3.2661 \(\int \frac{x^{-1+n}}{\sqrt{a+b x^n}} \, dx\)

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

[Out]

(2*Sqrt[a + b*x^n])/(b*n)

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Rubi [A]  time = 0.0058899, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {261} \[ \frac{2 \sqrt{a+b x^n}}{b n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + n)/Sqrt[a + b*x^n],x]

[Out]

(2*Sqrt[a + b*x^n])/(b*n)

Rule 261

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

Rubi steps

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

Mathematica [A]  time = 0.0039598, size = 19, normalized size = 1. \[ \frac{2 \sqrt{a+b x^n}}{b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[a + b*x^n])/(b*n)

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Maple [A]  time = 0.018, size = 18, normalized size = 1. \begin{align*} 2\,{\frac{\sqrt{a+b{x}^{n}}}{bn}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.931071, size = 23, normalized size = 1.21 \begin{align*} \frac{2 \, \sqrt{b x^{n} + a}}{b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(b*x^n + a)/(b*n)

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Fricas [A]  time = 0.99893, size = 34, normalized size = 1.79 \begin{align*} \frac{2 \, \sqrt{b x^{n} + a}}{b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(b*x^n + a)/(b*n)

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Sympy [A]  time = 7.0506, size = 41, normalized size = 2.16 \begin{align*} \begin{cases} \frac{\log{\left (x \right )}}{\sqrt{a}} & \text{for}\: b = 0 \wedge n = 0 \\\frac{\log{\left (x \right )}}{\sqrt{a + b}} & \text{for}\: n = 0 \\\frac{x^{n}}{\sqrt{a} n} & \text{for}\: b = 0 \\\frac{2 \sqrt{a + b x^{n}}}{b n} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((log(x)/sqrt(a), Eq(b, 0) & Eq(n, 0)), (log(x)/sqrt(a + b), Eq(n, 0)), (x**n/(sqrt(a)*n), Eq(b, 0)),
(2*sqrt(a + b*x**n)/(b*n), True))

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Giac [A]  time = 1.11718, size = 23, normalized size = 1.21 \begin{align*} \frac{2 \, \sqrt{b x^{n} + a}}{b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(b*x^n + a)/(b*n)