3.890 \(\int \frac{\sqrt{\frac{(-1+x^2)^2}{x+x^3}}}{1+x^2} \, dx\)

Optimal. Leaf size=33 \[ \frac{2 x \sqrt{\frac{\left (1-x^2\right )^2}{x^3+x}}}{1-x^2} \]

[Out]

(2*x*Sqrt[(1 - x^2)^2/(x + x^3)])/(1 - x^2)

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Rubi [A]  time = 0.192576, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6719, 2056, 449} \[ \frac{2 x \sqrt{\frac{\left (1-x^2\right )^2}{x^3+x}}}{1-x^2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[(-1 + x^2)^2/(x + x^3)]/(1 + x^2),x]

[Out]

(2*x*Sqrt[(1 - x^2)^2/(x + x^3)])/(1 - x^2)

Rule 6719

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F
racPart[p])*w^(n*FracPart[p])), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rule 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 449

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

Rubi steps

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

Mathematica [A]  time = 0.0115261, size = 29, normalized size = 0.88 \[ -\frac{2 x \sqrt{\frac{\left (x^2-1\right )^2}{x^3+x}}}{x^2-1} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[(-1 + x^2)^2/(x + x^3)]/(1 + x^2),x]

[Out]

(-2*x*Sqrt[(-1 + x^2)^2/(x + x^3)])/(-1 + x^2)

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Maple [A]  time = 0.003, size = 34, normalized size = 1. \begin{align*} -2\,{\frac{x}{ \left ( x-1 \right ) \left ( 1+x \right ) }\sqrt{{\frac{ \left ({x}^{2}-1 \right ) ^{2}}{x \left ({x}^{2}+1 \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2-1)^2/(x^3+x))^(1/2)/(x^2+1),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{{\left (x^{2} - 1\right )}^{2}}{x^{3} + x}}}{x^{2} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-1)^2/(x^3+x))^(1/2)/(x^2+1),x, algorithm="maxima")

[Out]

integrate(sqrt((x^2 - 1)^2/(x^3 + x))/(x^2 + 1), x)

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Fricas [A]  time = 1.46514, size = 68, normalized size = 2.06 \begin{align*} -\frac{2 \, x \sqrt{\frac{x^{4} - 2 \, x^{2} + 1}{x^{3} + x}}}{x^{2} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-1)^2/(x^3+x))^(1/2)/(x^2+1),x, algorithm="fricas")

[Out]

-2*x*sqrt((x^4 - 2*x^2 + 1)/(x^3 + x))/(x^2 - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{\left (x - 1\right )^{2} \left (x + 1\right )^{2}}{x^{3} + x}}}{x^{2} + 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**2-1)**2/(x**3+x))**(1/2)/(x**2+1),x)

[Out]

Integral(sqrt((x - 1)**2*(x + 1)**2/(x**3 + x))/(x**2 + 1), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{{\left (x^{2} - 1\right )}^{2}}{x^{3} + x}}}{x^{2} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-1)^2/(x^3+x))^(1/2)/(x^2+1),x, algorithm="giac")

[Out]

integrate(sqrt((x^2 - 1)^2/(x^3 + x))/(x^2 + 1), x)