\(\int \frac {\sqrt {1+x}}{(-1+x)^{5/2}} \, dx\) [832]

Optimal result

Integrand size = 15, antiderivative size = 18 \[ \int \frac {\sqrt {1+x}}{(-1+x)^{5/2}} \, dx=-\frac {(1+x)^{3/2}}{3 (-1+x)^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {37} \[ \int \frac {\sqrt {1+x}}{(-1+x)^{5/2}} \, dx=-\frac {(x+1)^{3/2}}{3 (x-1)^{3/2}} \]

[In]

Int[Sqrt[1 + x]/(-1 + x)^(5/2),x]

[Out]

-1/3*(1 + x)^(3/2)/(-1 + x)^(3/2)

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 {(1+x)^{3/2}}{3 (-1+x)^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+x}}{(-1+x)^{5/2}} \, dx=-\frac {(1+x)^{3/2}}{3 (-1+x)^{3/2}} \]

[In]

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

[Out]

-1/3*(1 + x)^(3/2)/(-1 + x)^(3/2)

Maple [A] (verified)

Time = 1.75 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.72

method	result	size
gosper	\(-\frac {\left (1+x \right )^{\frac {3}{2}}}{3 \left (-1+x \right )^{\frac {3}{2}}}\)	\(13\)
risch	\(-\frac {x^{2}+2 x +1}{3 \left (-1+x \right )^{\frac {3}{2}} \sqrt {1+x}}\)	\(21\)
default	\(-\frac {2 \sqrt {1+x}}{3 \left (-1+x \right )^{\frac {3}{2}}}-\frac {\sqrt {1+x}}{3 \sqrt {-1+x}}\)	\(26\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (12) = 24\).

Time = 0.22 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.72 \[ \int \frac {\sqrt {1+x}}{(-1+x)^{5/2}} \, dx=-\frac {{\left (x + 1\right )}^{\frac {3}{2}} \sqrt {x - 1} + x^{2} - 2 \, x + 1}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} \]

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.79 (sec) , antiderivative size = 60, normalized size of antiderivative = 3.33 \[ \int \frac {\sqrt {1+x}}{(-1+x)^{5/2}} \, dx=\begin {cases} - \frac {\left (x + 1\right )^{\frac {3}{2}}}{3 \sqrt {x - 1} \left (x + 1\right ) - 6 \sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\\frac {i \left (x + 1\right )^{\frac {3}{2}}}{3 \sqrt {1 - x} \left (x + 1\right ) - 6 \sqrt {1 - x}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (12) = 24\).

Time = 0.21 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.89 \[ \int \frac {\sqrt {1+x}}{(-1+x)^{5/2}} \, dx=-\frac {2 \, \sqrt {x^{2} - 1}}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} - \frac {\sqrt {x^{2} - 1}}{3 \, {\left (x - 1\right )}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {1+x}}{(-1+x)^{5/2}} \, dx=-\frac {{\left (x + 1\right )}^{\frac {3}{2}}}{3 \, {\left (x - 1\right )}^{\frac {3}{2}}} \]

[In]

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

[Out]

-1/3*(x + 1)^(3/2)/(x - 1)^(3/2)

Mupad [B] (verification not implemented)

Time = 1.11 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.50 \[ \int \frac {\sqrt {1+x}}{(-1+x)^{5/2}} \, dx=-\frac {x\,\sqrt {x+1}+\sqrt {x+1}}{\left (3\,x-3\right )\,\sqrt {x-1}} \]

[In]

int((x + 1)^(1/2)/(x - 1)^(5/2),x)

[Out]

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