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

Optimal. Leaf size=24 \[ \frac{2 x^{m+1} \sqrt{b x^n}}{2 m+n+2} \]

[Out]

(2*x^(1 + m)*Sqrt[b*x^n])/(2 + 2*m + n)

_______________________________________________________________________________________

Rubi [A]  time = 0.017473, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2 x^{m+1} \sqrt{b x^n}}{2 m+n+2} \]

Antiderivative was successfully verified.

[In]  Int[x^m*Sqrt[b*x^n],x]

[Out]

(2*x^(1 + m)*Sqrt[b*x^n])/(2 + 2*m + n)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.38072, size = 29, normalized size = 1.21 \[ \frac{2 x^{- \frac{n}{2}} x^{m + \frac{n}{2} + 1} \sqrt{b x^{n}}}{2 m + n + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.00817237, size = 25, normalized size = 1.04 \[ \frac{x^{m+1} \sqrt{b x^n}}{m+\frac{n}{2}+1} \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*Sqrt[b*x^n],x]

[Out]

(x^(1 + m)*Sqrt[b*x^n])/(1 + m + n/2)

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 23, normalized size = 1. \[ 2\,{\frac{{x}^{1+m}\sqrt{b{x}^{n}}}{2+2\,m+n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(b*x^n)^(1/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.45467, size = 30, normalized size = 1.25 \[ \frac{2 \, \sqrt{b} x x^{m} \sqrt{x^{n}}}{2 \, m + n + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(b)*x*x^m*sqrt(x^n)/(2*m + n + 2)

_______________________________________________________________________________________

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^m,x, algorithm="fricas")

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.227212, size = 34, normalized size = 1.42 \[ \frac{2 \, \sqrt{b} x e^{\left (m{\rm ln}\left (x\right ) + \frac{1}{2} \, n{\rm ln}\left (x\right )\right )}}{2 \, m + n + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(b)*x*e^(m*ln(x) + 1/2*n*ln(x))/(2*m + n + 2)