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

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

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Rubi [A]  time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {2}{3} \sqrt {x+1} \sqrt {x^2-x+1} \end {gather*}

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*}

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Mathematica [A]  time = 0.02, size = 23, normalized size = 1.00 \begin {gather*} \frac {2}{3} \sqrt {x+1} \sqrt {x^2-x+1} \end {gather*}

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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IntegrateAlgebraic [F]  time = 150.83, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2}{\sqrt {1+x} \sqrt {1-x+x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.38, size = 17, normalized size = 0.74 \begin {gather*} \frac {2}{3} \, \sqrt {x^{2} - x + 1} \sqrt {x + 1} \end {gather*}

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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giac [A]  time = 0.18, size = 18, normalized size = 0.78 \begin {gather*} \frac {2}{3} \, \sqrt {{\left (x + 1\right )}^{2} - 3 \, x} \sqrt {x + 1} \end {gather*}

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)

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maple [A]  time = 0.00, size = 18, normalized size = 0.78 \begin {gather*} \frac {2 \sqrt {x +1}\, \sqrt {x^{2}-x +1}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.97, size = 22, normalized size = 0.96 \begin {gather*} \frac {2 \, {\left (x^{3} + 1\right )}}{3 \, \sqrt {x^{2} - x + 1} \sqrt {x + 1}} \end {gather*}

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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mupad [B]  time = 0.15, size = 9, normalized size = 0.39 \begin {gather*} \frac {2\,\sqrt {x^3+1}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(x^3 + 1)^(1/2))/3

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\sqrt {x + 1} \sqrt {x^{2} - x + 1}}\, dx \end {gather*}

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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