3.2691 \(\int \frac{x^{-1-\frac{n}{2}}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=26 \[ -\frac{2 x^{-n/2} \sqrt{a+b x^n}}{a n} \]

[Out]

(-2*Sqrt[a + b*x^n])/(a*n*x^(n/2))

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Rubi [A]  time = 0.00625, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {264} \[ -\frac{2 x^{-n/2} \sqrt{a+b x^n}}{a n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - n/2)/Sqrt[a + b*x^n],x]

[Out]

(-2*Sqrt[a + b*x^n])/(a*n*x^(n/2))

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0074433, size = 26, normalized size = 1. \[ -\frac{2 x^{-n/2} \sqrt{a+b x^n}}{a n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[a + b*x^n])/(a*n*x^(n/2))

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Maple [F]  time = 0.077, size = 0, normalized size = 0. \begin{align*} \int{{x}^{-1-{\frac{n}{2}}}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{-\frac{1}{2} \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^(-1/2*n - 1)/sqrt(b*x^n + a), x)

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1-1/2*n)/(a+b*x**n)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{-\frac{1}{2} \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^(-1/2*n - 1)/sqrt(b*x^n + a), x)