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

Optimal. Leaf size=63 \[ -\frac {2}{3} \sqrt {1-x^2} x^2-\frac {1}{24} (21 x+32) \sqrt {1-x^2}-\frac {1}{4} \sqrt {1-x^2} x^3+\frac {7}{8} \sin ^{-1}(x) \]

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Rubi [A]  time = 0.08, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1809, 833, 780, 216} \begin {gather*} -\frac {1}{4} \sqrt {1-x^2} x^3-\frac {2}{3} \sqrt {1-x^2} x^2-\frac {1}{24} (21 x+32) \sqrt {1-x^2}+\frac {7}{8} \sin ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2*(1 + x)^2)/Sqrt[1 - x^2],x]

[Out]

(-2*x^2*Sqrt[1 - x^2])/3 - (x^3*Sqrt[1 - x^2])/4 - ((32 + 21*x)*Sqrt[1 - x^2])/24 + (7*ArcSin[x])/8

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 833

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

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^2 (1+x)^2}{\sqrt {1-x^2}} \, dx &=-\frac {1}{4} x^3 \sqrt {1-x^2}-\frac {1}{4} \int \frac {(-7-8 x) x^2}{\sqrt {1-x^2}} \, dx\\ &=-\frac {2}{3} x^2 \sqrt {1-x^2}-\frac {1}{4} x^3 \sqrt {1-x^2}+\frac {1}{12} \int \frac {x (16+21 x)}{\sqrt {1-x^2}} \, dx\\ &=-\frac {2}{3} x^2 \sqrt {1-x^2}-\frac {1}{4} x^3 \sqrt {1-x^2}-\frac {1}{24} (32+21 x) \sqrt {1-x^2}+\frac {7}{8} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=-\frac {2}{3} x^2 \sqrt {1-x^2}-\frac {1}{4} x^3 \sqrt {1-x^2}-\frac {1}{24} (32+21 x) \sqrt {1-x^2}+\frac {7}{8} \sin ^{-1}(x)\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 37, normalized size = 0.59 \begin {gather*} \frac {7}{8} \sin ^{-1}(x)-\frac {1}{24} \sqrt {1-x^2} \left (6 x^3+16 x^2+21 x+32\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/24*(Sqrt[1 - x^2]*(32 + 21*x + 16*x^2 + 6*x^3)) + (7*ArcSin[x])/8

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IntegrateAlgebraic [A]  time = 0.22, size = 53, normalized size = 0.84 \begin {gather*} \frac {7}{4} \tan ^{-1}\left (\frac {x}{\sqrt {1-x^2}-1}\right )+\frac {1}{24} \sqrt {1-x^2} \left (-6 x^3-16 x^2-21 x-32\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - x^2]*(-32 - 21*x - 16*x^2 - 6*x^3))/24 + (7*ArcTan[x/(-1 + Sqrt[1 - x^2])])/4

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fricas [A]  time = 0.38, size = 45, normalized size = 0.71 \begin {gather*} -\frac {1}{24} \, {\left (6 \, x^{3} + 16 \, x^{2} + 21 \, x + 32\right )} \sqrt {-x^{2} + 1} - \frac {7}{4} \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(6*x^3 + 16*x^2 + 21*x + 32)*sqrt(-x^2 + 1) - 7/4*arctan((sqrt(-x^2 + 1) - 1)/x)

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giac [A]  time = 0.19, size = 30, normalized size = 0.48 \begin {gather*} -\frac {1}{24} \, {\left ({\left (2 \, {\left (3 \, x + 8\right )} x + 21\right )} x + 32\right )} \sqrt {-x^{2} + 1} + \frac {7}{8} \, \arcsin \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*((2*(3*x + 8)*x + 21)*x + 32)*sqrt(-x^2 + 1) + 7/8*arcsin(x)

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maple [A]  time = 0.01, size = 57, normalized size = 0.90 \begin {gather*} -\frac {\sqrt {-x^{2}+1}\, x^{3}}{4}-\frac {2 \sqrt {-x^{2}+1}\, x^{2}}{3}-\frac {7 \sqrt {-x^{2}+1}\, x}{8}+\frac {7 \arcsin \relax (x )}{8}-\frac {4 \sqrt {-x^{2}+1}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(-x^2+1)^(1/2)*x^3-7/8*(-x^2+1)^(1/2)*x+7/8*arcsin(x)-2/3*(-x^2+1)^(1/2)*x^2-4/3*(-x^2+1)^(1/2)

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maxima [A]  time = 0.97, size = 56, normalized size = 0.89 \begin {gather*} -\frac {1}{4} \, \sqrt {-x^{2} + 1} x^{3} - \frac {2}{3} \, \sqrt {-x^{2} + 1} x^{2} - \frac {7}{8} \, \sqrt {-x^{2} + 1} x - \frac {4}{3} \, \sqrt {-x^{2} + 1} + \frac {7}{8} \, \arcsin \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(-x^2 + 1)*x^3 - 2/3*sqrt(-x^2 + 1)*x^2 - 7/8*sqrt(-x^2 + 1)*x - 4/3*sqrt(-x^2 + 1) + 7/8*arcsin(x)

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mupad [B]  time = 0.03, size = 31, normalized size = 0.49 \begin {gather*} \frac {7\,\mathrm {asin}\relax (x)}{8}-\sqrt {1-x^2}\,\left (\frac {x^3}{4}+\frac {2\,x^2}{3}+\frac {7\,x}{8}+\frac {4}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(7*asin(x))/8 - (1 - x^2)^(1/2)*((7*x)/8 + (2*x^2)/3 + x^3/4 + 4/3)

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sympy [A]  time = 0.81, size = 60, normalized size = 0.95 \begin {gather*} - \frac {x^{3} \sqrt {1 - x^{2}}}{4} - \frac {2 x^{2} \sqrt {1 - x^{2}}}{3} - \frac {7 x \sqrt {1 - x^{2}}}{8} - \frac {4 \sqrt {1 - x^{2}}}{3} + \frac {7 \operatorname {asin}{\relax (x )}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x**3*sqrt(1 - x**2)/4 - 2*x**2*sqrt(1 - x**2)/3 - 7*x*sqrt(1 - x**2)/8 - 4*sqrt(1 - x**2)/3 + 7*asin(x)/8

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