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

Optimal result

Integrand size = 7, antiderivative size = 16 \[ \int \left (b x^n\right )^p \, dx=\frac {x \left (b x^n\right )^p}{1+n p} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {15, 30} \[ \int \left (b x^n\right )^p \, dx=\frac {x \left (b x^n\right )^p}{n p+1} \]

[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*} \text {integral}& = \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] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \left (b x^n\right )^p \, dx=\frac {x \left (b x^n\right )^p}{1+n p} \]

[In]

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

[Out]

(x*(b*x^n)^p)/(1 + n*p)

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int \left (b x^n\right )^p \, dx=\frac {x e^{\left (n p \log \left (x\right ) + p \log \left (b\right )\right )}}{n p + 1} \]

[In]

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

[Out]

x*e^(n*p*log(x) + p*log(b))/(n*p + 1)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (12) = 24\).

Time = 0.40 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.81 \[ \int \left (b x^n\right )^p \, dx=\begin {cases} \frac {x \left (b x^{n}\right )^{p}}{n p + 1} & \text {for}\: n \neq - \frac {1}{p} \\x \left (b x^{- \frac {1}{p}}\right )^{p} \log {\left (x \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \left (b x^n\right )^p \, dx=\frac {b^{p} x {\left (x^{n}\right )}^{p}}{n p + 1} \]

[In]

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

[Out]

b^p*x*(x^n)^p/(n*p + 1)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int \left (b x^n\right )^p \, dx=\frac {x e^{\left (n p \log \left (x\right ) + p \log \left (b\right )\right )}}{n p + 1} \]

[In]

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

[Out]

x*e^(n*p*log(x) + p*log(b))/(n*p + 1)

Mupad [B] (verification not implemented)

Time = 5.75 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \left (b x^n\right )^p \, dx=\frac {x\,{\left (b\,x^n\right )}^p}{n\,p+1} \]

[In]

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

[Out]

(x*(b*x^n)^p)/(n*p + 1)