3.571 \(\int a^x b^{-x} x^2 \, dx\)
Optimal. Leaf size=61 \[ \frac {x^2 a^x b^{-x}}{\log (a)-\log (b)}-\frac {2 x a^x b^{-x}}{(\log (a)-\log (b))^2}+\frac {2 a^x b^{-x}}{(\log (a)-\log (b))^3} \]
[Out]
2*a^x/(b^x)/(ln(a)-ln(b))^3-2*a^x*x/(b^x)/(ln(a)-ln(b))^2+a^x*x^2/(b^x)/(ln(a)-ln(b))
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Rubi [A] time = 0.07, antiderivative size = 61, normalized size of antiderivative = 1.00,
number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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*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 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 = 43, normalized size = 0.70 \[ \frac {a^x b^{-x} \left (x^2 (\log (a)-\log (b))^2-2 x (\log (a)-\log (b))+2\right )}{(\log (a)-\log (b))^3} \]
Antiderivative was successfully verified.
[In]
Integrate[(a^x*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)
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fricas [A] time = 0.41, size = 75, normalized size = 1.23 \[ \frac {{\left (x^{2} \log \relax (a)^{2} + x^{2} \log \relax (b)^{2} - 2 \, x \log \relax (a) - 2 \, {\left (x^{2} \log \relax (a) - x\right )} \log \relax (b) + 2\right )} a^{x}}{{\left (\log \relax (a)^{3} - 3 \, \log \relax (a)^{2} \log \relax (b) + 3 \, \log \relax (a) \log \relax (b)^{2} - \log \relax (b)^{3}\right )} b^{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[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)
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giac [B] time = 0.36, size = 1859, normalized size = 30.48 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[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*((pi
^2*i*x^2*sgn(a)*sgn(b) - 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(ab
s(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*l
og(abs(b))*sgn(b) - 4*i*x*log(abs(a)) + 4*i*x*log(abs(b)) + 2*pi*x*sgn(a) - 2*pi*x*sgn(b) + 4*i)*e^(1/2*(pi*(s
gn(a) - 1) - pi*(sgn(b) - 1))*i*x)/(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) - 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)*s
gn(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) + (pi^2*i*x^2*sgn(a)*sgn(b) - 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*i*x*log(abs(a)) + 4*i*x*log(abs(b)) - 2*pi*x*sgn
(a) + 2*pi*x*sgn(b) + 4*i)*e^(-1/2*(pi*(sgn(a) - 1) - pi*(sgn(b) - 1))*i*x)/(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) - 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))*sg
n(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))*e^(x*(log(abs(a)) - log(abs(b))))
/i
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maple [A] time = 0.01, size = 73, normalized size = 1.20 \[ \frac {\left (x^{2} \ln \relax (a )^{2}-2 x^{2} \ln \relax (a ) \ln \relax (b )+x^{2} \ln \relax (b )^{2}-2 x \ln \relax (a )+2 x \ln \relax (b )+2\right ) a^{x} b^{-x}}{\left (\ln \relax (a )-\ln \relax (b )\right ) \left (\ln \relax (a )^{2}-2 \ln \relax (a ) \ln \relax (b )+\ln \relax (b )^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(a^x*x^2/(b^x),x)
[Out]
(x^2*ln(a)^2-2*x^2*ln(a)*ln(b)+x^2*ln(b)^2-2*x*ln(a)+2*x*ln(b)+2)*a^x/(ln(a)-ln(b))/(ln(a)^2-2*ln(a)*ln(b)+ln(
b)^2)/(b^x)
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maxima [A] time = 0.97, size = 72, normalized size = 1.18 \[ \frac {{\left ({\left (\log \relax (a)^{2} - 2 \, \log \relax (a) \log \relax (b) + \log \relax (b)^{2}\right )} x^{2} - 2 \, x {\left (\log \relax (a) - \log \relax (b)\right )} + 2\right )} e^{\left (x \log \relax (a) - x \log \relax (b)\right )}}{\log \relax (a)^{3} - 3 \, \log \relax (a)^{2} \log \relax (b) + 3 \, \log \relax (a) \log \relax (b)^{2} - \log \relax (b)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(a^x*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)
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mupad [B] time = 3.56, size = 43, normalized size = 0.70 \[ \frac {a^x\,\left (x^2\,{\left (\ln \relax (a)-\ln \relax (b)\right )}^2-2\,x\,\left (\ln \relax (a)-\ln \relax (b)\right )+2\right )}{b^x\,{\left (\ln \relax (a)-\ln \relax (b)\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[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)
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sympy [A] time = 1.74, size = 333, normalized size = 5.46 \[ \begin {cases} \frac {a^{x} x^{2} \log {\relax (a )}^{2}}{b^{x} \log {\relax (a )}^{3} - 3 b^{x} \log {\relax (a )}^{2} \log {\relax (b )} + 3 b^{x} \log {\relax (a )} \log {\relax (b )}^{2} - b^{x} \log {\relax (b )}^{3}} - \frac {2 a^{x} x^{2} \log {\relax (a )} \log {\relax (b )}}{b^{x} \log {\relax (a )}^{3} - 3 b^{x} \log {\relax (a )}^{2} \log {\relax (b )} + 3 b^{x} \log {\relax (a )} \log {\relax (b )}^{2} - b^{x} \log {\relax (b )}^{3}} + \frac {a^{x} x^{2} \log {\relax (b )}^{2}}{b^{x} \log {\relax (a )}^{3} - 3 b^{x} \log {\relax (a )}^{2} \log {\relax (b )} + 3 b^{x} \log {\relax (a )} \log {\relax (b )}^{2} - b^{x} \log {\relax (b )}^{3}} - \frac {2 a^{x} x \log {\relax (a )}}{b^{x} \log {\relax (a )}^{3} - 3 b^{x} \log {\relax (a )}^{2} \log {\relax (b )} + 3 b^{x} \log {\relax (a )} \log {\relax (b )}^{2} - b^{x} \log {\relax (b )}^{3}} + \frac {2 a^{x} x \log {\relax (b )}}{b^{x} \log {\relax (a )}^{3} - 3 b^{x} \log {\relax (a )}^{2} \log {\relax (b )} + 3 b^{x} \log {\relax (a )} \log {\relax (b )}^{2} - b^{x} \log {\relax (b )}^{3}} + \frac {2 a^{x}}{b^{x} \log {\relax (a )}^{3} - 3 b^{x} \log {\relax (a )}^{2} \log {\relax (b )} + 3 b^{x} \log {\relax (a )} \log {\relax (b )}^{2} - b^{x} \log {\relax (b )}^{3}} & \text {for}\: a \neq b \\\frac {x^{3}}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[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))
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