3.400 \(\int (a x^m)^r (b x^n)^s (c x^p)^t \, dx\)

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

[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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Rubi [A]  time = 0.0155787, antiderivative size = 36, normalized size of antiderivative = 1., 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} \[ \frac{x \left (a x^m\right )^r \left (b x^n\right )^s \left (c x^p\right )^t}{m r+n s+p t+1} \]

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*}

Mathematica [A]  time = 0.012172, size = 36, normalized size = 1. \[ \frac{x \left (a x^m\right )^r \left (b x^n\right )^s \left (c x^p\right )^t}{m r+n s+p t+1} \]

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 = 0.001, size = 37, normalized size = 1. \begin{align*}{\frac{x \left ( a{x}^{m} \right ) ^{r} \left ( b{x}^{n} \right ) ^{s} \left ( c{x}^{p} \right ) ^{t}}{mr+ns+pt+1}} \end{align*}

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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Maxima [A]  time = 1.44166, size = 59, normalized size = 1.64 \begin{align*} \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{align*}

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 = 0.989281, size = 134, normalized size = 3.72 \begin{align*} \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{align*}

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(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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Giac [A]  time = 1.15558, size = 59, normalized size = 1.64 \begin{align*} \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{align*}

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)