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]

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

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Rubi [A]  time = 0.05, antiderivative size = 41, normalized size of antiderivative = 1.00, 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 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]

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 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])

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*}

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Mathematica [A]  time = 0.02, 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]

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

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fricas [A]  time = 0.95, size = 38, normalized size = 0.93 \[ -\frac {1}{3} \, {\left (x^{2} + 3 \, x + 5\right )} \sqrt {-x^{2} + 1} - 2 \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \]

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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giac [A]  time = 0.22, size = 21, normalized size = 0.51 \[ -\frac {1}{3} \, {\left ({\left (x + 3\right )} x + 5\right )} \sqrt {-x^{2} + 1} + \arcsin \relax (x) \]

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)

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maple [A]  time = 0.00, size = 41, normalized size = 1.00 \[ -\frac {\sqrt {-x^{2}+1}\, x^{2}}{3}-\sqrt {-x^{2}+1}\, x +\arcsin \relax (x )-\frac {5 \sqrt {-x^{2}+1}}{3} \]

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

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maxima [A]  time = 0.97, size = 40, normalized size = 0.98 \[ -\frac {1}{3} \, \sqrt {-x^{2} + 1} x^{2} - \sqrt {-x^{2} + 1} x - \frac {5}{3} \, \sqrt {-x^{2} + 1} + \arcsin \relax (x) \]

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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mupad [B]  time = 0.03, size = 22, normalized size = 0.54 \[ \mathrm {asin}\relax (x)-\sqrt {1-x^2}\,\left (\frac {x^2}{3}+x+\frac {5}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(x + 1)^2)/(1 - x^2)^(1/2),x)

[Out]

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

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sympy [A]  time = 0.41, size = 37, normalized size = 0.90 \[ - \frac {x^{2} \sqrt {1 - x^{2}}}{3} - x \sqrt {1 - x^{2}} - \frac {5 \sqrt {1 - x^{2}}}{3} + \operatorname {asin}{\relax (x )} \]

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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