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*(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]))

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Rubi [A]  time = 0.0697103, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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 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 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]

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*}

Mathematica [A]  time = 0.0281628, size = 43, normalized size = 0.7 \[ \frac{a^x b^{-x} \left (x^2 (\log (a)-\log (b))^2-2 x (\log (a)-\log (b))+2\right )}{(\log (a)-\log (b))^3} \]

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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Maple [A]  time = 0.007, size = 73, normalized size = 1.2 \begin{align*}{\frac{ \left ( \left ( \ln \left ( a \right ) \right ) ^{2}{x}^{2}-2\,\ln \left ( a \right ) \ln \left ( b \right ){x}^{2}+ \left ( \ln \left ( b \right ) \right ) ^{2}{x}^{2}-2\,\ln \left ( a \right ) x+2\,\ln \left ( b \right ) x+2 \right ){a}^{x}}{ \left ( \ln \left ( a \right ) -\ln \left ( b \right ) \right ) \left ( \left ( \ln \left ( a \right ) \right ) ^{2}-2\,\ln \left ( a \right ) \ln \left ( b \right ) + \left ( \ln \left ( b \right ) \right ) ^{2} \right ){b}^{x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Maxima [A]  time = 1.03328, size = 97, normalized size = 1.59 \begin{align*} \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}} \end{align*}

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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Fricas [A]  time = 1.31193, size = 200, normalized size = 3.28 \begin{align*} \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}} \end{align*}

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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Sympy [A]  time = 1.66017, size = 333, normalized size = 5.46 \begin{align*} \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} \end{align*}

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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Giac [B]  time = 1.31603, size = 2510, normalized size = 41.15 \begin{align*} \text{result too large to display} \end{align*}

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