3.2716 \(\int \frac{(b x^n)^p}{x} \, dx\)

Optimal. Leaf size=14 \[ \frac{\left (b x^n\right )^p}{n p} \]

[Out]

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

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Rubi [A]  time = 0.0038183, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {15, 30} \[ \frac{\left (b x^n\right )^p}{n p} \]

Antiderivative was successfully verified.

[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*} \int \frac{\left (b x^n\right )^p}{x} \, dx &=\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]  time = 0.0011997, size = 14, normalized size = 1. \[ \frac{\left (b x^n\right )^p}{n p} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 15, normalized size = 1.1 \begin{align*}{\frac{ \left ( b{x}^{n} \right ) ^{p}}{np}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.959177, size = 20, normalized size = 1.43 \begin{align*} \frac{b^{p}{\left (x^{n}\right )}^{p}}{n p} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.34491, size = 45, normalized size = 3.21 \begin{align*} \frac{e^{\left (n p \log \left (x\right ) + p \log \left (b\right )\right )}}{n p} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(n*p*log(x) + p*log(b))/(n*p)

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Sympy [A]  time = 0.27713, size = 22, normalized size = 1.57 \begin{align*} \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{b^{p} \left (x^{n}\right )^{p}}{n p} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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**p*(x**n)**p/(n*p), T
rue))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (b x^{n}\right )^{p}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^n)^p/x, x)