3.2714 \(\int x (b x^n)^p \, dx\)

Optimal. Leaf size=18 \[ \frac {x^2 \left (b x^n\right )^p}{n p+2} \]

[Out]

x^2*(b*x^n)^p/(n*p+2)

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {15, 30} \[ \frac {x^2 \left (b x^n\right )^p}{n p+2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(b*x^n)^p)/(2 + 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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int x \left (b x^n\right )^p \, dx &=\left (x^{-n p} \left (b x^n\right )^p\right ) \int x^{1+n p} \, dx\\ &=\frac {x^2 \left (b x^n\right )^p}{2+n p}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 18, normalized size = 1.00 \[ \frac {x^2 \left (b x^n\right )^p}{n p+2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(b*x^n)^p)/(2 + n*p)

________________________________________________________________________________________

fricas [A]  time = 0.88, size = 22, normalized size = 1.22 \[ \frac {x^{2} e^{\left (n p \log \relax (x) + p \log \relax (b)\right )}}{n p + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2*e^(n*p*log(x) + p*log(b))/(n*p + 2)

________________________________________________________________________________________

giac [A]  time = 0.17, size = 22, normalized size = 1.22 \[ \frac {x^{2} e^{\left (n p \log \relax (x) + p \log \relax (b)\right )}}{n p + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2*e^(n*p*log(x) + p*log(b))/(n*p + 2)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 19, normalized size = 1.06 \[ \frac {x^{2} \left (b \,x^{n}\right )^{p}}{n p +2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(n*p+2)*x^2*(b*x^n)^p

________________________________________________________________________________________

maxima [A]  time = 0.57, size = 19, normalized size = 1.06 \[ \frac {b^{p} x^{2} {\left (x^{n}\right )}^{p}}{n p + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^p*x^2*(x^n)^p/(n*p + 2)

________________________________________________________________________________________

mupad [B]  time = 1.20, size = 18, normalized size = 1.00 \[ \frac {x^2\,{\left (b\,x^n\right )}^p}{n\,p+2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^2*(b*x^n)^p)/(n*p + 2)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} \frac {b^{p} x^{2} \left (x^{n}\right )^{p}}{n p + 2} & \text {for}\: n \neq - \frac {2}{p} \\\int x \left (b x^{- \frac {2}{p}}\right )^{p}\, dx & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((b**p*x**2*(x**n)**p/(n*p + 2), Ne(n, -2/p)), (Integral(x*(b*x**(-2/p))**p, x), True))

________________________________________________________________________________________