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 = 0.0627915, 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 \[ \frac{a^x b^x c^x}{\log (a)+\log (b)+\log (c)} \]

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])

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Rubi in Sympy [A]  time = 9.47671, size = 22, normalized size = 1.16 \[ \frac{e^{x \left (\log{\left (a \right )} + \log{\left (b \right )} + \log{\left (c \right )}\right )}}{\log{\left (a \right )} + \log{\left (b \right )} + \log{\left (c \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(a**x*b**x*c**x,x)

[Out]

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

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Mathematica [A]  time = 0.00793654, size = 19, normalized size = 1. \[ \frac{a^x b^x c^x}{\log (a)+\log (b)+\log (c)} \]

Antiderivative was successfully verified.

[In]  Integrate[a^x*b^x*c^x,x]

[Out]

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

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Maple [A]  time = 0.007, size = 20, normalized size = 1.1 \[{\frac{{a}^{x}{b}^{x}{c}^{x}}{\ln \left ( a \right ) +\ln \left ( b \right ) +\ln \left ( c \right ) }} \]

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. \[ \text{Exception raised: ValueError} \]

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 = 0.248315, size = 26, normalized size = 1.37 \[ \frac{a^{x} b^{x} c^{x}}{\log \left (a\right ) + \log \left (b\right ) + \log \left (c\right )} \]

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 = 3.82703, size = 41, normalized size = 2.16 \[ \begin{cases} \frac{a^{x} b^{x} c^{x}}{\log{\left (a \right )} + \log{\left (b \right )} + \log{\left (c \right )}} & \text{for}\: a \neq \frac{1}{b c} \\\tilde{\infty } b^{x} c^{x} \left (\frac{1}{b}\right )^{x} \left (\frac{1}{c}\right )^{x} & \text{otherwise} \end{cases} \]

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/XCAS [A]  time = 0.254901, size = 429, normalized size = 22.58 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(a^x*b^x*c^x,x, algorithm="giac")

[Out]

2*(2*(ln(abs(a)) + ln(abs(b)) + ln(abs(c)))*cos(-1/2*pi*x*sign(a) - 1/2*pi*x*sig
n(b) - 1/2*pi*x*sign(c) + 3/2*pi*x)/((3*pi - pi*sign(a) - pi*sign(b) - pi*sign(c
))^2 + 4*(ln(abs(a)) + ln(abs(b)) + ln(abs(c)))^2) + (3*pi - pi*sign(a) - pi*sig
n(b) - pi*sign(c))*sin(-1/2*pi*x*sign(a) - 1/2*pi*x*sign(b) - 1/2*pi*x*sign(c) +
3/2*pi*x)/((3*pi - pi*sign(a) - pi*sign(b) - pi*sign(c))^2 + 4*(ln(abs(a)) + ln
(abs(b)) + ln(abs(c)))^2))*e^(x*(ln(abs(a)) + ln(abs(b)) + ln(abs(c)))) - (i*e^(
1/2*(pi*(sign(a) - 1) + pi*(sign(b) - 1) + pi*(sign(c) - 1))*i*x)/(pi*i*sign(a)
+ pi*i*sign(b) + pi*i*sign(c) - 3*pi*i + 2*ln(abs(a)) + 2*ln(abs(b)) + 2*ln(abs(
c))) + i*e^(-1/2*(pi*(sign(a) - 1) + pi*(sign(b) - 1) + pi*(sign(c) - 1))*i*x)/(
pi*i*sign(a) + pi*i*sign(b) + pi*i*sign(c) - 3*pi*i - 2*ln(abs(a)) - 2*ln(abs(b)
) - 2*ln(abs(c))))*e^(x*(ln(abs(a)) + ln(abs(b)) + ln(abs(c))))/i