\(\int a^x b^{-x} x^2 \, dx\) [571]

Optimal result

Integrand size = 12, antiderivative size = 61 \[ \int a^x b^{-x} x^2 \, dx=\frac {2 a^x b^{-x}}{(\log (a)-\log (b))^3}-\frac {2 a^x b^{-x} x}{(\log (a)-\log (b))^2}+\frac {a^x b^{-x} x^2}{\log (a)-\log (b)} \]

[Out]

2*a^x/(b^x)/(ln(a)-ln(b))^3-2*a^x*x/(b^x)/(ln(a)-ln(b))^2+a^x*x^2/(b^x)/(ln(a)-ln(b))

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2325, 2207, 2225} \[ \int a^x b^{-x} x^2 \, dx=\frac {x^2 a^x b^{-x}}{\log (a)-\log (b)}-\frac {2 x a^x b^{-x}}{(\log (a)-\log (b))^2}+\frac {2 a^x b^{-x}}{(\log (a)-\log (b))^3} \]

[In]

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

[Out]

(2*a^x)/(b^x*(Log[a] - Log[b])^3) - (2*a^x*x)/(b^x*(Log[a] - Log[b])^2) + (a^x*x^2)/(b^x*(Log[a] - Log[b]))

Rule 2207

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] &&  !TrueQ[$UseGamma]

Rule 2225

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 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 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*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.70 \[ \int a^x b^{-x} x^2 \, dx=\frac {a^x b^{-x} \left (2-2 x (\log (a)-\log (b))+x^2 (\log (a)-\log (b))^2\right )}{(\log (a)-\log (b))^3} \]

[In]

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

[Out]

(a^x*(2 - 2*x*(Log[a] - Log[b]) + x^2*(Log[a] - Log[b])^2))/(b^x*(Log[a] - Log[b])^3)

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.92

method	result	size
risch	\(\frac {\left (\ln \left (a \right )^{2} x^{2}-2 \ln \left (a \right ) \ln \left (b \right ) x^{2}+\ln \left (b \right )^{2} x^{2}-2 \ln \left (a \right ) x +2 \ln \left (b \right ) x +2\right ) a^{x} b^{-x}}{\left (\ln \left (a \right )-\ln \left (b \right )\right )^{3}}\)	\(56\)
gosper	\(\frac {\left (\ln \left (a \right )^{2} x^{2}-2 \ln \left (a \right ) \ln \left (b \right ) x^{2}+\ln \left (b \right )^{2} x^{2}-2 \ln \left (a \right ) x +2 \ln \left (b \right ) x +2\right ) a^{x} b^{-x}}{\left (\ln \left (a \right )-\ln \left (b \right )\right ) \left (\ln \left (a \right )^{2}-2 \ln \left (a \right ) \ln \left (b \right )+\ln \left (b \right )^{2}\right )}\)	\(73\)
meijerg	\(-\frac {2-\frac {\left (3 x^{2} \ln \left (a \right )^{2} \left (1-\frac {\ln \left (b \right )}{\ln \left (a \right )}\right )^{2}-6 x \ln \left (a \right ) \left (1-\frac {\ln \left (b \right )}{\ln \left (a \right )}\right )+6\right ) {\mathrm e}^{x \ln \left (a \right ) \left (1-\frac {\ln \left (b \right )}{\ln \left (a \right )}\right )}}{3}}{\ln \left (a \right )^{3} \left (1-\frac {\ln \left (b \right )}{\ln \left (a \right )}\right )^{3}}\)	\(76\)
norman	\(\left (\frac {x^{2} {\mathrm e}^{\ln \left (a \right ) x}}{\ln \left (a \right )-\ln \left (b \right )}-\frac {2 x \,{\mathrm e}^{\ln \left (a \right ) x}}{\ln \left (a \right )^{2}-2 \ln \left (a \right ) \ln \left (b \right )+\ln \left (b \right )^{2}}+\frac {2 \,{\mathrm e}^{\ln \left (a \right ) x}}{\left (\ln \left (a \right )^{2}-2 \ln \left (a \right ) \ln \left (b \right )+\ln \left (b \right )^{2}\right ) \left (\ln \left (a \right )-\ln \left (b \right )\right )}\right ) {\mathrm e}^{-\ln \left (b \right ) x}\)	\(86\)
parallelrisch	\(-\frac {\left (-\ln \left (a \right )^{2} x^{2} a^{x}+2 \ln \left (a \right ) \ln \left (b \right ) x^{2} a^{x}-\ln \left (b \right )^{2} x^{2} a^{x}+2 x \,a^{x} \ln \left (a \right )-2 x \,a^{x} \ln \left (b \right )-2 a^{x}\right ) b^{-x}}{\left (\ln \left (a \right )-\ln \left (b \right )\right ) \left (\ln \left (a \right )^{2}-2 \ln \left (a \right ) \ln \left (b \right )+\ln \left (b \right )^{2}\right )}\)	\(92\)

