\(\int \frac {1}{(1-x)^{3/2} \sqrt {1+x}} \, dx\) [1111]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {37} \[ \int \frac {1}{(1-x)^{3/2} \sqrt {1+x}} \, dx=\frac {\sqrt {x+1}}{\sqrt {1-x}} \]

[In]

Int[1/((1 - x)^(3/2)*Sqrt[1 + x]),x]

[Out]

Sqrt[1 + x]/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]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+x}}{\sqrt {1-x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(1-x)^{3/2} \sqrt {1+x}} \, dx=\frac {\sqrt {1+x}}{\sqrt {1-x}} \]

[In]

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

[Out]

Sqrt[1 + x]/Sqrt[1 - x]

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82

method	result	size
gosper	\(\frac {\sqrt {1+x}}{\sqrt {1-x}}\)	\(14\)
default	\(\frac {\sqrt {1+x}}{\sqrt {1-x}}\)	\(14\)
risch	\(\frac {\sqrt {1+x}\, \sqrt {\left (1+x \right ) \left (1-x \right )}}{\sqrt {-\left (-1+x \right ) \left (1+x \right )}\, \sqrt {1-x}}\)	\(35\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.35 \[ \int \frac {1}{(1-x)^{3/2} \sqrt {1+x}} \, dx=\frac {x - \sqrt {x + 1} \sqrt {-x + 1} - 1}{x - 1} \]

[In]

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

[Out]

(x - sqrt(x + 1)*sqrt(-x + 1) - 1)/(x - 1)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.57 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.82 \[ \int \frac {1}{(1-x)^{3/2} \sqrt {1+x}} \, dx=\begin {cases} \frac {1}{\sqrt {-1 + \frac {2}{x + 1}}} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\- \frac {i}{\sqrt {1 - \frac {2}{x + 1}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(1-x)^{3/2} \sqrt {1+x}} \, dx=-\frac {\sqrt {-x^{2} + 1}}{x - 1} \]

[In]

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

[Out]

-sqrt(-x^2 + 1)/(x - 1)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {1}{(1-x)^{3/2} \sqrt {1+x}} \, dx=-\frac {\sqrt {x + 1} \sqrt {-x + 1}}{x - 1} \]

[In]

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

[Out]

-sqrt(x + 1)*sqrt(-x + 1)/(x - 1)

Mupad [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(1-x)^{3/2} \sqrt {1+x}} \, dx=\frac {\sqrt {x+1}}{\sqrt {1-x}} \]

[In]

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

[Out]

(x + 1)^(1/2)/(1 - x)^(1/2)