3.1129 \(\int \frac{(1-x)^{3/2}}{(1+x)^{5/2}} \, dx\)

Optimal. Leaf size=41 \[ -\frac{2 (1-x)^{3/2}}{3 (x+1)^{3/2}}+\frac{2 \sqrt{1-x}}{\sqrt{x+1}}+\sin ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.0055739, antiderivative size = 41, 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 = {47, 41, 216} \[ -\frac{2 (1-x)^{3/2}}{3 (x+1)^{3/2}}+\frac{2 \sqrt{1-x}}{\sqrt{x+1}}+\sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(1 - x)^(3/2)/(1 + x)^(5/2),x]

[Out]

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

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 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}}{(1+x)^{5/2}} \, dx &=-\frac{2 (1-x)^{3/2}}{3 (1+x)^{3/2}}-\int \frac{\sqrt{1-x}}{(1+x)^{3/2}} \, dx\\ &=-\frac{2 (1-x)^{3/2}}{3 (1+x)^{3/2}}+\frac{2 \sqrt{1-x}}{\sqrt{1+x}}+\int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx\\ &=-\frac{2 (1-x)^{3/2}}{3 (1+x)^{3/2}}+\frac{2 \sqrt{1-x}}{\sqrt{1+x}}+\int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=-\frac{2 (1-x)^{3/2}}{3 (1+x)^{3/2}}+\frac{2 \sqrt{1-x}}{\sqrt{1+x}}+\sin ^{-1}(x)\\ \end{align*}

Mathematica [A]  time = 0.0538163, size = 49, normalized size = 1.2 \[ \frac{-8 x^2+4 x+4}{3 \sqrt{1-x} (x+1)^{3/2}}-2 \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - x)^(3/2)/(1 + x)^(5/2),x]

[Out]

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

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Maple [B]  time = 0.015, size = 73, normalized size = 1.8 \begin{align*} -{\frac{8\,{x}^{2}-4\,x-4}{3}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) } \left ( 1+x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{- \left ( 1+x \right ) \left ( -1+x \right ) }}}{\frac{1}{\sqrt{1-x}}}}+{\arcsin \left ( x \right ) \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)^(5/2),x)

[Out]

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

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Maxima [B]  time = 1.5003, size = 89, normalized size = 2.17 \begin{align*} -\frac{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}{3 \,{\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}} - \frac{2 \, \sqrt{-x^{2} + 1}}{3 \,{\left (x^{2} + 2 \, x + 1\right )}} + \frac{7 \, \sqrt{-x^{2} + 1}}{3 \,{\left (x + 1\right )}} + \arcsin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(3/2)/(1+x)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.81969, size = 188, normalized size = 4.59 \begin{align*} \frac{2 \,{\left (2 \, x^{2} + 2 \,{\left (2 \, x + 1\right )} \sqrt{x + 1} \sqrt{-x + 1} - 3 \,{\left (x^{2} + 2 \, x + 1\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 4 \, x + 2\right )}}{3 \,{\left (x^{2} + 2 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(3/2)/(1+x)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [C]  time = 5.09716, size = 126, normalized size = 3.07 \begin{align*} \begin{cases} \frac{8 \sqrt{-1 + \frac{2}{x + 1}}}{3} - \frac{4 \sqrt{-1 + \frac{2}{x + 1}}}{3 \left (x + 1\right )} + i \log{\left (\frac{1}{x + 1} \right )} + i \log{\left (x + 1 \right )} + 2 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} & \text{for}\: \frac{2}{\left |{x + 1}\right |} > 1 \\\frac{8 i \sqrt{1 - \frac{2}{x + 1}}}{3} - \frac{4 i \sqrt{1 - \frac{2}{x + 1}}}{3 \left (x + 1\right )} + i \log{\left (\frac{1}{x + 1} \right )} - 2 i \log{\left (\sqrt{1 - \frac{2}{x + 1}} + 1 \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)**(3/2)/(1+x)**(5/2),x)

[Out]

Piecewise((8*sqrt(-1 + 2/(x + 1))/3 - 4*sqrt(-1 + 2/(x + 1))/(3*(x + 1)) + I*log(1/(x + 1)) + I*log(x + 1) + 2
*asin(sqrt(2)*sqrt(x + 1)/2), 2/Abs(x + 1) > 1), (8*I*sqrt(1 - 2/(x + 1))/3 - 4*I*sqrt(1 - 2/(x + 1))/(3*(x +
1)) + I*log(1/(x + 1)) - 2*I*log(sqrt(1 - 2/(x + 1)) + 1), True))

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Giac [B]  time = 1.1396, size = 138, normalized size = 3.37 \begin{align*} \frac{{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}}{12 \,{\left (x + 1\right )}^{\frac{3}{2}}} - \frac{5 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}}{4 \, \sqrt{x + 1}} + \frac{{\left (x + 1\right )}^{\frac{3}{2}}{\left (\frac{15 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{2}}{x + 1} - 1\right )}}{12 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}} + 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((1-x)^(3/2)/(1+x)^(5/2),x, algorithm="giac")

[Out]

1/12*(sqrt(2) - sqrt(-x + 1))^3/(x + 1)^(3/2) - 5/4*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) + 1/12*(x + 1)^(3/2)*
(15*(sqrt(2) - sqrt(-x + 1))^2/(x + 1) - 1)/(sqrt(2) - sqrt(-x + 1))^3 + 2*arcsin(1/2*sqrt(2)*sqrt(x + 1))