[In]

int(a^x*x^2/(b^x),x,method=_RETURNVERBOSE)

[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/(ln(a)-ln(b))^3/(b^x)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.23 \[ \int a^x b^{-x} x^2 \, dx=\frac {{\left (x^{2} \log \left (a\right )^{2} + x^{2} \log \left (b\right )^{2} - 2 \, x \log \left (a\right ) - 2 \, {\left (x^{2} \log \left (a\right ) - x\right )} \log \left (b\right ) + 2\right )} a^{x}}{{\left (\log \left (a\right )^{3} - 3 \, \log \left (a\right )^{2} \log \left (b\right ) + 3 \, \log \left (a\right ) \log \left (b\right )^{2} - \log \left (b\right )^{3}\right )} b^{x}} \]

[In]

integrate(a^x*x^2/(b^x),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/((log(a)^3 - 3*log(a)^2*log(b)
+ 3*log(a)*log(b)^2 - log(b)^3)*b^x)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (51) = 102\).

Time = 0.43 (sec) , antiderivative size = 333, normalized size of antiderivative = 5.46 \[ \int a^x b^{-x} x^2 \, dx=\begin {cases} \frac {a^{x} x^{2} \log {\left (a \right )}^{2}}{b^{x} \log {\left (a \right )}^{3} - 3 b^{x} \log {\left (a \right )}^{2} \log {\left (b \right )} + 3 b^{x} \log {\left (a \right )} \log {\left (b \right )}^{2} - b^{x} \log {\left (b \right )}^{3}} - \frac {2 a^{x} x^{2} \log {\left (a \right )} \log {\left (b \right )}}{b^{x} \log {\left (a \right )}^{3} - 3 b^{x} \log {\left (a \right )}^{2} \log {\left (b \right )} + 3 b^{x} \log {\left (a \right )} \log {\left (b \right )}^{2} - b^{x} \log {\left (b \right )}^{3}} + \frac {a^{x} x^{2} \log {\left (b \right )}^{2}}{b^{x} \log {\left (a \right )}^{3} - 3 b^{x} \log {\left (a \right )}^{2} \log {\left (b \right )} + 3 b^{x} \log {\left (a \right )} \log {\left (b \right )}^{2} - b^{x} \log {\left (b \right )}^{3}} - \frac {2 a^{x} x \log {\left (a \right )}}{b^{x} \log {\left (a \right )}^{3} - 3 b^{x} \log {\left (a \right )}^{2} \log {\left (b \right )} + 3 b^{x} \log {\left (a \right )} \log {\left (b \right )}^{2} - b^{x} \log {\left (b \right )}^{3}} + \frac {2 a^{x} x \log {\left (b \right )}}{b^{x} \log {\left (a \right )}^{3} - 3 b^{x} \log {\left (a \right )}^{2} \log {\left (b \right )} + 3 b^{x} \log {\left (a \right )} \log {\left (b \right )}^{2} - b^{x} \log {\left (b \right )}^{3}} + \frac {2 a^{x}}{b^{x} \log {\left (a \right )}^{3} - 3 b^{x} \log {\left (a \right )}^{2} \log {\left (b \right )} + 3 b^{x} \log {\left (a \right )} \log {\left (b \right )}^{2} - b^{x} \log {\left (b \right )}^{3}} & \text {for}\: a \neq b \\\frac {x^{3}}{3} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((a**x*x**2*log(a)**2/(b**x*log(a)**3 - 3*b**x*log(a)**2*log(b) + 3*b**x*log(a)*log(b)**2 - b**x*log(
b)**3) - 2*a**x*x**2*log(a)*log(b)/(b**x*log(a)**3 - 3*b**x*log(a)**2*log(b) + 3*b**x*log(a)*log(b)**2 - b**x*
log(b)**3) + a**x*x**2*log(b)**2/(b**x*log(a)**3 - 3*b**x*log(a)**2*log(b) + 3*b**x*log(a)*log(b)**2 - b**x*lo
g(b)**3) - 2*a**x*x*log(a)/(b**x*log(a)**3 - 3*b**x*log(a)**2*log(b) + 3*b**x*log(a)*log(b)**2 - b**x*log(b)**
3) + 2*a**x*x*log(b)/(b**x*log(a)**3 - 3*b**x*log(a)**2*log(b) + 3*b**x*log(a)*log(b)**2 - b**x*log(b)**3) + 2
*a**x/(b**x*log(a)**3 - 3*b**x*log(a)**2*log(b) + 3*b**x*log(a)*log(b)**2 - b**x*log(b)**3), Ne(a, b)), (x**3/
3, True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.18 \[ \int a^x b^{-x} x^2 \, dx=\frac {{\left ({\left (\log \left (a\right )^{2} - 2 \, \log \left (a\right ) \log \left (b\right ) + \log \left (b\right )^{2}\right )} x^{2} - 2 \, x {\left (\log \left (a\right ) - \log \left (b\right )\right )} + 2\right )} e^{\left (x \log \left (a\right ) - x \log \left (b\right )\right )}}{\log \left (a\right )^{3} - 3 \, \log \left (a\right )^{2} \log \left (b\right ) + 3 \, \log \left (a\right ) \log \left (b\right )^{2} - \log \left (b\right )^{3}} \]

