3.569 \(\int a^x b^x c^x \, dx\)

Optimal. Leaf size=19 \[ \frac{a^x b^x c^x}{\log (a)+\log (b)+\log (c)} \]

[Out]

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

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Rubi [A]  time = 0.0410007, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2287, 2194} \[ \frac{a^x b^x c^x}{\log (a)+\log (b)+\log (c)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0162939, size = 21, normalized size = 1.11 \[ \frac{e^{x (\log (a)+\log (b)+\log (c))}}{\log (a)+\log (b)+\log (c)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

E^(x*(Log[a] + Log[b] + Log[c]))/(Log[a] + Log[b] + Log[c])

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Maple [A]  time = 0.005, size = 20, normalized size = 1.1 \begin{align*}{\frac{{a}^{x}{b}^{x}{c}^{x}}{\ln \left ( a \right ) +\ln \left ( b \right ) +\ln \left ( c \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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*c^x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.2876, size = 54, normalized size = 2.84 \begin{align*} \frac{a^{x} b^{x} c^{x}}{\log \left (a\right ) + \log \left (b\right ) + \log \left (c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^x*b^x*c^x/(log(a) + log(b) + log(c))

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Sympy [A]  time = 2.65143, size = 41, normalized size = 2.16 \begin{align*} \begin{cases} \frac{a^{x} b^{x} c^{x}}{\log{\left (a \right )} + \log{\left (b \right )} + \log{\left (c \right )}} & \text{for}\: a \neq \frac{1}{b c} \\\tilde{\infty } b^{x} c^{x} \left (\frac{1}{b}\right )^{x} \left (\frac{1}{c}\right )^{x} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.47958, size = 429, normalized size = 22.58 \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*c^x,x, algorithm="giac")

[Out]

2*(2*(log(abs(a)) + log(abs(b)) + log(abs(c)))*cos(-1/2*pi*x*sgn(a) - 1/2*pi*x*sgn(b) - 1/2*pi*x*sgn(c) + 3/2*
pi*x)/((3*pi - pi*sgn(a) - pi*sgn(b) - pi*sgn(c))^2 + 4*(log(abs(a)) + log(abs(b)) + log(abs(c)))^2) + (3*pi -
pi*sgn(a) - pi*sgn(b) - pi*sgn(c))*sin(-1/2*pi*x*sgn(a) - 1/2*pi*x*sgn(b) - 1/2*pi*x*sgn(c) + 3/2*pi*x)/((3*p
i - pi*sgn(a) - pi*sgn(b) - pi*sgn(c))^2 + 4*(log(abs(a)) + log(abs(b)) + log(abs(c)))^2))*e^(x*(log(abs(a)) +
log(abs(b)) + log(abs(c)))) - (i*e^(1/2*(pi*(sgn(a) - 1) + pi*(sgn(b) - 1) + pi*(sgn(c) - 1))*i*x)/(pi*i*sgn(
a) + pi*i*sgn(b) + pi*i*sgn(c) - 3*pi*i + 2*log(abs(a)) + 2*log(abs(b)) + 2*log(abs(c))) + i*e^(-1/2*(pi*(sgn(
a) - 1) + pi*(sgn(b) - 1) + pi*(sgn(c) - 1))*i*x)/(pi*i*sgn(a) + pi*i*sgn(b) + pi*i*sgn(c) - 3*pi*i - 2*log(ab
s(a)) - 2*log(abs(b)) - 2*log(abs(c))))*e^(x*(log(abs(a)) + log(abs(b)) + log(abs(c))))/i