3.145 \(\int \frac{1}{\sqrt{b x^n}} \, dx\)

Optimal. Leaf size=19 \[ \frac{2 x}{(2-n) \sqrt{b x^n}} \]

[Out]

(2*x)/((2 - n)*Sqrt[b*x^n])

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Rubi [A]  time = 0.0039118, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {15, 30} \[ \frac{2 x}{(2-n) \sqrt{b x^n}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[b*x^n],x]

[Out]

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

Mathematica [A]  time = 0.0025948, size = 17, normalized size = 0.89 \[ -\frac{2 x}{(n-2) \sqrt{b x^n}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[b*x^n],x]

[Out]

(-2*x)/((-2 + n)*Sqrt[b*x^n])

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Maple [A]  time = 0., size = 16, normalized size = 0.8 \begin{align*} -2\,{\frac{x}{ \left ( -2+n \right ) \sqrt{b{x}^{n}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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

[Out]

Exception raised: UnboundLocalError

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

[Out]

Exception raised: TypeError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x^{n}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(b*x^n), x)