3.131 \(\int x^2 \sqrt{b x^n} \, dx\)

Optimal. Leaf size=19 \[ \frac{2 x^3 \sqrt{b x^n}}{n+6} \]

[Out]

(2*x^3*Sqrt[b*x^n])/(6 + n)

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Rubi [A]  time = 0.0050222, antiderivative size = 19, 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} \[ \frac{2 x^3 \sqrt{b x^n}}{n+6} \]

Antiderivative was successfully verified.

[In]

Int[x^2*Sqrt[b*x^n],x]

[Out]

(2*x^3*Sqrt[b*x^n])/(6 + 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^2 \sqrt{b x^n} \, dx &=\left (x^{-n/2} \sqrt{b x^n}\right ) \int x^{2+\frac{n}{2}} \, dx\\ &=\frac{2 x^3 \sqrt{b x^n}}{6+n}\\ \end{align*}

Mathematica [A]  time = 0.0032968, size = 19, normalized size = 1. \[ \frac{2 x^3 \sqrt{b x^n}}{n+6} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Sqrt[b*x^n],x]

[Out]

(2*x^3*Sqrt[b*x^n])/(6 + n)

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Maple [A]  time = 0.001, size = 18, normalized size = 1. \begin{align*} 2\,{\frac{{x}^{3}\sqrt{b{x}^{n}}}{6+n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x^3*(b*x^n)^(1/2)/(6+n)

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

[Out]

Exception raised: ValueError

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

[Out]

Exception raised: UnboundLocalError

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

[Out]

Exception raised: TypeError

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Giac [A]  time = 1.26267, size = 24, normalized size = 1.26 \begin{align*} \frac{2 \, \sqrt{b} x^{3} x^{\frac{1}{2} \, n}}{n + 6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(b)*x^3*x^(1/2*n)/(n + 6)