\(\int \frac {\sqrt {1+x}}{(1-x)^{9/2}} \, dx\) [1072]

Optimal result

Integrand size = 17, antiderivative size = 61 \[ \int \frac {\sqrt {1+x}}{(1-x)^{9/2}} \, dx=\frac {(1+x)^{3/2}}{7 (1-x)^{7/2}}+\frac {2 (1+x)^{3/2}}{35 (1-x)^{5/2}}+\frac {2 (1+x)^{3/2}}{105 (1-x)^{3/2}} \]

[Out]

1/7*(1+x)^(3/2)/(1-x)^(7/2)+2/35*(1+x)^(3/2)/(1-x)^(5/2)+2/105*(1+x)^(3/2)/(1-x)^(3/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {47, 37} \[ \int \frac {\sqrt {1+x}}{(1-x)^{9/2}} \, dx=\frac {2 (x+1)^{3/2}}{105 (1-x)^{3/2}}+\frac {2 (x+1)^{3/2}}{35 (1-x)^{5/2}}+\frac {(x+1)^{3/2}}{7 (1-x)^{7/2}} \]

[In]

Int[Sqrt[1 + x]/(1 - x)^(9/2),x]

[Out]

(1 + x)^(3/2)/(7*(1 - x)^(7/2)) + (2*(1 + x)^(3/2))/(35*(1 - x)^(5/2)) + (2*(1 + x)^(3/2))/(105*(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}}{7 (1-x)^{7/2}}+\frac {2}{7} \int \frac {\sqrt {1+x}}{(1-x)^{7/2}} \, dx \\ & = \frac {(1+x)^{3/2}}{7 (1-x)^{7/2}}+\frac {2 (1+x)^{3/2}}{35 (1-x)^{5/2}}+\frac {2}{35} \int \frac {\sqrt {1+x}}{(1-x)^{5/2}} \, dx \\ & = \frac {(1+x)^{3/2}}{7 (1-x)^{7/2}}+\frac {2 (1+x)^{3/2}}{35 (1-x)^{5/2}}+\frac {2 (1+x)^{3/2}}{105 (1-x)^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.49 \[ \int \frac {\sqrt {1+x}}{(1-x)^{9/2}} \, dx=\frac {(1+x)^{3/2} \left (23-10 x+2 x^2\right )}{105 (1-x)^{7/2}} \]

[In]

Integrate[Sqrt[1 + x]/(1 - x)^(9/2),x]

[Out]

((1 + x)^(3/2)*(23 - 10*x + 2*x^2))/(105*(1 - x)^(7/2))

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.41

method	result	size
gosper	\(\frac {\left (1+x \right )^{\frac {3}{2}} \left (2 x^{2}-10 x +23\right )}{105 \left (1-x \right )^{\frac {7}{2}}}\)	\(25\)
default	\(\frac {2 \sqrt {1+x}}{7 \left (1-x \right )^{\frac {7}{2}}}-\frac {\sqrt {1+x}}{35 \left (1-x \right )^{\frac {5}{2}}}-\frac {2 \sqrt {1+x}}{105 \left (1-x \right )^{\frac {3}{2}}}-\frac {2 \sqrt {1+x}}{105 \sqrt {1-x}}\)	\(58\)
risch	\(-\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (2 x^{4}-6 x^{3}+5 x^{2}+36 x +23\right )}{105 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{3} \sqrt {-\left (-1+x \right ) \left (1+x \right )}}\)	\(61\)

[In]

int((1+x)^(1/2)/(1-x)^(9/2),x,method=_RETURNVERBOSE)

[Out]

1/105/(1-x)^(7/2)*(1+x)^(3/2)*(2*x^2-10*x+23)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.15 \[ \int \frac {\sqrt {1+x}}{(1-x)^{9/2}} \, dx=\frac {23 \, x^{4} - 92 \, x^{3} + 138 \, x^{2} + {\left (2 \, x^{3} - 8 \, x^{2} + 13 \, x + 23\right )} \sqrt {x + 1} \sqrt {-x + 1} - 92 \, x + 23}{105 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} \]

[In]

integrate((1+x)^(1/2)/(1-x)^(9/2),x, algorithm="fricas")

[Out]

