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

Optimal. Leaf size=88 \[ -\frac{1}{4} \sqrt{1-x} x (x+1)^{5/2}-\frac{1}{6} \sqrt{1-x} (x+1)^{5/2}-\frac{7}{24} \sqrt{1-x} (x+1)^{3/2}-\frac{7}{8} \sqrt{1-x} \sqrt{x+1}+\frac{7}{8} \sin ^{-1}(x) \]

[Out]

(-7*Sqrt[1 - x]*Sqrt[1 + x])/8 - (7*Sqrt[1 - x]*(1 + x)^(3/2))/24 - (Sqrt[1 - x]*(1 + x)^(5/2))/6 - (Sqrt[1 -
x]*x*(1 + x)^(5/2))/4 + (7*ArcSin[x])/8

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Rubi [A]  time = 0.0187907, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {90, 80, 50, 41, 216} \[ -\frac{1}{4} \sqrt{1-x} x (x+1)^{5/2}-\frac{1}{6} \sqrt{1-x} (x+1)^{5/2}-\frac{7}{24} \sqrt{1-x} (x+1)^{3/2}-\frac{7}{8} \sqrt{1-x} \sqrt{x+1}+\frac{7}{8} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*Sqrt[1 - x]*Sqrt[1 + x])/8 - (7*Sqrt[1 - x]*(1 + x)^(3/2))/24 - (Sqrt[1 - x]*(1 + x)^(5/2))/6 - (Sqrt[1 -
x]*x*(1 + x)^(5/2))/4 + (7*ArcSin[x])/8

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

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 (1+x)^{3/2}}{\sqrt{1-x}} \, dx &=-\frac{1}{4} \sqrt{1-x} x (1+x)^{5/2}-\frac{1}{4} \int \frac{(-1-2 x) (1+x)^{3/2}}{\sqrt{1-x}} \, dx\\ &=-\frac{1}{6} \sqrt{1-x} (1+x)^{5/2}-\frac{1}{4} \sqrt{1-x} x (1+x)^{5/2}+\frac{7}{12} \int \frac{(1+x)^{3/2}}{\sqrt{1-x}} \, dx\\ &=-\frac{7}{24} \sqrt{1-x} (1+x)^{3/2}-\frac{1}{6} \sqrt{1-x} (1+x)^{5/2}-\frac{1}{4} \sqrt{1-x} x (1+x)^{5/2}+\frac{7}{8} \int \frac{\sqrt{1+x}}{\sqrt{1-x}} \, dx\\ &=-\frac{7}{8} \sqrt{1-x} \sqrt{1+x}-\frac{7}{24} \sqrt{1-x} (1+x)^{3/2}-\frac{1}{6} \sqrt{1-x} (1+x)^{5/2}-\frac{1}{4} \sqrt{1-x} x (1+x)^{5/2}+\frac{7}{8} \int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx\\ &=-\frac{7}{8} \sqrt{1-x} \sqrt{1+x}-\frac{7}{24} \sqrt{1-x} (1+x)^{3/2}-\frac{1}{6} \sqrt{1-x} (1+x)^{5/2}-\frac{1}{4} \sqrt{1-x} x (1+x)^{5/2}+\frac{7}{8} \int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=-\frac{7}{8} \sqrt{1-x} \sqrt{1+x}-\frac{7}{24} \sqrt{1-x} (1+x)^{3/2}-\frac{1}{6} \sqrt{1-x} (1+x)^{5/2}-\frac{1}{4} \sqrt{1-x} x (1+x)^{5/2}+\frac{7}{8} \sin ^{-1}(x)\\ \end{align*}

Mathematica [A]  time = 0.0297277, size = 51, normalized size = 0.58 \[ -\frac{1}{24} \sqrt{1-x^2} \left (6 x^3+16 x^2+21 x+32\right )-\frac{7}{4} \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 80, normalized size = 0.9 \begin{align*}{\frac{1}{24}\sqrt{1-x}\sqrt{1+x} \left ( -6\,{x}^{3}\sqrt{-{x}^{2}+1}-16\,{x}^{2}\sqrt{-{x}^{2}+1}-21\,x\sqrt{-{x}^{2}+1}+21\,\arcsin \left ( x \right ) -32\,\sqrt{-{x}^{2}+1} \right ){\frac{1}{\sqrt{-{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(1+x)^(1/2)*(1-x)^(1/2)*(-6*x^3*(-x^2+1)^(1/2)-16*x^2*(-x^2+1)^(1/2)-21*x*(-x^2+1)^(1/2)+21*arcsin(x)-32*
(-x^2+1)^(1/2))/(-x^2+1)^(1/2)

