3.1108 \(\int \frac{(1-x)^{3/2}}{\sqrt{1+x}} \, dx\)

Optimal. Leaf size=47 \[ \frac{1}{2} \sqrt{x+1} (1-x)^{3/2}+\frac{3}{2} \sqrt{x+1} \sqrt{1-x}+\frac{3}{2} \sin ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.0069494, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {50, 41, 216} \[ \frac{1}{2} \sqrt{x+1} (1-x)^{3/2}+\frac{3}{2} \sqrt{x+1} \sqrt{1-x}+\frac{3}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(1 - x)^(3/2)/Sqrt[1 + x],x]

[Out]

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

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

Mathematica [A]  time = 0.0200142, size = 47, normalized size = 1. \[ \frac{\sqrt{x+1} \left (x^2-5 x+4\right )}{2 \sqrt{1-x}}-3 \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 57, normalized size = 1.2 \begin{align*}{\frac{1}{2} \left ( 1-x \right ) ^{{\frac{3}{2}}}\sqrt{1+x}}+{\frac{3}{2}\sqrt{1-x}\sqrt{1+x}}+{\frac{3\,\arcsin \left ( x \right ) }{2}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.50223, size = 38, normalized size = 0.81 \begin{align*} -\frac{1}{2} \, \sqrt{-x^{2} + 1} x + 2 \, \sqrt{-x^{2} + 1} + \frac{3}{2} \, \arcsin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.55196, size = 113, normalized size = 2.4 \begin{align*} -\frac{1}{2} \, \sqrt{x + 1}{\left (x - 4\right )} \sqrt{-x + 1} - 3 \, \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((1-x)^(3/2)/(1+x)^(1/2),x, algorithm="fricas")

[Out]

-1/2*sqrt(x + 1)*(x - 4)*sqrt(-x + 1) - 3*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x)

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Sympy [A]  time = 2.95077, size = 139, normalized size = 2.96 \begin{align*} \begin{cases} - 3 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} - \frac{i \left (x + 1\right )^{\frac{5}{2}}}{2 \sqrt{x - 1}} + \frac{7 i \left (x + 1\right )^{\frac{3}{2}}}{2 \sqrt{x - 1}} - \frac{5 i \sqrt{x + 1}}{\sqrt{x - 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\3 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} + \frac{\left (x + 1\right )^{\frac{5}{2}}}{2 \sqrt{1 - x}} - \frac{7 \left (x + 1\right )^{\frac{3}{2}}}{2 \sqrt{1 - x}} + \frac{5 \sqrt{x + 1}}{\sqrt{1 - x}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-3*I*acosh(sqrt(2)*sqrt(x + 1)/2) - I*(x + 1)**(5/2)/(2*sqrt(x - 1)) + 7*I*(x + 1)**(3/2)/(2*sqrt(x
- 1)) - 5*I*sqrt(x + 1)/sqrt(x - 1), Abs(x + 1)/2 > 1), (3*asin(sqrt(2)*sqrt(x + 1)/2) + (x + 1)**(5/2)/(2*sq
rt(1 - x)) - 7*(x + 1)**(3/2)/(2*sqrt(1 - x)) + 5*sqrt(x + 1)/sqrt(1 - x), True))

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Giac [A]  time = 1.07467, size = 59, normalized size = 1.26 \begin{align*} -\frac{1}{2} \, \sqrt{x + 1}{\left (x - 2\right )} \sqrt{-x + 1} + \sqrt{x + 1} \sqrt{-x + 1} + 3 \, \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((1-x)^(3/2)/(1+x)^(1/2),x, algorithm="giac")

[Out]

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