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)

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Rubi [A]  time = 0.0072219, 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, Rules used = {15, 30} \[ \frac{2 x^{m+1} \sqrt{b x^n}}{2 m+n+2} \]

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)

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int x^m \sqrt{b x^n} \, dx &=\left (x^{-n/2} \sqrt{b x^n}\right ) \int x^{m+\frac{n}{2}} \, dx\\ &=\frac{2 x^{1+m} \sqrt{b x^n}}{2+2 m+n}\\ \end{align*}

Mathematica [A]  time = 0.0041527, 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)

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Maple [A]  time = 0.002, size = 23, normalized size = 1. \begin{align*} 2\,{\frac{{x}^{1+m}\sqrt{b{x}^{n}}}{2+2\,m+n}} \end{align*}

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)

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Maxima [A]  time = 1.02723, size = 30, normalized size = 1.25 \begin{align*} \frac{2 \, \sqrt{b} x x^{m} \sqrt{x^{n}}}{2 \, m + n + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.19987, size = 30, normalized size = 1.25 \begin{align*} \frac{2 \, \sqrt{b} x x^{m} x^{\frac{1}{2} \, n}}{2 \, m + n + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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