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

Optimal. Leaf size=26 \[ \frac{2 x^{m+1}}{(2 m-n+2) \sqrt{b x^n}} \]

[Out]

(2*x^(1 + m))/((2 + 2*m - n)*Sqrt[b*x^n])

________________________________________________________________________________________

Rubi [A]  time = 0.0075513, antiderivative size = 26, 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} \[ \frac{2 x^{m+1}}{(2 m-n+2) \sqrt{b x^n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.005874, size = 25, normalized size = 0.96 \[ \frac{x^{m+1}}{\left (m-\frac{n}{2}+1\right ) \sqrt{b x^n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^(1 + m)/((1 + m - n/2)*Sqrt[b*x^n])

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 25, normalized size = 1. \begin{align*} 2\,{\frac{{x}^{1+m}}{ \left ( 2+2\,m-n \right ) \sqrt{b{x}^{n}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x^(1+m)/(2+2*m-n)/(b*x^n)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 0.994444, size = 32, normalized size = 1.23 \begin{align*} \frac{2 \, x x^{m}}{\sqrt{b}{\left (2 \, m - n + 2\right )} \sqrt{x^{n}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x*x^m/(sqrt(b)*(2*m - n + 2)*sqrt(x^n))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m/sqrt(b*x^n), x)