3.2.76 \(\int (b x^n)^p \, dx\) [176]

Optimal. Leaf size=16 \[ \frac {x \left (b x^n\right )^p}{1+n p} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {15, 30} \begin {gather*} \frac {x \left (b x^n\right )^p}{n p+1} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} \frac {x \left (b x^n\right )^p}{1+n p} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(b*x^n)^p)/(1 + n*p)

________________________________________________________________________________________

Maple [A]
time = 0.03, size = 17, normalized size = 1.06

method	result	size
gosper	\(\frac {x \left (b \,x^{n}\right )^{p}}{n p +1}\)	\(17\)
norman	\(\frac {x \,{\mathrm e}^{p \ln \left (b \,{\mathrm e}^{n \ln \left (x \right )}\right )}}{n p +1}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^n)^p,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.33, size = 17, normalized size = 1.06 \begin {gather*} \frac {b^{p} x {\left (x^{n}\right )}^{p}}{n p + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^p*x*(x^n)^p/(n*p + 1)

________________________________________________________________________________________

Fricas [A]
time = 0.36, size = 20, normalized size = 1.25 \begin {gather*} \frac {x e^{\left (n p \log \left (x\right ) + p \log \left (b\right )\right )}}{n p + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^(n*p*log(x) + p*log(b))/(n*p + 1)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 1.37, size = 20, normalized size = 1.25 \begin {gather*} \frac {x e^{\left (n p \log \left (x\right ) + p \log \left (b\right )\right )}}{n p + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^(n*p*log(x) + p*log(b))/(n*p + 1)

________________________________________________________________________________________

Mupad [B]
time = 1.19, size = 16, normalized size = 1.00 \begin {gather*} \frac {x\,{\left (b\,x^n\right )}^p}{n\,p+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*(b*x^n)^p)/(n*p + 1)

________________________________________________________________________________________