3.238 \(\int \left (a x^m\right )^r \left (b x^n\right )^s \, dx\)

Optimal. Leaf size=26 \[ \frac{x \left (a x^m\right )^r \left (b x^n\right )^s}{m r+n s+1} \]

[Out]

(x*(a*x^m)^r*(b*x^n)^s)/(1 + m*r + n*s)

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Rubi [A]  time = 0.0207064, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{x \left (a x^m\right )^r \left (b x^n\right )^s}{m r+n s+1} \]

Antiderivative was successfully verified.

[In]  Int[(a*x^m)^r*(b*x^n)^s,x]

[Out]

(x*(a*x^m)^r*(b*x^n)^s)/(1 + m*r + n*s)

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Rubi in Sympy [A]  time = 7.1562, size = 41, normalized size = 1.58 \[ \frac{x^{- m r} x^{- n s} x^{m r + n s + 1} \left (a x^{m}\right )^{r} \left (b x^{n}\right )^{s}}{m r + n s + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*x**m)**r*(b*x**n)**s,x)

[Out]

x**(-m*r)*x**(-n*s)*x**(m*r + n*s + 1)*(a*x**m)**r*(b*x**n)**s/(m*r + n*s + 1)

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Mathematica [A]  time = 0.010642, size = 26, normalized size = 1. \[ \frac{x \left (a x^m\right )^r \left (b x^n\right )^s}{m r+n s+1} \]

Antiderivative was successfully verified.

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

[Out]

(x*(a*x^m)^r*(b*x^n)^s)/(1 + m*r + n*s)

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Maple [A]  time = 0.003, size = 27, normalized size = 1. \[{\frac{x \left ( a{x}^{m} \right ) ^{r} \left ( b{x}^{n} \right ) ^{s}}{mr+ns+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*x^m)^r*(b*x^n)^s,x)

[Out]

x*(a*x^m)^r*(b*x^n)^s/(m*r+n*s+1)

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Maxima [A]  time = 0.722567, size = 43, normalized size = 1.65 \[ \frac{a^{r} b^{s} x e^{\left (r \log \left (x^{m}\right ) + s \log \left (x^{n}\right )\right )}}{m r + n s + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x^m)^r*(b*x^n)^s,x, algorithm="maxima")

[Out]

a^r*b^s*x*e^(r*log(x^m) + s*log(x^n))/(m*r + n*s + 1)

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Fricas [A]  time = 0.333926, size = 43, normalized size = 1.65 \[ \frac{x e^{\left (m r \log \left (x\right ) + n s \log \left (x\right ) + r \log \left (a\right ) + s \log \left (b\right )\right )}}{m r + n s + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x^m)^r*(b*x^n)^s,x, algorithm="fricas")

[Out]

x*e^(m*r*log(x) + n*s*log(x) + r*log(a) + s*log(b))/(m*r + n*s + 1)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x**m)**r*(b*x**n)**s,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.265955, size = 43, normalized size = 1.65 \[ \frac{x e^{\left (m r{\rm ln}\left (x\right ) + n s{\rm ln}\left (x\right ) + r{\rm ln}\left (a\right ) + s{\rm ln}\left (b\right )\right )}}{m r + n s + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*e^(m*r*ln(x) + n*s*ln(x) + r*ln(a) + s*ln(b))/(m*r + n*s + 1)