3.398 \(\int (a x^m)^r \, dx\)

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 = 0.0046085, 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, Rules used = {15, 30} \[ \frac{x \left (a x^m\right )^r}{m r+1} \]

Antiderivative was successfully verified.

[In]

Int[(a*x^m)^r,x]

[Out]

(x*(a*x^m)^r)/(1 + m*r)

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \left (a x^m\right )^r \, dx &=\left (x^{-m r} \left (a x^m\right )^r\right ) \int x^{m r} \, dx\\ &=\frac{x \left (a x^m\right )^r}{1+m r}\\ \end{align*}

Mathematica [A]  time = 0.0031748, 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 \begin{align*}{\frac{x \left ( a{x}^{m} \right ) ^{r}}{mr+1}} \end{align*}

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 = 1.31621, size = 23, normalized size = 1.44 \begin{align*} \frac{a^{r} x{\left (x^{m}\right )}^{r}}{m r + 1} \end{align*}

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 = 1.00261, size = 53, normalized size = 3.31 \begin{align*} \frac{x e^{\left (m r \log \left (x\right ) + r \log \left (a\right )\right )}}{m r + 1} \end{align*}

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. \begin{align*} \text{Exception raised: TypeError} \end{align*}

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 [A]  time = 1.14182, size = 27, normalized size = 1.69 \begin{align*} \frac{x e^{\left (m r \log \left (x\right ) + r \log \left (a\right )\right )}}{m r + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^(m*r*log(x) + r*log(a))/(m*r + 1)