\(\int \frac {1}{(1-x)^{3/2} (1+x)^{5/2}} \, dx\) [1132]

Optimal result

Integrand size = 17, antiderivative size = 58 \[ \int \frac {1}{(1-x)^{3/2} (1+x)^{5/2}} \, dx=\frac {1}{\sqrt {1-x} (1+x)^{3/2}}-\frac {2 \sqrt {1-x}}{3 (1+x)^{3/2}}-\frac {2 \sqrt {1-x}}{3 \sqrt {1+x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {47, 37} \[ \int \frac {1}{(1-x)^{3/2} (1+x)^{5/2}} \, dx=-\frac {2 \sqrt {1-x}}{3 \sqrt {x+1}}-\frac {2 \sqrt {1-x}}{3 (x+1)^{3/2}}+\frac {1}{(x+1)^{3/2} \sqrt {1-x}} \]

[In]

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

[Out]

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

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

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.52 \[ \int \frac {1}{(1-x)^{3/2} (1+x)^{5/2}} \, dx=\frac {-1+2 x+2 x^2}{3 \sqrt {1-x} (1+x)^{3/2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.43

method	result	size
gosper	\(\frac {2 x^{2}+2 x -1}{3 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}\)	\(25\)
default	\(\frac {1}{\sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}-\frac {2 \sqrt {1-x}}{3 \left (1+x \right )^{\frac {3}{2}}}-\frac {2 \sqrt {1-x}}{3 \sqrt {1+x}}\)	\(43\)
risch	\(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (2 x^{2}+2 x -1\right )}{3 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}} \sqrt {-\left (-1+x \right ) \left (1+x \right )}}\)	\(46\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.84 \[ \int \frac {1}{(1-x)^{3/2} (1+x)^{5/2}} \, dx=-\frac {x^{3} + x^{2} + {\left (2 \, x^{2} + 2 \, x - 1\right )} \sqrt {x + 1} \sqrt {-x + 1} - x - 1}{3 \, {\left (x^{3} + x^{2} - x - 1\right )}} \]

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 5.13 (sec) , antiderivative size = 167, normalized size of antiderivative = 2.88 \[ \int \frac {1}{(1-x)^{3/2} (1+x)^{5/2}} \, dx=\begin {cases} - \frac {2 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2}}{- 6 x + 3 \left (x + 1\right )^{2} - 6} + \frac {2 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )}{- 6 x + 3 \left (x + 1\right )^{2} - 6} + \frac {\sqrt {-1 + \frac {2}{x + 1}}}{- 6 x + 3 \left (x + 1\right )^{2} - 6} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\- \frac {2 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2}}{- 6 x + 3 \left (x + 1\right )^{2} - 6} + \frac {2 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )}{- 6 x + 3 \left (x + 1\right )^{2} - 6} + \frac {i \sqrt {1 - \frac {2}{x + 1}}}{- 6 x + 3 \left (x + 1\right )^{2} - 6} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-2*sqrt(-1 + 2/(x + 1))*(x + 1)**2/(-6*x + 3*(x + 1)**2 - 6) + 2*sqrt(-1 + 2/(x + 1))*(x + 1)/(-6*x
+ 3*(x + 1)**2 - 6) + sqrt(-1 + 2/(x + 1))/(-6*x + 3*(x + 1)**2 - 6), 1/Abs(x + 1) > 1/2), (-2*I*sqrt(1 - 2/(
x + 1))*(x + 1)**2/(-6*x + 3*(x + 1)**2 - 6) + 2*I*sqrt(1 - 2/(x + 1))*(x + 1)/(-6*x + 3*(x + 1)**2 - 6) + I*s
qrt(1 - 2/(x + 1))/(-6*x + 3*(x + 1)**2 - 6), True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.66 \[ \int \frac {1}{(1-x)^{3/2} (1+x)^{5/2}} \, dx=\frac {2 \, x}{3 \, \sqrt {-x^{2} + 1}} - \frac {1}{3 \, {\left (\sqrt {-x^{2} + 1} x + \sqrt {-x^{2} + 1}\right )}} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (42) = 84\).

Time = 0.30 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.86 \[ \int \frac {1}{(1-x)^{3/2} (1+x)^{5/2}} \, dx=\frac {{\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}}{96 \, {\left (x + 1\right )}^{\frac {3}{2}}} + \frac {7 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}}{32 \, \sqrt {x + 1}} - \frac {\sqrt {x + 1} \sqrt {-x + 1}}{4 \, {\left (x - 1\right )}} - \frac {{\left (x + 1\right )}^{\frac {3}{2}} {\left (\frac {21 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{2}}{x + 1} + 1\right )}}{96 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}} \]

[In]

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

[Out]

1/96*(sqrt(2) - sqrt(-x + 1))^3/(x + 1)^(3/2) + 7/32*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - 1/4*sqrt(x + 1)*sq
rt(-x + 1)/(x - 1) - 1/96*(x + 1)^(3/2)*(21*(sqrt(2) - sqrt(-x + 1))^2/(x + 1) + 1)/(sqrt(2) - sqrt(-x + 1))^3

Mupad [B] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.83 \[ \int \frac {1}{(1-x)^{3/2} (1+x)^{5/2}} \, dx=-\frac {2\,x\,\sqrt {1-x}-\sqrt {1-x}+2\,x^2\,\sqrt {1-x}}{\left (3\,x^2-3\right )\,\sqrt {x+1}} \]

[In]

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

[Out]

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