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/(m*r+1)

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Rubi [A]  time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 1.00 \[ \frac {x \left (a x^m\right )^r}{m r+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(a*x^m)^r)/(1 + m*r)

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fricas [A]  time = 0.45, size = 20, normalized size = 1.25 \[ \frac {x e^{\left (m r \log \relax (x) + r \log \relax (a)\right )}}{m r + 1} \]

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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giac [A]  time = 0.21, size = 20, normalized size = 1.25 \[ \frac {x e^{\left (m r \log \relax (x) + r \log \relax (a)\right )}}{m r + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^(m*r*log(x) + r*log(a))/(m*r + 1)

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maple [A]  time = 0.00, size = 17, normalized size = 1.06 \[ \frac {x \left (a \,x^{m}\right )^{r}}{m r +1} \]

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 = 1.48, size = 17, normalized size = 1.06 \[ \frac {a^{r} x {\left (x^{m}\right )}^{r}}{m r + 1} \]

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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mupad [B]  time = 3.40, size = 16, normalized size = 1.00 \[ \frac {x\,{\left (a\,x^m\right )}^r}{m\,r+1} \]

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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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} \frac {a^{r} x \left (x^{m}\right )^{r}}{m r + 1} & \text {for}\: m \neq - \frac {1}{r} \\\int \left (a x^{- \frac {1}{r}}\right )^{r}\, dx & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x**m)**r,x)

[Out]

Piecewise((a**r*x*(x**m)**r/(m*r + 1), Ne(m, -1/r)), (Integral((a*x**(-1/r))**r, x), True))

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