\(\int \frac {(b x^n)^p}{x} \, dx\) [2716]

Optimal result

Integrand size = 11, antiderivative size = 14 \[ \int \frac {\left (b x^n\right )^p}{x} \, dx=\frac {\left (b x^n\right )^p}{n p} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {15, 30} \[ \int \frac {\left (b x^n\right )^p}{x} \, dx=\frac {\left (b x^n\right )^p}{n p} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {\left (b x^n\right )^p}{x} \, dx=\frac {\left (b x^n\right )^p}{n p} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07

method	result	size
gosper	\(\frac {\left (b \,x^{n}\right )^{p}}{n p}\)	\(15\)
derivativedivides	\(\frac {\left (b \,x^{n}\right )^{p}}{n p}\)	\(15\)
default	\(\frac {\left (b \,x^{n}\right )^{p}}{n p}\)	\(15\)
parallelrisch	\(\frac {\left (b \,x^{n}\right )^{p}}{n p}\)	\(15\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.29 \[ \int \frac {\left (b x^n\right )^p}{x} \, dx=\frac {e^{\left (n p \log \left (x\right ) + p \log \left (b\right )\right )}}{n p} \]

[In]

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

[Out]

e^(n*p*log(x) + p*log(b))/(n*p)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 20 vs. \(2 (8) = 16\).

Time = 0.12 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.43 \[ \int \frac {\left (b x^n\right )^p}{x} \, dx=\begin {cases} \log {\left (x \right )} & \text {for}\: n = 0 \wedge p = 0 \\b^{p} \log {\left (x \right )} & \text {for}\: n = 0 \\\log {\left (x \right )} & \text {for}\: p = 0 \\\frac {\left (b x^{n}\right )^{p}}{n p} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((log(x), Eq(n, 0) & Eq(p, 0)), (b**p*log(x), Eq(n, 0)), (log(x), Eq(p, 0)), ((b*x**n)**p/(n*p), True
))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07 \[ \int \frac {\left (b x^n\right )^p}{x} \, dx=\frac {b^{p} {\left (x^{n}\right )}^{p}}{n p} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {\left (b x^n\right )^p}{x} \, dx=\frac {\left (b x^{n}\right )^{p}}{n p} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (b x^n\right )^p}{x} \, dx=\int \frac {{\left (b\,x^n\right )}^p}{x} \,d x \]

[In]

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

[Out]

int((b*x^n)^p/x, x)