3.133 \(\int \sqrt{b x^n} \, dx\)

Optimal. Leaf size=17 \[ \frac{2 x \sqrt{b x^n}}{n+2} \]

[Out]

(2*x*Sqrt[b*x^n])/(2 + n)

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Rubi [A]  time = 0.0138313, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{2 x \sqrt{b x^n}}{n+2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[b*x^n],x]

[Out]

(2*x*Sqrt[b*x^n])/(2 + n)

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Rubi in Sympy [A]  time = 1.98656, size = 24, normalized size = 1.41 \[ \frac{2 x^{- \frac{n}{2}} x^{\frac{n}{2} + 1} \sqrt{b x^{n}}}{n + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**(-n/2)*x**(n/2 + 1)*sqrt(b*x**n)/(n + 2)

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Mathematica [A]  time = 0.00337966, size = 17, normalized size = 1. \[ \frac{2 x \sqrt{b x^n}}{n+2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[b*x^n],x]

[Out]

(2*x*Sqrt[b*x^n])/(2 + n)

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Maple [A]  time = 0.002, size = 16, normalized size = 0.9 \[ 2\,{\frac{x\sqrt{b{x}^{n}}}{2+n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^n)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^n),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^n),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**n)**(1/2),x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [A]  time = 0.227862, size = 23, normalized size = 1.35 \[ \frac{2 \, \sqrt{b} x e^{\left (\frac{1}{2} \, n{\rm ln}\left (x\right )\right )}}{n + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^n),x, algorithm="giac")

[Out]

2*sqrt(b)*x*e^(1/2*n*ln(x))/(n + 2)