\(\int \frac {1}{\sqrt {1-x} (1+x)^{5/2}} \, dx\) [1131]

Optimal result

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

[Out]

-1/3*(1-x)^(1/2)/(1+x)^(3/2)-1/3*(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 = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {47, 37} \[ \int \frac {1}{\sqrt {1-x} (1+x)^{5/2}} \, dx=-\frac {\sqrt {1-x}}{3 \sqrt {x+1}}-\frac {\sqrt {1-x}}{3 (x+1)^{3/2}} \]

[In]

Int[1/(Sqrt[1 - x]*(1 + x)^(5/2)),x]

[Out]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.61 \[ \int \frac {1}{\sqrt {1-x} (1+x)^{5/2}} \, dx=\frac {(-2-x) \sqrt {1-x}}{3 (1+x)^{3/2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.44

method	result	size
gosper	\(-\frac {\left (2+x \right ) \sqrt {1-x}}{3 \left (1+x \right )^{\frac {3}{2}}}\)	\(18\)
default	\(-\frac {\sqrt {1-x}}{3 \left (1+x \right )^{\frac {3}{2}}}-\frac {\sqrt {1-x}}{3 \sqrt {1+x}}\)	\(30\)
risch	\(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (x^{2}+x -2\right )}{3 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}} \sqrt {-\left (-1+x \right ) \left (1+x \right )}}\)	\(42\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\sqrt {1-x} (1+x)^{5/2}} \, dx=-\frac {2 \, x^{2} + {\left (x + 2\right )} \sqrt {x + 1} \sqrt {-x + 1} + 4 \, x + 2}{3 \, {\left (x^{2} + 2 \, x + 1\right )}} \]

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.70 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.61 \[ \int \frac {1}{\sqrt {1-x} (1+x)^{5/2}} \, dx=\begin {cases} - \frac {\sqrt {-1 + \frac {2}{x + 1}}}{3} - \frac {\sqrt {-1 + \frac {2}{x + 1}}}{3 \left (x + 1\right )} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\- \frac {i \sqrt {1 - \frac {2}{x + 1}}}{3} - \frac {i \sqrt {1 - \frac {2}{x + 1}}}{3 \left (x + 1\right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\sqrt {1-x} (1+x)^{5/2}} \, dx=-\frac {\sqrt {-x^{2} + 1}}{3 \, {\left (x^{2} + 2 \, x + 1\right )}} - \frac {\sqrt {-x^{2} + 1}}{3 \, {\left (x + 1\right )}} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (29) = 58\).

Time = 0.31 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.17 \[ \int \frac {1}{\sqrt {1-x} (1+x)^{5/2}} \, dx=\frac {{\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}}{48 \, {\left (x + 1\right )}^{\frac {3}{2}}} + \frac {3 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}}{16 \, \sqrt {x + 1}} - \frac {{\left (x + 1\right )}^{\frac {3}{2}} {\left (\frac {9 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{2}}{x + 1} + 1\right )}}{48 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}} \]

[In]

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

[Out]

1/48*(sqrt(2) - sqrt(-x + 1))^3/(x + 1)^(3/2) + 3/16*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - 1/48*(x + 1)^(3/2)
*(9*(sqrt(2) - sqrt(-x + 1))^2/(x + 1) + 1)/(sqrt(2) - sqrt(-x + 1))^3

Mupad [B] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\sqrt {1-x} (1+x)^{5/2}} \, dx=-\frac {x\,\sqrt {1-x}+2\,\sqrt {1-x}}{\left (3\,x+3\right )\,\sqrt {x+1}} \]

[In]

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

[Out]

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