3.173 \(\int (c x)^m (b x^n)^p \, dx\)

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

[Out]

(c*x)^(1+m)*(b*x^n)^p/c/(n*p+m+1)

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 22, normalized size of antiderivative = 0.85, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {15, 20, 30} \[ \frac {x (c x)^m \left (b x^n\right )^p}{m+n p+1} \]

Antiderivative was successfully verified.

[In]

Int[(c*x)^m*(b*x^n)^p,x]

[Out]

(x*(c*x)^m*(b*x^n)^p)/(1 + m + n*p)

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 20

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[(b^IntPart[n]*(b*v)^FracPart[n])/(a^IntPart[n
]*(a*v)^FracPart[n]), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]
&&  !IntegerQ[m + n]

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

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 22, normalized size = 0.85 \[ \frac {x (c x)^m \left (b x^n\right )^p}{m+n p+1} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*x)^m*(b*x^n)^p,x]

[Out]

(x*(c*x)^m*(b*x^n)^p)/(1 + m + n*p)

________________________________________________________________________________________

fricas [A]  time = 0.70, size = 29, normalized size = 1.12 \[ \frac {x e^{\left (n p \log \relax (x) + p \log \relax (b) + m \log \relax (c) + m \log \relax (x)\right )}}{n p + m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^(n*p*log(x) + p*log(b) + m*log(c) + m*log(x))/(n*p + m + 1)

________________________________________________________________________________________

giac [A]  time = 0.21, size = 29, normalized size = 1.12 \[ \frac {x e^{\left (n p \log \relax (x) + p \log \relax (b) + m \log \relax (c) + m \log \relax (x)\right )}}{n p + m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^(n*p*log(x) + p*log(b) + m*log(c) + m*log(x))/(n*p + m + 1)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 23, normalized size = 0.88 \[ \frac {x \left (b \,x^{n}\right )^{p} \left (c x \right )^{m}}{n p +m +1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x)^m*(b*x^n)^p,x)

[Out]

x*(c*x)^m*(b*x^n)^p/(n*p+m+1)

________________________________________________________________________________________

maxima [A]  time = 1.48, size = 28, normalized size = 1.08 \[ \frac {b^{p} c^{m} x e^{\left (m \log \relax (x) + p \log \left (x^{n}\right )\right )}}{n p + m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^p*c^m*x*e^(m*log(x) + p*log(x^n))/(n*p + m + 1)

________________________________________________________________________________________

mupad [B]  time = 1.01, size = 22, normalized size = 0.85 \[ \frac {x\,{\left (b\,x^n\right )}^p\,{\left (c\,x\right )}^m}{m+n\,p+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^n)^p*(c*x)^m,x)

[Out]

(x*(b*x^n)^p*(c*x)^m)/(m + n*p + 1)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)**m*(b*x**n)**p,x)

[Out]

Piecewise((b**p*c**m*x*x**m*(x**n)**p/(m + n*p + 1), Ne(m, -n*p - 1)), (Integral((b*x**n)**p*(c*x)**(-n*p - 1)
, x), True))

________________________________________________________________________________________