\(\int a^x b^x x \, dx\) [564]

Optimal result

Integrand size = 8, antiderivative size = 31 \[ \int a^x b^x x \, dx=-\frac {a^x b^x}{(\log (a)+\log (b))^2}+\frac {a^x b^x x}{\log (a)+\log (b)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2325, 2207, 2225} \[ \int a^x b^x x \, dx=\frac {x a^x b^x}{\log (a)+\log (b)}-\frac {a^x b^x}{(\log (a)+\log (b))^2} \]

[In]

Int[a^x*b^x*x,x]

[Out]

-((a^x*b^x)/(Log[a] + Log[b])^2) + (a^x*b^x*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 \, dx \\ & = \frac {a^x b^x x}{\log (a)+\log (b)}-\frac {\int e^{x (\log (a)+\log (b))} \, dx}{\log (a)+\log (b)} \\ & = -\frac {a^x b^x}{(\log (a)+\log (b))^2}+\frac {a^x b^x x}{\log (a)+\log (b)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int a^x b^x x \, dx=a^x b^x \left (-\frac {1}{(\log (a)+\log (b))^2}+\frac {x}{\log (a)+\log (b)}\right ) \]

[In]

Integrate[a^x*b^x*x,x]

[Out]

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

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81

method	result	size
gosper	\(\frac {\left (\ln \left (a \right ) x +\ln \left (b \right ) x -1\right ) a^{x} b^{x}}{\left (\ln \left (a \right )+\ln \left (b \right )\right )^{2}}\)	\(25\)
risch	\(\frac {\left (\ln \left (a \right ) x +\ln \left (b \right ) x -1\right ) a^{x} b^{x}}{\left (\ln \left (a \right )+\ln \left (b \right )\right )^{2}}\)	\(25\)
parallelrisch	\(\frac {b^{x} x \,a^{x} \ln \left (a \right )+b^{x} x \,a^{x} \ln \left (b \right )-a^{x} b^{x}}{\left (\ln \left (a \right )+\ln \left (b \right )\right )^{2}}\)	\(38\)
norman	\(\frac {x \,{\mathrm e}^{\ln \left (a \right ) x} {\mathrm e}^{\ln \left (b \right ) x}}{\ln \left (a \right )+\ln \left (b \right )}-\frac {{\mathrm e}^{\ln \left (a \right ) x} {\mathrm e}^{\ln \left (b \right ) x}}{\left (\ln \left (a \right )+\ln \left (b \right )\right )^{2}}\)	\(40\)
meijerg	\(\frac {1-\frac {\left (2-2 x \ln \left (b \right ) \left (1+\frac {\ln \left (a \right )}{\ln \left (b \right )}\right )\right ) {\mathrm e}^{x \ln \left (b \right ) \left (1+\frac {\ln \left (a \right )}{\ln \left (b \right )}\right )}}{2}}{\ln \left (b \right )^{2} \left (1+\frac {\ln \left (a \right )}{\ln \left (b \right )}\right )^{2}}\)	\(51\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int a^x b^x x \, dx=\frac {{\left (x \log \left (a\right ) + x \log \left (b\right ) - 1\right )} a^{x} b^{x}}{\log \left (a\right )^{2} + 2 \, \log \left (a\right ) \log \left (b\right ) + \log \left (b\right )^{2}} \]

[In]

integrate(a^x*b^x*x,x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (29) = 58\).

Time = 0.37 (sec) , antiderivative size = 190, normalized size of antiderivative = 6.13 \[ \int a^x b^x x \, dx=\begin {cases} \frac {a^{x} b^{x} x \log {\left (a \right )}}{\log {\left (a \right )}^{2} + 2 \log {\left (a \right )} \log {\left (b \right )} + \log {\left (b \right )}^{2}} + \frac {a^{x} b^{x} x \log {\left (b \right )}}{\log {\left (a \right )}^{2} + 2 \log {\left (a \right )} \log {\left (b \right )} + \log {\left (b \right )}^{2}} - \frac {a^{x} b^{x}}{\log {\left (a \right )}^{2} + 2 \log {\left (a \right )} \log {\left (b \right )} + \log {\left (b \right )}^{2}} & \text {for}\: a \neq \frac {1}{b} \\\frac {b^{x} x \left (\frac {1}{b}\right )^{x} \log {\left (\frac {1}{b} \right )}}{\log {\left (\frac {1}{b} \right )}^{2} + 2 \log {\left (\frac {1}{b} \right )} \log {\left (b \right )} + \log {\left (b \right )}^{2}} + \frac {b^{x} x \left (\frac {1}{b}\right )^{x} \log {\left (b \right )}}{\log {\left (\frac {1}{b} \right )}^{2} + 2 \log {\left (\frac {1}{b} \right )} \log {\left (b \right )} + \log {\left (b \right )}^{2}} - \frac {b^{x} \left (\frac {1}{b}\right )^{x}}{\log {\left (\frac {1}{b} \right )}^{2} + 2 \log {\left (\frac {1}{b} \right )} \log {\left (b \right )} + \log {\left (b \right )}^{2}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((a**x*b**x*x*log(a)/(log(a)**2 + 2*log(a)*log(b) + log(b)**2) + a**x*b**x*x*log(b)/(log(a)**2 + 2*lo
g(a)*log(b) + log(b)**2) - a**x*b**x/(log(a)**2 + 2*log(a)*log(b) + log(b)**2), Ne(a, 1/b)), (b**x*x*(1/b)**x*
log(1/b)/(log(1/b)**2 + 2*log(1/b)*log(b) + log(b)**2) + b**x*x*(1/b)**x*log(b)/(log(1/b)**2 + 2*log(1/b)*log(
b) + log(b)**2) - b**x*(1/b)**x/(log(1/b)**2 + 2*log(1/b)*log(b) + log(b)**2), True))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.19 \[ \int a^x b^x x \, dx=\frac {{\left (x {\left (\log \left (a\right ) + \log \left (b\right )\right )} - 1\right )} e^{\left (x \log \left (a\right ) + x \log \left (b\right )\right )}}{\log \left (a\right )^{2} + 2 \, \log \left (a\right ) \log \left (b\right ) + \log \left (b\right )^{2}} \]

[In]

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

[Out]

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

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.31 (sec) , antiderivative size = 994, normalized size of antiderivative = 32.06 \[ \int a^x b^x x \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int a^x b^x x \, dx=\frac {a^x\,b^x\,\left (x\,\left (\ln \left (a\right )+\ln \left (b\right )\right )-1\right )}{{\left (\ln \left (a\right )+\ln \left (b\right )\right )}^2} \]

[In]

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

[Out]

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