3.504 \(\int \frac{x^2}{\sqrt{1+x} \sqrt{1-x+x^2}} \, dx\)

Optimal. Leaf size=23 \[ \frac{2}{3} \sqrt{x+1} \sqrt{x^2-x+1} \]

[Out]

(2*Sqrt[1 + x]*Sqrt[1 - x + x^2])/3

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Rubi [A]  time = 0.0187211, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {913} \[ \frac{2}{3} \sqrt{x+1} \sqrt{x^2-x+1} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(Sqrt[1 + x]*Sqrt[1 - x + x^2]),x]

[Out]

(2*Sqrt[1 + x]*Sqrt[1 - x + x^2])/3

Rule 913

Int[(x_)^2*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^
(m + 1)*(a + b*x + c*x^2)^(p + 1))/(c*e*(m + 2*p + 3)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*e*(m +
p + 2) + 2*c*d*(p + 1), 0] && EqQ[b*d*(p + 1) + a*e*(m + 1), 0] && NeQ[m + 2*p + 3, 0]

Rubi steps

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

Mathematica [A]  time = 0.0255489, size = 23, normalized size = 1. \[ \frac{2}{3} \sqrt{x+1} \sqrt{x^2-x+1} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(Sqrt[1 + x]*Sqrt[1 - x + x^2]),x]

[Out]

(2*Sqrt[1 + x]*Sqrt[1 - x + x^2])/3

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Maple [A]  time = 0.043, size = 18, normalized size = 0.8 \begin{align*}{\frac{2}{3}\sqrt{1+x}\sqrt{{x}^{2}-x+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(1+x)^(1/2)/(x^2-x+1)^(1/2),x)

[Out]

2/3*(1+x)^(1/2)*(x^2-x+1)^(1/2)

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Maxima [A]  time = 1.47163, size = 30, normalized size = 1.3 \begin{align*} \frac{2 \,{\left (x^{3} + 1\right )}}{3 \, \sqrt{x^{2} - x + 1} \sqrt{x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(1+x)^(1/2)/(x^2-x+1)^(1/2),x, algorithm="maxima")

[Out]

2/3*(x^3 + 1)/(sqrt(x^2 - x + 1)*sqrt(x + 1))

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Fricas [A]  time = 1.68422, size = 47, normalized size = 2.04 \begin{align*} \frac{2}{3} \, \sqrt{x^{2} - x + 1} \sqrt{x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(1+x)^(1/2)/(x^2-x+1)^(1/2),x, algorithm="fricas")

[Out]

2/3*sqrt(x^2 - x + 1)*sqrt(x + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\sqrt{x + 1} \sqrt{x^{2} - x + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(1+x)**(1/2)/(x**2-x+1)**(1/2),x)

[Out]

Integral(x**2/(sqrt(x + 1)*sqrt(x**2 - x + 1)), x)

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Giac [A]  time = 1.17004, size = 24, normalized size = 1.04 \begin{align*} \frac{2}{3} \, \sqrt{{\left (x + 1\right )}^{2} - 3 \, x} \sqrt{x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(1+x)^(1/2)/(x^2-x+1)^(1/2),x, algorithm="giac")

[Out]

2/3*sqrt((x + 1)^2 - 3*x)*sqrt(x + 1)