3.4.100 \(\int (a x^m)^r (b x^n)^s (c x^p)^t \, dx\) [400]

Optimal. Leaf size=36 \[ \frac {x \left (a x^m\right )^r \left (b x^n\right )^s \left (c x^p\right )^t}{1+m r+n s+p t} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {15, 30} \begin {gather*} \frac {x \left (a x^m\right )^r \left (b x^n\right )^s \left (c x^p\right )^t}{m r+n s+p t+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*x^m)^r*(b*x^n)^s*(c*x^p)^t,x]

[Out]

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

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

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Mathematica [A]
time = 0.01, size = 36, normalized size = 1.00 \begin {gather*} \frac {x \left (a x^m\right )^r \left (b x^n\right )^s \left (c x^p\right )^t}{1+m r+n s+p t} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*x^m)^r*(b*x^n)^s*(c*x^p)^t,x]

[Out]

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

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Maple [A]
time = 24.71, size = 37, normalized size = 1.03

method	result	size
gosper	\(\frac {x \left (a \,x^{m}\right )^{r} \left (b \,x^{n}\right )^{s} \left (c \,x^{p}\right )^{t}}{m r +n s +p t +1}\)	\(37\)
risch	\(\frac {x \,{\mathrm e}^{-\frac {i \pi \mathrm {csgn}\left (i a \,x^{m}\right )^{3} r}{2}-\frac {i \mathrm {csgn}\left (i c \,x^{p}\right )^{3} \pi t}{2}+\frac {i \mathrm {csgn}\left (i c \,x^{p}\right )^{2} \mathrm {csgn}\left (i x^{p}\right ) \pi t}{2}-\frac {i \pi \,\mathrm {csgn}\left (i a \,x^{m}\right ) \mathrm {csgn}\left (i a \right ) \mathrm {csgn}\left (i x^{m}\right ) r}{2}+r \ln \left (a \right )+\ln \left (x^{m}\right ) r +\frac {i \pi \mathrm {csgn}\left (i b \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right ) s}{2}-\frac {i \mathrm {csgn}\left (i c \,x^{p}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{p}\right ) \pi t}{2}+\frac {i \pi \mathrm {csgn}\left (i a \,x^{m}\right )^{2} \mathrm {csgn}\left (i a \right ) r}{2}+\frac {i \pi \mathrm {csgn}\left (i b \,x^{n}\right )^{2} \mathrm {csgn}\left (i b \right ) s}{2}+s \ln \left (b \right )+\ln \left (x^{n}\right ) s +\frac {i \pi \mathrm {csgn}\left (i a \,x^{m}\right )^{2} \mathrm {csgn}\left (i x^{m}\right ) r}{2}-\frac {i \pi \,\mathrm {csgn}\left (i b \,x^{n}\right ) \mathrm {csgn}\left (i b \right ) \mathrm {csgn}\left (i x^{n}\right ) s}{2}-\frac {i \pi \mathrm {csgn}\left (i b \,x^{n}\right )^{3} s}{2}+\frac {i \mathrm {csgn}\left (i c \,x^{p}\right )^{2} \mathrm {csgn}\left (i c \right ) \pi t}{2}+\ln \left (c \right ) t +\ln \left (x^{p}\right ) t}}{m r +n s +p t +1}\)	\(294\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 44, normalized size = 1.22 \begin {gather*} \frac {a^{r} b^{s} c^{t} x e^{\left (r \log \left (x^{m}\right ) + s \log \left (x^{n}\right ) + t \log \left (x^{p}\right )\right )}}{m r + n s + p t + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^m)^r*(b*x^n)^s*(c*x^p)^t,x, algorithm="maxima")

[Out]

a^r*b^s*c^t*x*e^(r*log(x^m) + s*log(x^n) + t*log(x^p))/(m*r + n*s + p*t + 1)

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Fricas [A]
time = 0.37, size = 44, normalized size = 1.22 \begin {gather*} \frac {x e^{\left (m r \log \left (x\right ) + n s \log \left (x\right ) + p t \log \left (x\right ) + r \log \left (a\right ) + s \log \left (b\right ) + t \log \left (c\right )\right )}}{m r + n s + p t + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^m)^r*(b*x^n)^s*(c*x^p)^t,x, algorithm="fricas")

[Out]

x*e^(m*r*log(x) + n*s*log(x) + p*t*log(x) + r*log(a) + s*log(b) + t*log(c))/(m*r + n*s + p*t + 1)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4369 deep

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Giac [A]
time = 4.48, size = 44, normalized size = 1.22 \begin {gather*} \frac {x e^{\left (m r \log \left (x\right ) + n s \log \left (x\right ) + p t \log \left (x\right ) + r \log \left (a\right ) + s \log \left (b\right ) + t \log \left (c\right )\right )}}{m r + n s + p t + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^m)^r*(b*x^n)^s*(c*x^p)^t,x, algorithm="giac")

[Out]

x*e^(m*r*log(x) + n*s*log(x) + p*t*log(x) + r*log(a) + s*log(b) + t*log(c))/(m*r + n*s + p*t + 1)

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Mupad [B]
time = 3.10, size = 36, normalized size = 1.00 \begin {gather*} \frac {x\,{\left (a\,x^m\right )}^r\,{\left (b\,x^n\right )}^s\,{\left (c\,x^p\right )}^t}{m\,r+n\,s+p\,t+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^m)^r*(b*x^n)^s*(c*x^p)^t,x)

[Out]

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

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