3.237 \(\int \left (a x^m\right )^r \, dx\)

Optimal. Leaf size=16 \[ \frac{x \left (a x^m\right )^r}{m r+1} \]

[Out]

(x*(a*x^m)^r)/(1 + m*r)

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Rubi [A]  time = 0.0126771, antiderivative size = 16, 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 \[ \frac{x \left (a x^m\right )^r}{m r+1} \]

Antiderivative was successfully verified.

[In]  Int[(a*x^m)^r,x]

[Out]

(x*(a*x^m)^r)/(1 + m*r)

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Rubi in Sympy [A]  time = 2.02921, size = 22, normalized size = 1.38 \[ \frac{x^{- m r} x^{m r + 1} \left (a x^{m}\right )^{r}}{m r + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*x**m)**r,x)

[Out]

x**(-m*r)*x**(m*r + 1)*(a*x**m)**r/(m*r + 1)

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Mathematica [A]  time = 0.00507077, size = 16, normalized size = 1. \[ \frac{x \left (a x^m\right )^r}{m r+1} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*x^m)^r,x]

[Out]

(x*(a*x^m)^r)/(1 + m*r)

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Maple [A]  time = 0.002, size = 17, normalized size = 1.1 \[{\frac{x \left ( a{x}^{m} \right ) ^{r}}{mr+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*x^m)^r,x)

[Out]

x*(a*x^m)^r/(m*r+1)

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Maxima [A]  time = 0.705845, size = 23, normalized size = 1.44 \[ \frac{a^{r} x{\left (x^{m}\right )}^{r}}{m r + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^r*x*(x^m)^r/(m*r + 1)

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Fricas [A]  time = 0.279681, size = 27, normalized size = 1.69 \[ \frac{x e^{\left (m r \log \left (x\right ) + r \log \left (a\right )\right )}}{m r + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*e^(m*r*log(x) + r*log(a))/(m*r + 1)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x**m)**r,x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [A]  time = 0.262485, size = 27, normalized size = 1.69 \[ \frac{x e^{\left (m r{\rm ln}\left (x\right ) + r{\rm ln}\left (a\right )\right )}}{m r + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*e^(m*r*ln(x) + r*ln(a))/(m*r + 1)