3.4.99 \(\int (a x^m)^r (b x^n)^s \, dx\) [399]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {x \left (a x^m\right )^r \left (b x^n\right )^s}{m r+n s+1} \end {gather*}

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*}

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Mathematica [A]
time = 0.00, size = 26, normalized size = 1.00 \begin {gather*} \frac {x \left (a x^m\right )^r \left (b x^n\right )^s}{1+m r+n s} \end {gather*}

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 = 0.31, size = 27, normalized size = 1.04

method	result	size
gosper	\(\frac {x \left (a \,x^{m}\right )^{r} \left (b \,x^{n}\right )^{s}}{m r +n s +1}\)	\(27\)
risch	\(\frac {x \,{\mathrm e}^{-\frac {i \pi \mathrm {csgn}\left (i a \,x^{m}\right )^{3} r}{2}+\frac {i \pi \mathrm {csgn}\left (i a \,x^{m}\right )^{2} \mathrm {csgn}\left (i a \right ) r}{2}+\frac {i \pi \mathrm {csgn}\left (i a \,x^{m}\right )^{2} \mathrm {csgn}\left (i x^{m}\right ) r}{2}-\frac {i \pi \,\mathrm {csgn}\left (i a \,x^{m}\right ) \mathrm {csgn}\left (i a \right ) \mathrm {csgn}\left (i x^{m}\right ) r}{2}+r \ln \left (a \right )+\ln \left (x^{m}\right ) r -\frac {i \pi \mathrm {csgn}\left (i b \,x^{n}\right )^{3} s}{2}+\frac {i \pi \mathrm {csgn}\left (i b \,x^{n}\right )^{2} \mathrm {csgn}\left (i b \right ) s}{2}+\frac {i \pi \mathrm {csgn}\left (i b \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right ) s}{2}-\frac {i \pi \,\mathrm {csgn}\left (i b \,x^{n}\right ) \mathrm {csgn}\left (i b \right ) \mathrm {csgn}\left (i x^{n}\right ) s}{2}+s \ln \left (b \right )+\ln \left (x^{n}\right ) s}}{m r +n s +1}\)	\(199\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^m)^r*(b*x^n)^s,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 32, normalized size = 1.23 \begin {gather*} \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 {gather*}

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 = 0.33, size = 32, normalized size = 1.23 \begin {gather*} \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 {gather*}

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 4.37, size = 32, normalized size = 1.23 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 2.92, size = 26, normalized size = 1.00 \begin {gather*} \frac {x\,{\left (a\,x^m\right )}^r\,{\left (b\,x^n\right )}^s}{m\,r+n\,s+1} \end {gather*}

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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