn(a) + 3*pi*i*log(abs(a))^2*sgn(a) + 6*pi*i*log
(abs(a))*log(abs(b))*sgn(a) + 3*pi*i*log(abs(b))^2*sgn(a) - 4*pi^3*i*sgn(b) + 3*pi*i*log(abs(a))^2*sgn(b) + 6*
pi*i*log(abs(a))*log(abs(b))*sgn(b) + 3*pi*i*log(abs(b))^2*sgn(b) - 3*pi^2*log(abs(a))*sgn(a)*sgn(b) - 3*pi^2*
log(abs(b))*sgn(a)*sgn(b) + 5*pi^3*i - 6*pi*i*log(abs(a))^2 - 12*pi*i*log(abs(a))*log(abs(b)) - 6*pi*i*log(abs
(b))^2 + 6*pi^2*log(abs(a))*sgn(a) + 6*pi^2*log(abs(b))*sgn(a) + 6*pi^2*log(abs(a))*sgn(b) + 6*pi^2*log(abs(b)
)*sgn(b) - 9*pi^2*log(abs(a)) + 2*log(abs(a))^3 - 9*pi^2*log(abs(b)) + 6*log(abs(a))^2*log(abs(b)) + 6*log(abs
(a))*log(abs(b))^2 + 2*log(abs(b))^3) + (pi^2*i*x^2*sgn(a)*sgn(b) - 2*pi^2*i*x^2*sgn(a) - 2*pi^2*i*x^2*sgn(b)
+ 3*pi^2*i*x^2 - 2*i*x^2*log(abs(a))^2 - 4*i*x^2*log(abs(a))*log(abs(b)) - 2*i*x^2*log(abs(b))^2 - 2*pi*x^2*lo
g(abs(a))*sgn(a) - 2*pi*x^2*log(abs(b))*sgn(a) - 2*pi*x^2*log(abs(a))*sgn(b) - 2*pi*x^2*log(abs(b))*sgn(b) + 4
*pi*x^2*log(abs(a)) + 4*pi*x^2*log(abs(b)) + 4*i*x*log(abs(a)) + 4*i*x*log(abs(b)) + 2*pi*x*sgn(a) + 2*pi*x*sg
n(b) - 4*pi*x - 4*i)*e^(-1/2*(pi*(sgn(a) - 1) + pi*(sgn(b) - 1))*i*x)/(3*pi^3*i*sgn(a)*sgn(b) - 4*pi^3*i*sgn(a
) + 3*pi*i*log(abs(a))^2*sgn(a) + 6*pi*i*log(abs(a))*log(abs(b))*sgn(a) + 3*pi*i*log(abs(b))^2*sgn(a) - 4*pi^3
*i*sgn(b) + 3*pi*i*log(abs(a))^2*sgn(b) + 6*pi*i*log(abs(a))*log(abs(b))*sgn(b) + 3*pi*i*log(abs(b))^2*sgn(b)
+ 3*pi^2*log(abs(a))*sgn(a)*sgn(b) + 3*pi^2*log(abs(b))*sgn(a)*sgn(b) + 5*pi^3*i - 6*pi*i*log(abs(a))^2 - 12*p
i*i*log(abs(a))*log(abs(b)) - 6*pi*i*log(abs(b))^2 - 6*pi^2*log(abs(a))*sgn(a) - 6*pi^2*log(abs(b))*sgn(a) - 6
*pi^2*log(abs(a))*sgn(b) - 6*pi^2*log(abs(b))*sgn(b) + 9*pi^2*log(abs(a)) - 2*log(abs(a))^3 + 9*pi^2*log(abs(b
)) - 6*log(abs(a))^2*log(abs(b)) - 6*log(abs(a))*log(abs(b))^2 - 2*log(abs(b))^3))*e^(x*(log(abs(a)) + log(abs
(b))))/i
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maple [A] time = 0.01, size = 69, normalized size = 1.41 \[ \frac {\left (x^{2} \ln \relax (a )^{2}+2 x^{2} \ln \relax (a ) \ln \relax (b )+x^{2} \ln \relax (b )^{2}-2 x \ln \relax (a )-2 x \ln \relax (b )+2\right ) a^{x} b^{x}}{\left (\ln \relax (a )+\ln \relax (b )\right ) \left (\ln \relax (a )^{2}+2 \ln \relax (a ) \ln \relax (b )+\ln \relax (b )^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(a^x*b^x*x^2,x)
[Out]
(ln(a)^2*x^2+2*ln(a)*ln(b)*x^2+ln(b)^2*x^2-2*ln(a)*x-2*ln(b)*x+2)*a^x*b^x/(ln(a)+ln(b))/(ln(a)^2+2*ln(a)*ln(b)
+ln(b)^2)
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maxima [A] time = 0.92, size = 67, normalized size = 1.37 \[ \frac {{\left ({\left (\log \relax (a)^{2} + 2 \, \log \relax (a) \log \relax (b) + \log \relax (b)^{2}\right )} x^{2} - 2 \, x {\left (\log \relax (a) + \log \relax (b)\right )} + 2\right )} e^{\left (x \log \relax (a) + x \log \relax (b)\right )}}{\log \relax (a)^{3} + 3 \, \log \relax (a)^{2} \log \relax (b) + 3 \, \log \relax (a) \log \relax (b)^{2} + \log \relax (b)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(a^x*b^x*x^2,x, algorithm="maxima")
