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

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Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 19, normalized size = 1.00 \[ \frac {2 \sqrt {a+b x^n}}{b n} \]

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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fricas [A]  time = 0.49, size = 17, normalized size = 0.89 \[ \frac {2 \, \sqrt {b x^{n} + a}}{b n} \]

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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giac [A]  time = 0.16, size = 17, normalized size = 0.89 \[ \frac {2 \, \sqrt {b x^{n} + a}}{b n} \]

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)

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maple [A]  time = 0.02, size = 18, normalized size = 0.95 \[ \frac {2 \sqrt {b \,x^{n}+a}}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.64, size = 17, normalized size = 0.89 \[ \frac {2 \, \sqrt {b x^{n} + a}}{b n} \]

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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mupad [B]  time = 1.27, size = 17, normalized size = 0.89 \[ \frac {2\,\sqrt {a+b\,x^n}}{b\,n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(n - 1)/(a + b*x^n)^(1/2),x)

[Out]

(2*(a + b*x^n)^(1/2))/(b*n)

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sympy [A]  time = 7.71, size = 41, normalized size = 2.16 \[ \begin {cases} \frac {\log {\relax (x )}}{\sqrt {a}} & \text {for}\: b = 0 \wedge n = 0 \\\frac {\log {\relax (x )}}{\sqrt {a + b}} & \text {for}\: n = 0 \\\frac {x^{n}}{\sqrt {a} n} & \text {for}\: b = 0 \\\frac {2 \sqrt {a + b x^{n}}}{b n} & \text {otherwise} \end {cases} \]

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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