3.147 \(\int \frac{1}{x^2 \sqrt{b x^n}} \, dx\)

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

[Out]

-2/((2 + n)*x*Sqrt[b*x^n])

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Rubi [A]  time = 0.0038447, antiderivative size = 19, 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} \[ -\frac{2}{(n+2) x \sqrt{b x^n}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*Sqrt[b*x^n]),x]

[Out]

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

Mathematica [A]  time = 0.0043512, size = 19, normalized size = 1. \[ -\frac{2}{(n+2) x \sqrt{b x^n}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*Sqrt[b*x^n]),x]

[Out]

-2/((2 + n)*x*Sqrt[b*x^n])

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Maple [A]  time = 0.002, size = 18, normalized size = 1. \begin{align*} -2\,{\frac{1}{ \left ( 2+n \right ) x\sqrt{b{x}^{n}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(2+n)/x/(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/x^2/(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/x^2/(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/x**2/(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}} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(b*x^n)*x^2), x)