3.399 \(\int (a x^m)^r (b x^n)^s \, dx\)

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 = 0.0091378, 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, Rules used = {15, 30} \[ \frac{x \left (a x^m\right )^r \left (b x^n\right )^s}{m r+n s+1} \]

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)

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

Mathematica [A]  time = 0.0073769, 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 = 0.002, size = 27, normalized size = 1. \begin{align*}{\frac{x \left ( a{x}^{m} \right ) ^{r} \left ( b{x}^{n} \right ) ^{s}}{mr+ns+1}} \end{align*}

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 = 1.50721, size = 43, normalized size = 1.65 \begin{align*} \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} \end{align*}

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 = 1.01956, size = 93, normalized size = 3.58 \begin{align*} \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} \end{align*}

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(-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*(b*x**n)**s,x)

[Out]

Exception raised: TypeError

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Giac [A]  time = 1.17513, size = 43, normalized size = 1.65 \begin{align*} \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} \end{align*}

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