3.494 \(\int a^{m x} b^{n x} \, dx\)

Optimal. Leaf size=22 \[ \frac{a^{m x} b^{n x}}{m \log (a)+n \log (b)} \]

[Out]

(a^(m*x)*b^(n*x))/(m*Log[a] + n*Log[b])

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Rubi [A]  time = 0.0263098, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2287, 2194} \[ \frac{a^{m x} b^{n x}}{m \log (a)+n \log (b)} \]

Antiderivative was successfully verified.

[In]

Int[a^(m*x)*b^(n*x),x]

[Out]

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

Mathematica [A]  time = 0.0135011, size = 22, normalized size = 1. \[ \frac{a^{m x} b^{n x}}{m \log (a)+n \log (b)} \]

Antiderivative was successfully verified.

[In]

Integrate[a^(m*x)*b^(n*x),x]

[Out]

(a^(m*x)*b^(n*x))/(m*Log[a] + n*Log[b])

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Maple [A]  time = 0.004, size = 23, normalized size = 1.1 \begin{align*}{\frac{{a}^{mx}{b}^{nx}}{m\ln \left ( a \right ) +n\ln \left ( b \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^(m*x)*b^(n*x)/(m*ln(a)+n*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^(m*x)*b^(n*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.91508, size = 53, normalized size = 2.41 \begin{align*} \frac{a^{m x} b^{n x}}{m \log \left (a\right ) + n \log \left (b\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^(m*x)*b^(n*x)/(m*log(a) + n*log(b))

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Sympy [A]  time = 0.654854, size = 42, normalized size = 1.91 \begin{align*} \begin{cases} \frac{a^{m x} b^{n x}}{m \log{\left (a \right )} + n \log{\left (b \right )}} & \text{for}\: m \neq - \frac{n \log{\left (b \right )}}{\log{\left (a \right )}} \\b^{n x} x e^{- n x \log{\left (b \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**(m*x)*b**(n*x)/(m*log(a) + n*log(b)), Ne(m, -n*log(b)/log(a))), (b**(n*x)*x*exp(-n*x*log(b)), Tr
ue))

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Giac [C]  time = 1.14874, size = 439, normalized size = 19.95 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(2*(m*log(abs(a)) + n*log(abs(b)))*cos(-1/2*pi*m*x*sgn(a) - 1/2*pi*n*x*sgn(b) + 1/2*pi*m*x + 1/2*pi*n*x)/((p
i*m*sgn(a) + pi*n*sgn(b) - pi*m - pi*n)^2 + 4*(m*log(abs(a)) + n*log(abs(b)))^2) - (pi*m*sgn(a) + pi*n*sgn(b)
- pi*m - pi*n)*sin(-1/2*pi*m*x*sgn(a) - 1/2*pi*n*x*sgn(b) + 1/2*pi*m*x + 1/2*pi*n*x)/((pi*m*sgn(a) + pi*n*sgn(
b) - pi*m - pi*n)^2 + 4*(m*log(abs(a)) + n*log(abs(b)))^2))*e^((m*log(abs(a)) + n*log(abs(b)))*x) - 1/2*I*(-2*
I*e^(1/2*I*pi*m*x*sgn(a) + 1/2*I*pi*n*x*sgn(b) - 1/2*I*pi*m*x - 1/2*I*pi*n*x)/(I*pi*m*sgn(a) + I*pi*n*sgn(b) -
I*pi*m - I*pi*n + 2*m*log(abs(a)) + 2*n*log(abs(b))) + 2*I*e^(-1/2*I*pi*m*x*sgn(a) - 1/2*I*pi*n*x*sgn(b) + 1/
2*I*pi*m*x + 1/2*I*pi*n*x)/(-I*pi*m*sgn(a) - I*pi*n*sgn(b) + I*pi*m + I*pi*n + 2*m*log(abs(a)) + 2*n*log(abs(b
))))*e^((m*log(abs(a)) + n*log(abs(b)))*x)