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Maxima [A]  time = 1.55942, size = 76, normalized size = 0.86 \begin{align*} -\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 \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(1+x)^(3/2)/(1-x)^(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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Fricas [A]  time = 1.72598, size = 146, normalized size = 1.66 \begin{align*} -\frac{1}{24} \,{\left (6 \, x^{3} + 16 \, x^{2} + 21 \, x + 32\right )} \sqrt{x + 1} \sqrt{-x + 1} - \frac{7}{4} \, \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 111.667, size = 240, normalized size = 2.73 \begin{align*} 2 \left (\begin{cases} - \frac{x \sqrt{1 - x} \sqrt{x + 1}}{4} - \sqrt{1 - x} \sqrt{x + 1} + \frac{3 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{2} & \text{for}\: x \geq -1 \wedge x < 1 \end{cases}\right ) - 4 \left (\begin{cases} - \frac{3 x \sqrt{1 - x} \sqrt{x + 1}}{4} + \frac{\left (1 - x\right )^{\frac{3}{2}} \left (x + 1\right )^{\frac{3}{2}}}{6} - 2 \sqrt{1 - x} \sqrt{x + 1} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{2} & \text{for}\: x \geq -1 \wedge x < 1 \end{cases}\right ) + 2 \left (\begin{cases} - \frac{7 x \sqrt{1 - x} \sqrt{x + 1}}{4} + \frac{2 \left (1 - x\right )^{\frac{3}{2}} \left (x + 1\right )^{\frac{3}{2}}}{3} + \frac{\sqrt{1 - x} \sqrt{x + 1} \left (- 5 x - 2 \left (x + 1\right )^{3} + 6 \left (x + 1\right )^{2} - 4\right )}{16} - 4 \sqrt{1 - x} \sqrt{x + 1} + \frac{35 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )}}{8} & \text{for}\: x \geq -1 \wedge x < 1 \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*Piecewise((-x*sqrt(1 - x)*sqrt(x + 1)/4 - sqrt(1 - x)*sqrt(x + 1) + 3*asin(sqrt(2)*sqrt(x + 1)/2)/2, (x >= -
1) & (x < 1))) - 4*Piecewise((-3*x*sqrt(1 - x)*sqrt(x + 1)/4 + (1 - x)**(3/2)*(x + 1)**(3/2)/6 - 2*sqrt(1 - x)
*sqrt(x + 1) + 5*asin(sqrt(2)*sqrt(x + 1)/2)/2, (x >= -1) & (x < 1))) + 2*Piecewise((-7*x*sqrt(1 - x)*sqrt(x +
1)/4 + 2*(1 - x)**(3/2)*(x + 1)**(3/2)/3 + sqrt(1 - x)*sqrt(x + 1)*(-5*x - 2*(x + 1)**3 + 6*(x + 1)**2 - 4)/1
6 - 4*sqrt(1 - x)*sqrt(x + 1) + 35*asin(sqrt(2)*sqrt(x + 1)/2)/8, (x >= -1) & (x < 1)))

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Giac [A]  time = 1.22538, size = 62, normalized size = 0.7 \begin{align*} -\frac{1}{24} \,{\left ({\left (2 \,{\left (3 \, x + 2\right )}{\left (x + 1\right )} + 7\right )}{\left (x + 1\right )} + 21\right )} \sqrt{x + 1} \sqrt{-x + 1} + \frac{7}{4} \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*((2*(3*x + 2)*(x + 1) + 7)*(x + 1) + 21)*sqrt(x + 1)*sqrt(-x + 1) + 7/4*arcsin(1/2*sqrt(2)*sqrt(x + 1))