\(\int \frac {1}{(1-x)^{5/2} \sqrt {1+x}} \, dx\) [1112]

Optimal result

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

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

[In]

Int[1/((1 - x)^(5/2)*Sqrt[1 + x]),x]

[Out]

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

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.33 (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	\(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (x^{2}-x -2\right )}{3 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right ) \sqrt {-\left (-1+x \right ) \left (1+x \right )}}\)	\(49\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(1-x)^{5/2} \sqrt {1+x}} \, dx=\frac {2 \, x^{2} - \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} - 4 \, x + 2}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} \]

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.16 (sec) , antiderivative size = 128, normalized size of antiderivative = 3.12 \[ \int \frac {1}{(1-x)^{5/2} \sqrt {1+x}} \, dx=\begin {cases} \frac {x + 1}{3 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right ) - 6 \sqrt {-1 + \frac {2}{x + 1}}} - \frac {3}{3 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right ) - 6 \sqrt {-1 + \frac {2}{x + 1}}} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\- \frac {i \left (x + 1\right )}{3 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right ) - 6 \sqrt {1 - \frac {2}{x + 1}}} + \frac {3 i}{3 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right ) - 6 \sqrt {1 - \frac {2}{x + 1}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.54 \[ \int \frac {1}{(1-x)^{5/2} \sqrt {1+x}} \, dx=-\frac {\sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1}}{3 \, {\left (x - 1\right )}^{2}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.05 \[ \int \frac {1}{(1-x)^{5/2} \sqrt {1+x}} \, dx=\frac {x\,\sqrt {1-x}+2\,\sqrt {1-x}-x^2\,\sqrt {1-x}}{3\,{\left (x-1\right )}^2\,\sqrt {x+1}} \]

[In]

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

[Out]

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