\(\int \frac {\sqrt {1+x}}{(1-x)^{7/2}} \, dx\) [1071]

Optimal result

Integrand size = 17, antiderivative size = 41 \[ \int \frac {\sqrt {1+x}}{(1-x)^{7/2}} \, dx=\frac {(1+x)^{3/2}}{5 (1-x)^{5/2}}+\frac {(1+x)^{3/2}}{15 (1-x)^{3/2}} \]

[Out]

1/5*(1+x)^(3/2)/(1-x)^(5/2)+1/15*(1+x)^(3/2)/(1-x)^(3/2)

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} \[ \int \frac {\sqrt {1+x}}{(1-x)^{7/2}} \, dx=\frac {(x+1)^{3/2}}{15 (1-x)^{3/2}}+\frac {(x+1)^{3/2}}{5 (1-x)^{5/2}} \]

[In]

Int[Sqrt[1 + x]/(1 - x)^(7/2),x]

[Out]

(1 + x)^(3/2)/(5*(1 - x)^(5/2)) + (1 + x)^(3/2)/(15*(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]

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+x)^{3/2}}{5 (1-x)^{5/2}}+\frac {1}{5} \int \frac {\sqrt {1+x}}{(1-x)^{5/2}} \, dx \\ & = \frac {(1+x)^{3/2}}{5 (1-x)^{5/2}}+\frac {(1+x)^{3/2}}{15 (1-x)^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {1+x}}{(1-x)^{7/2}} \, dx=\frac {(4-x) (1+x)^{3/2}}{15 (1-x)^{5/2}} \]

[In]

Integrate[Sqrt[1 + x]/(1 - x)^(7/2),x]

[Out]

((4 - x)*(1 + x)^(3/2))/(15*(1 - x)^(5/2))

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.44

method	result	size
gosper	\(-\frac {\left (x -4\right ) \left (1+x \right )^{\frac {3}{2}}}{15 \left (1-x \right )^{\frac {5}{2}}}\)	\(18\)
default	\(\frac {2 \sqrt {1+x}}{5 \left (1-x \right )^{\frac {5}{2}}}-\frac {\sqrt {1+x}}{15 \left (1-x \right )^{\frac {3}{2}}}-\frac {\sqrt {1+x}}{15 \sqrt {1-x}}\)	\(44\)
risch	\(-\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (x^{3}-2 x^{2}-7 x -4\right )}{15 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{2} \sqrt {-\left (-1+x \right ) \left (1+x \right )}}\)	\(54\)

[In]

int((1+x)^(1/2)/(1-x)^(7/2),x,method=_RETURNVERBOSE)

[Out]

-1/15*(x-4)/(1-x)^(5/2)*(1+x)^(3/2)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.29 \[ \int \frac {\sqrt {1+x}}{(1-x)^{7/2}} \, dx=\frac {4 \, x^{3} - 12 \, x^{2} + {\left (x^{2} - 3 \, x - 4\right )} \sqrt {x + 1} \sqrt {-x + 1} + 12 \, x - 4}{15 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} \]

[In]

integrate((1+x)^(1/2)/(1-x)^(7/2),x, algorithm="fricas")

[Out]

1/15*(4*x^3 - 12*x^2 + (x^2 - 3*x - 4)*sqrt(x + 1)*sqrt(-x + 1) + 12*x - 4)/(x^3 - 3*x^2 + 3*x - 1)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 6.41 (sec) , antiderivative size = 172, normalized size of antiderivative = 4.20 \[ \int \frac {\sqrt {1+x}}{(1-x)^{7/2}} \, dx=\begin {cases} \frac {i \left (x + 1\right )^{\frac {5}{2}}}{15 \sqrt {x - 1} \left (x + 1\right )^{2} - 60 \sqrt {x - 1} \left (x + 1\right ) + 60 \sqrt {x - 1}} - \frac {5 i \left (x + 1\right )^{\frac {3}{2}}}{15 \sqrt {x - 1} \left (x + 1\right )^{2} - 60 \sqrt {x - 1} \left (x + 1\right ) + 60 \sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\- \frac {\left (x + 1\right )^{\frac {5}{2}}}{15 \sqrt {1 - x} \left (x + 1\right )^{2} - 60 \sqrt {1 - x} \left (x + 1\right ) + 60 \sqrt {1 - x}} + \frac {5 \left (x + 1\right )^{\frac {3}{2}}}{15 \sqrt {1 - x} \left (x + 1\right )^{2} - 60 \sqrt {1 - x} \left (x + 1\right ) + 60 \sqrt {1 - x}} & \text {otherwise} \end {cases} \]

[In]

integrate((1+x)**(1/2)/(1-x)**(7/2),x)

[Out]

Piecewise((I*(x + 1)**(5/2)/(15*sqrt(x - 1)*(x + 1)**2 - 60*sqrt(x - 1)*(x + 1) + 60*sqrt(x - 1)) - 5*I*(x + 1
)**(3/2)/(15*sqrt(x - 1)*(x + 1)**2 - 60*sqrt(x - 1)*(x + 1) + 60*sqrt(x - 1)), Abs(x + 1) > 2), (-(x + 1)**(5
/2)/(15*sqrt(1 - x)*(x + 1)**2 - 60*sqrt(1 - x)*(x + 1) + 60*sqrt(1 - x)) + 5*(x + 1)**(3/2)/(15*sqrt(1 - x)*(
x + 1)**2 - 60*sqrt(1 - x)*(x + 1) + 60*sqrt(1 - x)), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (29) = 58\).

Time = 0.21 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.56 \[ \int \frac {\sqrt {1+x}}{(1-x)^{7/2}} \, dx=-\frac {2 \, \sqrt {-x^{2} + 1}}{5 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac {\sqrt {-x^{2} + 1}}{15 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{15 \, {\left (x - 1\right )}} \]

[In]

integrate((1+x)^(1/2)/(1-x)^(7/2),x, algorithm="maxima")

[Out]

-2/5*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x - 1) - 1/15*sqrt(-x^2 + 1)/(x^2 - 2*x + 1) + 1/15*sqrt(-x^2 + 1)/(x - 1
)

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.54 \[ \int \frac {\sqrt {1+x}}{(1-x)^{7/2}} \, dx=\frac {{\left (x + 1\right )}^{\frac {3}{2}} {\left (x - 4\right )} \sqrt {-x + 1}}{15 \, {\left (x - 1\right )}^{3}} \]

[In]

integrate((1+x)^(1/2)/(1-x)^(7/2),x, algorithm="giac")

[Out]

1/15*(x + 1)^(3/2)*(x - 4)*sqrt(-x + 1)/(x - 1)^3

Mupad [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.22 \[ \int \frac {\sqrt {1+x}}{(1-x)^{7/2}} \, dx=-\frac {\sqrt {1-x}\,\left (\frac {x\,\sqrt {x+1}}{5}+\frac {4\,\sqrt {x+1}}{15}-\frac {x^2\,\sqrt {x+1}}{15}\right )}{x^3-3\,x^2+3\,x-1} \]

[In]

int((x + 1)^(1/2)/(1 - x)^(7/2),x)

[Out]

-((1 - x)^(1/2)*((x*(x + 1)^(1/2))/5 + (4*(x + 1)^(1/2))/15 - (x^2*(x + 1)^(1/2))/15))/(3*x - 3*x^2 + x^3 - 1)