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

Optimal. Leaf size=14 \[ \frac{a^x b^x}{\log (a)+\log (b)} \]

[Out]

(a^x*b^x)/(Log[a] + Log[b])

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Rubi [A]  time = 0.0123337, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2287, 2194} \[ \frac{a^x b^x}{\log (a)+\log (b)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^x*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 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 \, dx &=\int e^{x (\log (a)+\log (b))} \, dx\\ &=\frac{a^x b^x}{\log (a)+\log (b)}\\ \end{align*}

Mathematica [A]  time = 0.0055033, size = 14, normalized size = 1. \[ \frac{a^x b^x}{\log (a)+\log (b)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^x*b^x)/(Log[a] + Log[b])

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Maple [A]  time = 0.003, size = 15, normalized size = 1.1 \begin{align*}{\frac{{a}^{x}{b}^{x}}{\ln \left ( a \right ) +\ln \left ( b \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.24828, size = 36, normalized size = 2.57 \begin{align*} \frac{a^{x} b^{x}}{\log \left (a\right ) + \log \left (b\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^x*b^x/(log(a) + log(b))

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Sympy [A]  time = 0.564912, size = 24, normalized size = 1.71 \begin{align*} \begin{cases} \frac{a^{x} b^{x}}{\log{\left (a \right )} + \log{\left (b \right )}} & \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)

[Out]

Piecewise((a**x*b**x/(log(a) + log(b)), Ne(a, 1/b)), (zoo*b**x*(1/b)**x, True))

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Giac [B]  time = 1.25521, size = 327, normalized size = 23.36 \begin{align*} 2 \,{\left (\frac{2 \,{\left (\log \left ({\left | a \right |}\right ) + \log \left ({\left | b \right |}\right )\right )} \cos \left (-\frac{1}{2} \, \pi x \mathrm{sgn}\left (a\right ) - \frac{1}{2} \, \pi x \mathrm{sgn}\left (b\right ) + \pi x\right )}{{\left (2 \, \pi - \pi \mathrm{sgn}\left (a\right ) - \pi \mathrm{sgn}\left (b\right )\right )}^{2} + 4 \,{\left (\log \left ({\left | a \right |}\right ) + \log \left ({\left | b \right |}\right )\right )}^{2}} + \frac{{\left (2 \, \pi - \pi \mathrm{sgn}\left (a\right ) - \pi \mathrm{sgn}\left (b\right )\right )} \sin \left (-\frac{1}{2} \, \pi x \mathrm{sgn}\left (a\right ) - \frac{1}{2} \, \pi x \mathrm{sgn}\left (b\right ) + \pi x\right )}{{\left (2 \, \pi - \pi \mathrm{sgn}\left (a\right ) - \pi \mathrm{sgn}\left (b\right )\right )}^{2} + 4 \,{\left (\log \left ({\left | a \right |}\right ) + \log \left ({\left | b \right |}\right )\right )}^{2}}\right )} e^{\left (x{\left (\log \left ({\left | a \right |}\right ) + \log \left ({\left | b \right |}\right )\right )}\right )} - \frac{{\left (\frac{i e^{\left (\frac{1}{2} \,{\left (\pi{\left (\mathrm{sgn}\left (a\right ) - 1\right )} + \pi{\left (\mathrm{sgn}\left (b\right ) - 1\right )}\right )} i x\right )}}{\pi i \mathrm{sgn}\left (a\right ) + \pi i \mathrm{sgn}\left (b\right ) - 2 \, \pi i + 2 \, \log \left ({\left | a \right |}\right ) + 2 \, \log \left ({\left | b \right |}\right )} + \frac{i e^{\left (-\frac{1}{2} \,{\left (\pi{\left (\mathrm{sgn}\left (a\right ) - 1\right )} + \pi{\left (\mathrm{sgn}\left (b\right ) - 1\right )}\right )} i x\right )}}{\pi i \mathrm{sgn}\left (a\right ) + \pi i \mathrm{sgn}\left (b\right ) - 2 \, \pi i - 2 \, \log \left ({\left | a \right |}\right ) - 2 \, \log \left ({\left | b \right |}\right )}\right )} e^{\left (x{\left (\log \left ({\left | a \right |}\right ) + \log \left ({\left | b \right |}\right )\right )}\right )}}{i} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(2*(log(abs(a)) + log(abs(b)))*cos(-1/2*pi*x*sgn(a) - 1/2*pi*x*sgn(b) + pi*x)/((2*pi - pi*sgn(a) - pi*sgn(b)
)^2 + 4*(log(abs(a)) + log(abs(b)))^2) + (2*pi - pi*sgn(a) - pi*sgn(b))*sin(-1/2*pi*x*sgn(a) - 1/2*pi*x*sgn(b)
+ pi*x)/((2*pi - pi*sgn(a) - pi*sgn(b))^2 + 4*(log(abs(a)) + log(abs(b)))^2))*e^(x*(log(abs(a)) + log(abs(b))
)) - (i*e^(1/2*(pi*(sgn(a) - 1) + pi*(sgn(b) - 1))*i*x)/(pi*i*sgn(a) + pi*i*sgn(b) - 2*pi*i + 2*log(abs(a)) +
2*log(abs(b))) + i*e^(-1/2*(pi*(sgn(a) - 1) + pi*(sgn(b) - 1))*i*x)/(pi*i*sgn(a) + pi*i*sgn(b) - 2*pi*i - 2*lo
g(abs(a)) - 2*log(abs(b))))*e^(x*(log(abs(a)) + log(abs(b))))/i