\(\int \frac {(1-x)^{3/2}}{(1+x)^{5/2}} \, dx\) [1129]

Optimal result

Integrand size = 17, antiderivative size = 41 \[ \int \frac {(1-x)^{3/2}}{(1+x)^{5/2}} \, dx=-\frac {2 (1-x)^{3/2}}{3 (1+x)^{3/2}}+\frac {2 \sqrt {1-x}}{\sqrt {1+x}}+\arcsin (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {49, 41, 222} \[ \int \frac {(1-x)^{3/2}}{(1+x)^{5/2}} \, dx=\arcsin (x)-\frac {2 (1-x)^{3/2}}{3 (x+1)^{3/2}}+\frac {2 \sqrt {1-x}}{\sqrt {x+1}} \]

[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 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 49

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 222

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*} \text {integral}& = -\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] (verified)

Time = 0.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.12 \[ \int \frac {(1-x)^{3/2}}{(1+x)^{5/2}} \, dx=\frac {4 \sqrt {1-x} (1+2 x)}{3 (1+x)^{3/2}}-2 \arctan \left (\frac {\sqrt {1-x^2}}{-1+x}\right ) \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(72\) vs. \(2(31)=62\).

Time = 0.18 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.78

method	result	size
risch	\(-\frac {4 \left (2 x^{2}-x -1\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{3 \left (1+x \right )^{\frac {3}{2}} \sqrt {-\left (-1+x \right ) \left (1+x \right )}\, \sqrt {1-x}}+\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{\sqrt {1+x}\, \sqrt {1-x}}\)	\(73\)

[In]

int((1-x)^(3/2)/(1+x)^(5/2),x,method=_RETURNVERBOSE)

[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)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (31) = 62\).

Time = 0.23 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.73 \[ \int \frac {(1-x)^{3/2}}{(1+x)^{5/2}} \, dx=\frac {2 \, {\left (2 \, x^{2} + 2 \, {\left (2 \, x + 1\right )} \sqrt {x + 1} \sqrt {-x + 1} - 3 \, {\left (x^{2} + 2 \, x + 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) + 4 \, x + 2\right )}}{3 \, {\left (x^{2} + 2 \, x + 1\right )}} \]

[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)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.91 (sec) , antiderivative size = 128, normalized size of antiderivative = 3.12 \[ \int \frac {(1-x)^{3/2}}{(1+x)^{5/2}} \, dx=\begin {cases} \frac {8 \sqrt {-1 + \frac {2}{x + 1}}}{3} - \frac {4 \sqrt {-1 + \frac {2}{x + 1}}}{3 \left (x + 1\right )} + i \log {\left (\frac {1}{x + 1} \right )} + i \log {\left (x + 1 \right )} + 2 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\\frac {8 i \sqrt {1 - \frac {2}{x + 1}}}{3} - \frac {4 i \sqrt {1 - \frac {2}{x + 1}}}{3 \left (x + 1\right )} + i \log {\left (\frac {1}{x + 1} \right )} - 2 i \log {\left (\sqrt {1 - \frac {2}{x + 1}} + 1 \right )} & \text {otherwise} \end {cases} \]

[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), 1/Abs(x + 1) > 1/2), (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))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (31) = 62\).

Time = 0.29 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.61 \[ \int \frac {(1-x)^{3/2}}{(1+x)^{5/2}} \, dx=-\frac {{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{3 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}} - \frac {2 \, \sqrt {-x^{2} + 1}}{3 \, {\left (x^{2} + 2 \, x + 1\right )}} + \frac {7 \, \sqrt {-x^{2} + 1}}{3 \, {\left (x + 1\right )}} + \arcsin \left (x\right ) \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (31) = 62\).

Time = 0.32 (sec) , antiderivative size = 102, normalized size of antiderivative = 2.49 \[ \int \frac {(1-x)^{3/2}}{(1+x)^{5/2}} \, dx=\frac {{\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}}{12 \, {\left (x + 1\right )}^{\frac {3}{2}}} - \frac {5 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}}{4 \, \sqrt {x + 1}} + \frac {{\left (x + 1\right )}^{\frac {3}{2}} {\left (\frac {15 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{2}}{x + 1} - 1\right )}}{12 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}} + 2 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]

[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))

Mupad [F(-1)]

Timed out. \[ \int \frac {(1-x)^{3/2}}{(1+x)^{5/2}} \, dx=\int \frac {{\left (1-x\right )}^{3/2}}{{\left (x+1\right )}^{5/2}} \,d x \]

[In]

int((1 - x)^(3/2)/(x + 1)^(5/2),x)

[Out]

int((1 - x)^(3/2)/(x + 1)^(5/2), x)