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

Optimal. Leaf size=61 \[ -\frac{1}{3} \sqrt{1-x} (x+1)^{5/2}-\frac{1}{3} \sqrt{1-x} (x+1)^{3/2}-\sqrt{1-x} \sqrt{x+1}+\sin ^{-1}(x) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0113053, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {80, 50, 41, 216} \[ -\frac{1}{3} \sqrt{1-x} (x+1)^{5/2}-\frac{1}{3} \sqrt{1-x} (x+1)^{3/2}-\sqrt{1-x} \sqrt{x+1}+\sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(x*(1 + x)^(3/2))/Sqrt[1 - x],x]

[Out]

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

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

Mathematica [A]  time = 0.0241145, size = 42, normalized size = 0.69 \[ -\frac{1}{3} \sqrt{1-x^2} \left (x^2+3 x+5\right )-2 \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 66, normalized size = 1.1 \begin{align*}{\frac{1}{3}\sqrt{1-x}\sqrt{1+x} \left ( -{x}^{2}\sqrt{-{x}^{2}+1}-3\,x\sqrt{-{x}^{2}+1}+3\,\arcsin \left ( x \right ) -5\,\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*(1+x)^(3/2)/(1-x)^(1/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.98909, size = 54, normalized size = 0.89 \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)^(3/2)/(1-x)^(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)

________________________________________________________________________________________

Fricas [A]  time = 1.78847, size = 124, normalized size = 2.03 \begin{align*} -\frac{1}{3} \,{\left (x^{2} + 3 \, x + 5\right )} \sqrt{x + 1} \sqrt{-x + 1} - 2 \, \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*(1+x)^(3/2)/(1-x)^(1/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 59.4705, size = 129, normalized size = 2.11 \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 ) + 2 \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(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))) + 2*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)))

________________________________________________________________________________________

Giac [A]  time = 1.29544, size = 50, normalized size = 0.82 \begin{align*} -\frac{1}{3} \,{\left ({\left (x + 2\right )}{\left (x + 1\right )} + 3\right )} \sqrt{x + 1} \sqrt{-x + 1} + 2 \, \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*(1+x)^(3/2)/(1-x)^(1/2),x, algorithm="giac")

[Out]

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