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

Optimal. Leaf size=16 \[ \frac{1}{3} x^2 \sqrt{b x^2} \]

[Out]

(x^2*Sqrt[b*x^2])/3

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Rubi [A]  time = 0.0016844, antiderivative size = 16, 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} \[ \frac{1}{3} x^2 \sqrt{b x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*Sqrt[b*x^2])/3

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^2} \, dx &=\frac{\sqrt{b x^2} \int x^2 \, dx}{x}\\ &=\frac{1}{3} x^2 \sqrt{b x^2}\\ \end{align*}

Mathematica [A]  time = 0.0010001, size = 16, normalized size = 1. \[ \frac{1}{3} x^2 \sqrt{b x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*Sqrt[b*x^2])/3

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Maple [A]  time = 0.001, size = 13, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}}{3}\sqrt{b{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^2*(b*x^2)^(1/2)

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Maxima [A]  time = 0.985859, size = 16, normalized size = 1. \begin{align*} \frac{\left (b x^{2}\right )^{\frac{3}{2}}}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(b*x^2)^(3/2)/b

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Fricas [A]  time = 1.22889, size = 28, normalized size = 1.75 \begin{align*} \frac{1}{3} \, \sqrt{b x^{2}} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sqrt(b*x^2)*x^2

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Sympy [A]  time = 0.213405, size = 15, normalized size = 0.94 \begin{align*} \frac{\sqrt{b} x^{2} \sqrt{x^{2}}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b)*x**2*sqrt(x**2)/3

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Giac [A]  time = 1.16882, size = 14, normalized size = 0.88 \begin{align*} \frac{1}{3} \, \sqrt{b} x^{3} \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sqrt(b)*x^3*sgn(x)