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

Optimal. Leaf size=19 \[ \frac{2 x^2 \sqrt{b x^n}}{n+4} \]

[Out]

(2*x^2*Sqrt[b*x^n])/(4 + n)

________________________________________________________________________________________

Rubi [A]  time = 0.0050968, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {15, 30} \[ \frac{2 x^2 \sqrt{b x^n}}{n+4} \]

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[b*x^n],x]

[Out]

(2*x^2*Sqrt[b*x^n])/(4 + 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 \sqrt{b x^n} \, dx &=\left (x^{-n/2} \sqrt{b x^n}\right ) \int x^{1+\frac{n}{2}} \, dx\\ &=\frac{2 x^2 \sqrt{b x^n}}{4+n}\\ \end{align*}

Mathematica [A]  time = 0.0022967, size = 19, normalized size = 1. \[ \frac{2 x^2 \sqrt{b x^n}}{n+4} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Sqrt[b*x^n],x]

[Out]

(2*x^2*Sqrt[b*x^n])/(4 + n)

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 18, normalized size = 1. \begin{align*} 2\,{\frac{{x}^{2}\sqrt{b{x}^{n}}}{4+n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x^2*(b*x^n)^(1/2)/(4+n)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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*(b*x^n)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

________________________________________________________________________________________

Giac [A]  time = 1.18315, size = 24, normalized size = 1.26 \begin{align*} \frac{2 \, \sqrt{b} x^{2} x^{\frac{1}{2} \, n}}{n + 4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(b)*x^2*x^(1/2*n)/(n + 4)