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3.564 \(\int a^x b^x x \, dx\)

Optimal. Leaf size=31 \[ \frac{x a^x b^x}{\log (a)+\log (b)}-\frac{a^x b^x}{(\log (a)+\log (b))^2} \]

[Out]

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

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Rubi [A]  time = 0.0286941, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {2287, 2176, 2194} \[ \frac{x a^x b^x}{\log (a)+\log (b)}-\frac{a^x b^x}{(\log (a)+\log (b))^2} \]

Antiderivative was successfully verified.

[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 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 \, dx &=\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]  time = 0.0123487, size = 26, normalized size = 0.84 \[ a^x b^x \left (\frac{x}{\log (a)+\log (b)}-\frac{1}{(\log (a)+\log (b))^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.005, size = 25, normalized size = 0.8 \begin{align*}{\frac{ \left ( \ln \left ( b \right ) x+\ln \left ( a \right ) x-1 \right ){a}^{x}{b}^{x}}{ \left ( \ln \left ( a \right ) +\ln \left ( b \right ) \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.99224, size = 50, normalized size = 1.61 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.357, size = 101, normalized size = 3.26 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 1.17821, size = 97, normalized size = 3.13 \begin{align*} \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} \\\tilde{\infty } b^{x} \left (\frac{1}{b}\right )^{x} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)), (zoo*b**x*(1/b)**
x, True))

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Giac [B]  time = 1.26627, size = 1377, normalized size = 44.42 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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*((2*i*x*log(abs(a)) + 2*i*x*log(abs(b)) - pi*x*sgn(a) - pi*x*sgn(b) + 2*pi*x - 2*i)*
e^(1/2*(pi*(sgn(a) - 1) + pi*(sgn(b) - 1))*i*x)/(2*pi*i*log(abs(a))*sgn(a) + 2*pi*i*log(abs(b))*sgn(a) + 2*pi*
i*log(abs(a))*sgn(b) + 2*pi*i*log(abs(b))*sgn(b) - pi^2*sgn(a)*sgn(b) - 4*pi*i*log(abs(a)) - 4*pi*i*log(abs(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*i*x*log(abs(a)) + 2*i*x*log(abs(b)) + pi*x*sgn(a) + pi*x*sgn(b) - 2*pi*x - 2*i)*e^(-1/2*(pi*(sgn(a) - 1) +
pi*(sgn(b) - 1))*i*x)/(2*pi*i*log(abs(a))*sgn(a) + 2*pi*i*log(abs(b))*sgn(a) + 2*pi*i*log(abs(a))*sgn(b) + 2*p
i*i*log(abs(b))*sgn(b) + pi^2*sgn(a)*sgn(b) - 4*pi*i*log(abs(a)) - 4*pi*i*log(abs(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))*e^(x*(log(abs(a)) + log(abs
(b))))/i

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