3.2654 \(\int \frac{x^{-1+n}}{\sqrt{a+b x^n}} \, dx\)

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 = 0.0246793, 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 \[ \frac{2 \sqrt{a+b x^n}}{b n} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[a + b*x^n])/(b*n)

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Rubi in Sympy [A]  time = 2.43325, size = 14, normalized size = 0.74 \[ \frac{2 \sqrt{a + b x^{n}}}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+n)/(a+b*x**n)**(1/2),x)

[Out]

2*sqrt(a + b*x**n)/(b*n)

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Mathematica [A]  time = 0.0137314, 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 = 0.029, size = 18, normalized size = 1. \[ 2\,{\frac{\sqrt{a+b{x}^{n}}}{bn}} \]

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 = 1.42855, size = 23, normalized size = 1.21 \[ \frac{2 \, \sqrt{b x^{n} + a}}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(n - 1)/sqrt(b*x^n + a),x, algorithm="maxima")

[Out]

2*sqrt(b*x^n + a)/(b*n)

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Fricas [A]  time = 0.231386, size = 23, normalized size = 1.21 \[ \frac{2 \, \sqrt{b x^{n} + a}}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(n - 1)/sqrt(b*x^n + a),x, algorithm="fricas")

[Out]

2*sqrt(b*x^n + a)/(b*n)

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Sympy [A]  time = 21.6292, size = 41, normalized size = 2.16 \[ \begin{cases} \frac{\log{\left (x \right )}}{\sqrt{a}} & \text{for}\: b = 0 \wedge n = 0 \\\frac{\log{\left (x \right )}}{\sqrt{a + b}} & \text{for}\: n = 0 \\\frac{x^{n}}{\sqrt{a} n} & \text{for}\: b = 0 \\\frac{2 \sqrt{a + b x^{n}}}{b n} & \text{otherwise} \end{cases} \]

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/XCAS [A]  time = 0.217321, size = 23, normalized size = 1.21 \[ \frac{2 \, \sqrt{b x^{n} + a}}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(n - 1)/sqrt(b*x^n + a),x, algorithm="giac")

[Out]

2*sqrt(b*x^n + a)/(b*n)