[In]

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

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.33 (sec) , antiderivative size = 1817, normalized size of antiderivative = 29.79 \[ \int a^x b^{-x} x^2 \, dx=\text {Too large to display} \]

[In]

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

[Out]

(((pi^2*x^2*sgn(a)*sgn(b) - 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) - 3*pi^2*log(abs(a)) + 2*log(abs(a))^3 + 3*pi^2*log(abs(b)) - 6*log(abs(a))^2*log(abs(b)) + 6*log(abs(a))*l
og(abs(b))^2 - 2*log(abs(b))^3)/((3*pi^2*log(abs(a))*sgn(a)*sgn(b) - 3*pi^2*log(abs(b))*sgn(a)*sgn(b) - 3*pi^2
*log(abs(a)) + 2*log(abs(a))^3 + 3*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^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) - pi^3*sgn(b) + 3*pi*log(abs(a))^2*sgn(b) - 6*pi*log(abs(a))*log(abs(b))*sgn(b) + 3*pi*l
og(abs(b))^2*sgn(b))^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) - pi*x*sgn(a) + pi*x*sgn(b))*(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) - pi^3*sgn(b) + 3*pi*log(abs(a))^2*sgn(b) - 6*pi*log(ab
s(a))*log(abs(b))*sgn(b) + 3*pi*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) - 3*pi^2*log(abs(a)) + 2*log(abs(a))^3 + 3*pi^2*log(abs(b)) - 6*log(abs(a))^2*log(abs(b)) + 6*lo
g(abs(a))*log(abs(b))^2 - 2*log(abs(b))^3)^2 + (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) - pi^3*sgn(b) + 3*pi*log(abs(a))^2*sgn(b) - 6*pi*log(abs(a))*log(a
bs(b))*sgn(b) + 3*pi*log(abs(b))^2*sgn(b))^2))*cos(-1/2*pi*x*sgn(a) + 1/2*pi*x*sgn(b)) + (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) - pi*x*sgn(a) + p
i*x*sgn(b))*(3*pi^2*log(abs(a))*sgn(a)*sgn(b) - 3*pi^2*log(abs(b))*sgn(a)*sgn(b) - 3*pi^2*log(abs(a)) + 2*log(
abs(a))^3 + 3*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^2*log(abs(a))*sgn(a)*sgn(b) - 3*pi^2*log(abs(b))*sgn(a)*sgn(b) - 3*pi^2*log(abs(a)) + 2*log(abs(a))^3 +
3*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^3*s
gn(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) - 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))^2) + (pi^2*
x^2*sgn(a)*sgn(b) - 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)*(pi^3*sgn(a) - 3*pi*log(abs(a))^2*sgn(a) + 6*pi*log(abs(a))*log(abs(b))*sg
n(a) - 3*pi*log(abs(b))^2*sgn(a) - 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))/((3*pi^2*log(abs(a))*sgn(a)*sgn(b) - 3*pi^2*log(abs(b))*sgn(a)*sgn(b) - 3*pi^2
