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

Optimal. Leaf size=41 \[ -\frac{1}{3} \sqrt{1-x^2} x^2-\frac{1}{3} (3 x+5) \sqrt{1-x^2}+\sin ^{-1}(x) \]

[Out]

-(x^2*Sqrt[1 - x^2])/3 - ((5 + 3*x)*Sqrt[1 - x^2])/3 + ArcSin[x]

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Rubi [A]  time = 0.0477031, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1809, 780, 216} \[ -\frac{1}{3} \sqrt{1-x^2} x^2-\frac{1}{3} (3 x+5) \sqrt{1-x^2}+\sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(x*(1 + x)^2)/Sqrt[1 - x^2],x]

[Out]

-(x^2*Sqrt[1 - x^2])/3 - ((5 + 3*x)*Sqrt[1 - x^2])/3 + ArcSin[x]

Rule 1809

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x,
Expon[Pq, x]]}, Simp[(f*(c*x)^(m + q - 1)*(a + b*x^2)^(p + 1))/(b*c^(q - 1)*(m + q + 2*p + 1)), x] + Dist[1/(
b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q
- a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x]
&& PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])

Rule 780

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*(a + c*x^2)^(p + 1))/(2*c*(p + 1)*(2*p + 3)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[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 (1+x)^2}{\sqrt{1-x^2}} \, dx &=-\frac{1}{3} x^2 \sqrt{1-x^2}-\frac{1}{3} \int \frac{(-5-6 x) x}{\sqrt{1-x^2}} \, dx\\ &=-\frac{1}{3} x^2 \sqrt{1-x^2}-\frac{1}{3} (5+3 x) \sqrt{1-x^2}+\int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=-\frac{1}{3} x^2 \sqrt{1-x^2}-\frac{1}{3} (5+3 x) \sqrt{1-x^2}+\sin ^{-1}(x)\\ \end{align*}

Mathematica [A]  time = 0.0185539, size = 26, normalized size = 0.63 \[ \sin ^{-1}(x)-\frac{1}{3} \sqrt{1-x^2} \left (x^2+3 x+5\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - x^2]*(5 + 3*x + x^2))/3 + ArcSin[x]

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Maple [A]  time = 0.047, size = 41, normalized size = 1. \begin{align*} -{\frac{{x}^{2}}{3}\sqrt{-{x}^{2}+1}}-{\frac{5}{3}\sqrt{-{x}^{2}+1}}-x\sqrt{-{x}^{2}+1}+\arcsin \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*x^2*(-x^2+1)^(1/2)-5/3*(-x^2+1)^(1/2)-x*(-x^2+1)^(1/2)+arcsin(x)

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Maxima [A]  time = 1.46933, size = 54, normalized size = 1.32 \begin{align*} -\frac{1}{3} \, \sqrt{-x^{2} + 1} x^{2} - \sqrt{-x^{2} + 1} x - \frac{5}{3} \, \sqrt{-x^{2} + 1} + \arcsin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*sqrt(-x^2 + 1)*x^2 - sqrt(-x^2 + 1)*x - 5/3*sqrt(-x^2 + 1) + arcsin(x)

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Fricas [A]  time = 1.8519, size = 97, normalized size = 2.37 \begin{align*} -\frac{1}{3} \,{\left (x^{2} + 3 \, x + 5\right )} \sqrt{-x^{2} + 1} - 2 \, \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(x^2 + 3*x + 5)*sqrt(-x^2 + 1) - 2*arctan((sqrt(-x^2 + 1) - 1)/x)

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Sympy [A]  time = 0.426946, size = 37, normalized size = 0.9 \begin{align*} - \frac{x^{2} \sqrt{1 - x^{2}}}{3} - x \sqrt{1 - x^{2}} - \frac{5 \sqrt{1 - x^{2}}}{3} + \operatorname{asin}{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x**2*sqrt(1 - x**2)/3 - x*sqrt(1 - x**2) - 5*sqrt(1 - x**2)/3 + asin(x)

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Giac [A]  time = 1.14691, size = 28, normalized size = 0.68 \begin{align*} -\frac{1}{3} \,{\left ({\left (x + 3\right )} x + 5\right )} \sqrt{-x^{2} + 1} + \arcsin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*((x + 3)*x + 5)*sqrt(-x^2 + 1) + arcsin(x)