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

Optimal. Leaf size=21 \[ \frac{2 x^2}{(4-n) \sqrt{b x^n}} \]

[Out]

(2*x^2)/((4 - n)*Sqrt[b*x^n])

________________________________________________________________________________________

Rubi [A]  time = 0.0040158, antiderivative size = 21, 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} \[ \frac{2 x^2}{(4-n) \sqrt{b x^n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^2)/((4 - 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}{\sqrt{b x^n}} \, dx &=\frac{x^{n/2} \int x^{1-\frac{n}{2}} \, dx}{\sqrt{b x^n}}\\ &=\frac{2 x^2}{(4-n) \sqrt{b x^n}}\\ \end{align*}

Mathematica [A]  time = 0.0036167, size = 19, normalized size = 0.9 \[ -\frac{2 x^2}{(n-4) \sqrt{b x^n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x^2)/((-4 + n)*Sqrt[b*x^n])

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 18, normalized size = 0.9 \begin{align*} -2\,{\frac{{x}^{2}}{ \left ( -4+n \right ) \sqrt{b{x}^{n}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x^2/(-4+n)/(b*x^n)^(1/2)

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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

[Out]

Exception raised: TypeError

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/sqrt(b*x^n), x)