3.134 \(\int \frac{\sqrt{b x^n}}{x} \, dx\)

Optimal. Leaf size=14 \[ \frac{2 \sqrt{b x^n}}{n} \]

[Out]

(2*Sqrt[b*x^n])/n

________________________________________________________________________________________

Rubi [A]  time = 0.0032852, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {15, 30} \[ \frac{2 \sqrt{b x^n}}{n} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*x^n]/x,x]

[Out]

(2*Sqrt[b*x^n])/n

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

Mathematica [A]  time = 0.0015657, size = 14, normalized size = 1. \[ \frac{2 \sqrt{b x^n}}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b*x^n]/x,x]

[Out]

(2*Sqrt[b*x^n])/n

________________________________________________________________________________________

Maple [A]  time = 0., size = 13, normalized size = 0.9 \begin{align*} 2\,{\frac{\sqrt{b{x}^{n}}}{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^n)^(1/2)/x,x)

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^n)^(1/2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.80259, size = 23, normalized size = 1.64 \begin{align*} \frac{2 \, \sqrt{b x^{n}}}{n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^n)^(1/2)/x,x, algorithm="fricas")

[Out]

2*sqrt(b*x^n)/n

________________________________________________________________________________________

Sympy [A]  time = 0.36662, size = 22, normalized size = 1.57 \begin{align*} \begin{cases} \frac{2 \sqrt{b} \sqrt{x^{n}}}{n} & \text{for}\: n \neq 0 \\\sqrt{b} \log{\left (x \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**n)**(1/2)/x,x)

[Out]

Piecewise((2*sqrt(b)*sqrt(x**n)/n, Ne(n, 0)), (sqrt(b)*log(x), True))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^n)^(1/2)/x,x, algorithm="giac")

[Out]

integrate(sqrt(b*x^n)/x, x)