[Out]
((log(a)^2 + 2*log(a)*log(b) + log(b)^2)*x^2 - 2*x*(log(a) + log(b)) + 2)*e^(x*log(a) + x*log(b))/(log(a)^3 +
3*log(a)^2*log(b) + 3*log(a)*log(b)^2 + log(b)^3)
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mupad [B] time = 0.05, size = 35, normalized size = 0.71 \[ \frac {a^x\,b^x\,\left (x^2\,{\left (\ln \relax (a)+\ln \relax (b)\right )}^2-2\,x\,\left (\ln \relax (a)+\ln \relax (b)\right )+2\right )}{{\left (\ln \relax (a)+\ln \relax (b)\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(a^x*b^x*x^2,x)
[Out]
(a^x*b^x*(x^2*(log(a) + log(b))^2 - 2*x*(log(a) + log(b)) + 2))/(log(a) + log(b))^3
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sympy [A] time = 2.29, size = 279, normalized size = 5.69 \[ \begin {cases} \frac {a^{x} b^{x} x^{2} \log {\relax (a )}^{2}}{\log {\relax (a )}^{3} + 3 \log {\relax (a )}^{2} \log {\relax (b )} + 3 \log {\relax (a )} \log {\relax (b )}^{2} + \log {\relax (b )}^{3}} + \frac {2 a^{x} b^{x} x^{2} \log {\relax (a )} \log {\relax (b )}}{\log {\relax (a )}^{3} + 3 \log {\relax (a )}^{2} \log {\relax (b )} + 3 \log {\relax (a )} \log {\relax (b )}^{2} + \log {\relax (b )}^{3}} + \frac {a^{x} b^{x} x^{2} \log {\relax (b )}^{2}}{\log {\relax (a )}^{3} + 3 \log {\relax (a )}^{2} \log {\relax (b )} + 3 \log {\relax (a )} \log {\relax (b )}^{2} + \log {\relax (b )}^{3}} - \frac {2 a^{x} b^{x} x \log {\relax (a )}}{\log {\relax (a )}^{3} + 3 \log {\relax (a )}^{2} \log {\relax (b )} + 3 \log {\relax (a )} \log {\relax (b )}^{2} + \log {\relax (b )}^{3}} - \frac {2 a^{x} b^{x} x \log {\relax (b )}}{\log {\relax (a )}^{3} + 3 \log {\relax (a )}^{2} \log {\relax (b )} + 3 \log {\relax (a )} \log {\relax (b )}^{2} + \log {\relax (b )}^{3}} + \frac {2 a^{x} b^{x}}{\log {\relax (a )}^{3} + 3 \log {\relax (a )}^{2} \log {\relax (b )} + 3 \log {\relax (a )} \log {\relax (b )}^{2} + \log {\relax (b )}^{3}} & \text {for}\: a \neq \frac {1}{b} \\\tilde {\infty } b^{x} \left (\frac {1}{b}\right )^{x} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(a**x*b**x*x**2,x)
[Out]
Piecewise((a**x*b**x*x**2*log(a)**2/(log(a)**3 + 3*log(a)**2*log(b) + 3*log(a)*log(b)**2 + log(b)**3) + 2*a**x
*b**x*x**2*log(a)*log(b)/(log(a)**3 + 3*log(a)**2*log(b) + 3*log(a)*log(b)**2 + log(b)**3) + a**x*b**x*x**2*lo
g(b)**2/(log(a)**3 + 3*log(a)**2*log(b) + 3*log(a)*log(b)**2 + log(b)**3) - 2*a**x*b**x*x*log(a)/(log(a)**3 +
3*log(a)**2*log(b) + 3*log(a)*log(b)**2 + log(b)**3) - 2*a**x*b**x*x*log(b)/(log(a)**3 + 3*log(a)**2*log(b) +
3*log(a)*log(b)**2 + log(b)**3) + 2*a**x*b**x/(log(a)**3 + 3*log(a)**2*log(b) + 3*log(a)*log(b)**2 + log(b)**3
), Ne(a, 1/b)), (zoo*b**x*(1/b)**x, True))
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