*log(abs(a)) + 2*log(abs(a))^3 + 3*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^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) - pi^3*sgn(b) + 3*pi*log(abs(a))^2*sgn(b) - 6*pi*log(abs(a))*log(abs(b))*sgn(b) + 3*pi*l
og(abs(b))^2*sgn(b))^2))*sin(-1/2*pi*x*sgn(a) + 1/2*pi*x*sgn(b)))*e^(x*(log(abs(a)) - log(abs(b)))) - 1/2*I*((
-I*pi^2*x^2*sgn(a)*sgn(b) + 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))*s
gn(b) + 2*pi*x^2*log(abs(b))*sgn(b) + I*pi^2*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*sgn(a) + 2*pi*x*sgn(b) + 4*I*x*log(abs(a)) - 4*I*x*log(abs(b)) - 4*I)*e^(1/2*I*p
i*x*sgn(a) - 1/2*I*pi*x*sgn(b))/(3*pi^2*log(abs(a))*sgn(a)*sgn(b) - 3*pi^2*log(abs(b))*sgn(a)*sgn(b) - I*pi^3*
sgn(a) + 3*I*pi*log(abs(a))^2*sgn(a) - 6*I*pi*log(abs(a))*log(abs(b))*sgn(a) + 3*I*pi*log(abs(b))^2*sgn(a) + I
*pi^3*sgn(b) - 3*I*pi*log(abs(a))^2*sgn(b) + 6*I*pi*log(abs(a))*log(abs(b))*sgn(b) - 3*I*pi*log(abs(b))^2*sgn(
b) - 3*pi^2*log(abs(a)) + 2*log(abs(a))^3 + 3*pi^2*log(abs(b)) - 6*log(abs(a))^2*log(abs(b)) + 6*log(abs(a))*l
og(abs(b))^2 - 2*log(abs(b))^3) - (-I*pi^2*x^2*sgn(a)*sgn(b) - 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) + I*pi^2*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*sgn(a) - 2*pi*x*sgn(b) + 4*I*x*log(abs(a)) -
4*I*x*log(abs(b)) - 4*I)*e^(-1/2*I*pi*x*sgn(a) + 1/2*I*pi*x*sgn(b))/(3*pi^2*log(abs(a))*sgn(a)*sgn(b) - 3*pi^2
*log(abs(b))*sgn(a)*sgn(b) + I*pi^3*sgn(a) - 3*I*pi*log(abs(a))^2*sgn(a) + 6*I*pi*log(abs(a))*log(abs(b))*sgn(
a) - 3*I*pi*log(abs(b))^2*sgn(a) - I*pi^3*sgn(b) + 3*I*pi*log(abs(a))^2*sgn(b) - 6*I*pi*log(abs(a))*log(abs(b)
)*sgn(b) + 3*I*pi*log(abs(b))^2*sgn(b) - 3*pi^2*log(abs(a)) + 2*log(abs(a))^3 + 3*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))))

Mupad [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.70 \[ \int a^x b^{-x} x^2 \, dx=\frac {a^x\,\left (x^2\,{\left (\ln \left (a\right )-\ln \left (b\right )\right )}^2-2\,x\,\left (\ln \left (a\right )-\ln \left (b\right )\right )+2\right )}{b^x\,{\left (\ln \left (a\right )-\ln \left (b\right )\right )}^3} \]

[In]

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

[Out]

(a^x*(x^2*(log(a) - log(b))^2 - 2*x*(log(a) - log(b)) + 2))/(b^x*(log(a) - log(b))^3)