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

Optimal. Leaf size=44 \[ -\frac{1}{2} \sqrt{1-(x+1)^2} x+\frac{3}{2} \sqrt{1-(x+1)^2}+\frac{3}{2} \sin ^{-1}(x+1) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0258208, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {371, 671, 641, 216} \[ -\frac{1}{2} \sqrt{1-(x+1)^2} x+\frac{3}{2} \sqrt{1-(x+1)^2}+\frac{3}{2} \sin ^{-1}(x+1) \]

Antiderivative was successfully verified.

[In]

Int[x^2/Sqrt[1 - (1 + x)^2],x]

[Out]

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

Rule 371

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coefficient[v, x, 0], d = Coefficient[v,
x, 1]}, Dist[1/d^(m + 1), Subst[Int[SimplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]
] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]

Rule 671

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

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

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

Mathematica [A]  time = 0.0214099, size = 51, normalized size = 1.16 \[ \frac{x \left (x^2-x-6\right )+6 \sqrt{x} \sqrt{x+2} \sinh ^{-1}\left (\frac{\sqrt{x}}{\sqrt{2}}\right )}{2 \sqrt{-x (x+2)}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/Sqrt[1 - (1 + x)^2],x]

[Out]

(x*(-6 - x + x^2) + 6*Sqrt[x]*Sqrt[2 + x]*ArcSinh[Sqrt[x]/Sqrt[2]])/(2*Sqrt[-(x*(2 + x))])

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 35, normalized size = 0.8 \begin{align*} -{\frac{x}{2}\sqrt{-{x}^{2}-2\,x}}+{\frac{3}{2}\sqrt{-{x}^{2}-2\,x}}+{\frac{3\,\arcsin \left ( 1+x \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.65103, size = 49, normalized size = 1.11 \begin{align*} -\frac{1}{2} \, \sqrt{-x^{2} - 2 \, x} x + \frac{3}{2} \, \sqrt{-x^{2} - 2 \, x} - \frac{3}{2} \, \arcsin \left (-x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.76868, size = 84, normalized size = 1.91 \begin{align*} -\frac{1}{2} \, \sqrt{-x^{2} - 2 \, x}{\left (x - 3\right )} - 3 \, \arctan \left (\frac{\sqrt{-x^{2} - 2 \, x}}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2/sqrt(-x*(x + 2)), x)

________________________________________________________________________________________

Giac [A]  time = 1.11091, size = 31, normalized size = 0.7 \begin{align*} -\frac{1}{2} \, \sqrt{-{\left (x + 1\right )}^{2} + 1}{\left (x - 3\right )} + \frac{3}{2} \, \arcsin \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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