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 = 0.0207525, 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 \[ \frac{a^x b^x}{\log (a)+\log (b)} \]

Antiderivative was successfully verified.

[In]  Int[a^x*b^x,x]

[Out]

(a^x*b^x)/(Log[a] + Log[b])

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Rubi in Sympy [A]  time = 3.52051, size = 15, normalized size = 1.07 \[ \frac{e^{x \left (\log{\left (a \right )} + \log{\left (b \right )}\right )}}{\log{\left (a \right )} + \log{\left (b \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(a**x*b**x,x)

[Out]

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

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Mathematica [A]  time = 0.00411626, 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 \[{\frac{{a}^{x}{b}^{x}}{\ln \left ( a \right ) +\ln \left ( b \right ) }} \]

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

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 = 0.243938, size = 19, normalized size = 1.36 \[ \frac{a^{x} b^{x}}{\log \left (a\right ) + \log \left (b\right )} \]

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 = 0.954187, size = 24, normalized size = 1.71 \[ \begin{cases} \frac{a^{x} b^{x}}{\log{\left (a \right )} + \log{\left (b \right )}} & \text{for}\: a \neq \frac{1}{b} \\\tilde{\infty } b^{x} \left (\frac{1}{b}\right )^{x} & \text{otherwise} \end{cases} \]

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/XCAS [A]  time = 0.243168, size = 327, normalized size = 23.36 \[ 2 \,{\left (\frac{2 \,{\left ({\rm ln}\left ({\left | a \right |}\right ) +{\rm ln}\left ({\left | b \right |}\right )\right )} \cos \left (-\frac{1}{2} \, \pi x{\rm sign}\left (a\right ) - \frac{1}{2} \, \pi x{\rm sign}\left (b\right ) + \pi x\right )}{{\left (2 \, \pi - \pi{\rm sign}\left (a\right ) - \pi{\rm sign}\left (b\right )\right )}^{2} + 4 \,{\left ({\rm ln}\left ({\left | a \right |}\right ) +{\rm ln}\left ({\left | b \right |}\right )\right )}^{2}} + \frac{{\left (2 \, \pi - \pi{\rm sign}\left (a\right ) - \pi{\rm sign}\left (b\right )\right )} \sin \left (-\frac{1}{2} \, \pi x{\rm sign}\left (a\right ) - \frac{1}{2} \, \pi x{\rm sign}\left (b\right ) + \pi x\right )}{{\left (2 \, \pi - \pi{\rm sign}\left (a\right ) - \pi{\rm sign}\left (b\right )\right )}^{2} + 4 \,{\left ({\rm ln}\left ({\left | a \right |}\right ) +{\rm ln}\left ({\left | b \right |}\right )\right )}^{2}}\right )} e^{\left (x{\left ({\rm ln}\left ({\left | a \right |}\right ) +{\rm ln}\left ({\left | b \right |}\right )\right )}\right )} - \frac{{\left (\frac{i e^{\left (\frac{1}{2} \,{\left (\pi{\left ({\rm sign}\left (a\right ) - 1\right )} + \pi{\left ({\rm sign}\left (b\right ) - 1\right )}\right )} i x\right )}}{\pi i{\rm sign}\left (a\right ) + \pi i{\rm sign}\left (b\right ) - 2 \, \pi i + 2 \,{\rm ln}\left ({\left | a \right |}\right ) + 2 \,{\rm ln}\left ({\left | b \right |}\right )} + \frac{i e^{\left (-\frac{1}{2} \,{\left (\pi{\left ({\rm sign}\left (a\right ) - 1\right )} + \pi{\left ({\rm sign}\left (b\right ) - 1\right )}\right )} i x\right )}}{\pi i{\rm sign}\left (a\right ) + \pi i{\rm sign}\left (b\right ) - 2 \, \pi i - 2 \,{\rm ln}\left ({\left | a \right |}\right ) - 2 \,{\rm ln}\left ({\left | b \right |}\right )}\right )} e^{\left (x{\left ({\rm ln}\left ({\left | a \right |}\right ) +{\rm ln}\left ({\left | b \right |}\right )\right )}\right )}}{i} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(2*(ln(abs(a)) + ln(abs(b)))*cos(-1/2*pi*x*sign(a) - 1/2*pi*x*sign(b) + pi*x)/
((2*pi - pi*sign(a) - pi*sign(b))^2 + 4*(ln(abs(a)) + ln(abs(b)))^2) + (2*pi - p
i*sign(a) - pi*sign(b))*sin(-1/2*pi*x*sign(a) - 1/2*pi*x*sign(b) + pi*x)/((2*pi
- pi*sign(a) - pi*sign(b))^2 + 4*(ln(abs(a)) + ln(abs(b)))^2))*e^(x*(ln(abs(a))
+ ln(abs(b)))) - (i*e^(1/2*(pi*(sign(a) - 1) + pi*(sign(b) - 1))*i*x)/(pi*i*sign
(a) + pi*i*sign(b) - 2*pi*i + 2*ln(abs(a)) + 2*ln(abs(b))) + i*e^(-1/2*(pi*(sign
(a) - 1) + pi*(sign(b) - 1))*i*x)/(pi*i*sign(a) + pi*i*sign(b) - 2*pi*i - 2*ln(a
bs(a)) - 2*ln(abs(b))))*e^(x*(ln(abs(a)) + ln(abs(b))))/i