1/105*(23*x^4 - 92*x^3 + 138*x^2 + (2*x^3 - 8*x^2 + 13*x + 23)*sqrt(x + 1)*sqrt(-x + 1) - 92*x + 23)/(x^4 - 4*
x^3 + 6*x^2 - 4*x + 1)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 20.65 (sec) , antiderivative size = 566, normalized size of antiderivative = 9.28 \[ \int \frac {\sqrt {1+x}}{(1-x)^{9/2}} \, dx=\begin {cases} \frac {2 i \left (x + 1\right )^{\frac {9}{2}}}{105 \sqrt {x - 1} \left (x + 1\right )^{4} - 840 \sqrt {x - 1} \left (x + 1\right )^{3} + 2520 \sqrt {x - 1} \left (x + 1\right )^{2} - 3360 \sqrt {x - 1} \left (x + 1\right ) + 1680 \sqrt {x - 1}} - \frac {18 i \left (x + 1\right )^{\frac {7}{2}}}{105 \sqrt {x - 1} \left (x + 1\right )^{4} - 840 \sqrt {x - 1} \left (x + 1\right )^{3} + 2520 \sqrt {x - 1} \left (x + 1\right )^{2} - 3360 \sqrt {x - 1} \left (x + 1\right ) + 1680 \sqrt {x - 1}} + \frac {63 i \left (x + 1\right )^{\frac {5}{2}}}{105 \sqrt {x - 1} \left (x + 1\right )^{4} - 840 \sqrt {x - 1} \left (x + 1\right )^{3} + 2520 \sqrt {x - 1} \left (x + 1\right )^{2} - 3360 \sqrt {x - 1} \left (x + 1\right ) + 1680 \sqrt {x - 1}} - \frac {70 i \left (x + 1\right )^{\frac {3}{2}}}{105 \sqrt {x - 1} \left (x + 1\right )^{4} - 840 \sqrt {x - 1} \left (x + 1\right )^{3} + 2520 \sqrt {x - 1} \left (x + 1\right )^{2} - 3360 \sqrt {x - 1} \left (x + 1\right ) + 1680 \sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\- \frac {2 \left (x + 1\right )^{\frac {9}{2}}}{105 \sqrt {1 - x} \left (x + 1\right )^{4} - 840 \sqrt {1 - x} \left (x + 1\right )^{3} + 2520 \sqrt {1 - x} \left (x + 1\right )^{2} - 3360 \sqrt {1 - x} \left (x + 1\right ) + 1680 \sqrt {1 - x}} + \frac {18 \left (x + 1\right )^{\frac {7}{2}}}{105 \sqrt {1 - x} \left (x + 1\right )^{4} - 840 \sqrt {1 - x} \left (x + 1\right )^{3} + 2520 \sqrt {1 - x} \left (x + 1\right )^{2} - 3360 \sqrt {1 - x} \left (x + 1\right ) + 1680 \sqrt {1 - x}} - \frac {63 \left (x + 1\right )^{\frac {5}{2}}}{105 \sqrt {1 - x} \left (x + 1\right )^{4} - 840 \sqrt {1 - x} \left (x + 1\right )^{3} + 2520 \sqrt {1 - x} \left (x + 1\right )^{2} - 3360 \sqrt {1 - x} \left (x + 1\right ) + 1680 \sqrt {1 - x}} + \frac {70 \left (x + 1\right )^{\frac {3}{2}}}{105 \sqrt {1 - x} \left (x + 1\right )^{4} - 840 \sqrt {1 - x} \left (x + 1\right )^{3} + 2520 \sqrt {1 - x} \left (x + 1\right )^{2} - 3360 \sqrt {1 - x} \left (x + 1\right ) + 1680 \sqrt {1 - x}} & \text {otherwise} \end {cases} \]

[In]

integrate((1+x)**(1/2)/(1-x)**(9/2),x)

[Out]

Piecewise((2*I*(x + 1)**(9/2)/(105*sqrt(x - 1)*(x + 1)**4 - 840*sqrt(x - 1)*(x + 1)**3 + 2520*sqrt(x - 1)*(x +
1)**2 - 3360*sqrt(x - 1)*(x + 1) + 1680*sqrt(x - 1)) - 18*I*(x + 1)**(7/2)/(105*sqrt(x - 1)*(x + 1)**4 - 840*
sqrt(x - 1)*(x + 1)**3 + 2520*sqrt(x - 1)*(x + 1)**2 - 3360*sqrt(x - 1)*(x + 1) + 1680*sqrt(x - 1)) + 63*I*(x
+ 1)**(5/2)/(105*sqrt(x - 1)*(x + 1)**4 - 840*sqrt(x - 1)*(x + 1)**3 + 2520*sqrt(x - 1)*(x + 1)**2 - 3360*sqrt
(x - 1)*(x + 1) + 1680*sqrt(x - 1)) - 70*I*(x + 1)**(3/2)/(105*sqrt(x - 1)*(x + 1)**4 - 840*sqrt(x - 1)*(x + 1
)**3 + 2520*sqrt(x - 1)*(x + 1)**2 - 3360*sqrt(x - 1)*(x + 1) + 1680*sqrt(x - 1)), Abs(x + 1) > 2), (-2*(x + 1
)**(9/2)/(105*sqrt(1 - x)*(x + 1)**4 - 840*sqrt(1 - x)*(x + 1)**3 + 2520*sqrt(1 - x)*(x + 1)**2 - 3360*sqrt(1
- x)*(x + 1) + 1680*sqrt(1 - x)) + 18*(x + 1)**(7/2)/(105*sqrt(1 - x)*(x + 1)**4 - 840*sqrt(1 - x)*(x + 1)**3
+ 2520*sqrt(1 - x)*(x + 1)**2 - 3360*sqrt(1 - x)*(x + 1) + 1680*sqrt(1 - x)) - 63*(x + 1)**(5/2)/(105*sqrt(1 -
x)*(x + 1)**4 - 840*sqrt(1 - x)*(x + 1)**3 + 2520*sqrt(1 - x)*(x + 1)**2 - 3360*sqrt(1 - x)*(x + 1) + 1680*sq
rt(1 - x)) + 70*(x + 1)**(3/2)/(105*sqrt(1 - x)*(x + 1)**4 - 840*sqrt(1 - x)*(x + 1)**3 + 2520*sqrt(1 - x)*(x
+ 1)**2 - 3360*sqrt(1 - x)*(x + 1) + 1680*sqrt(1 - x)), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (43) = 86\).

Time = 0.20 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.56 \[ \int \frac {\sqrt {1+x}}{(1-x)^{9/2}} \, dx=\frac {2 \, \sqrt {-x^{2} + 1}}{7 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{35 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac {2 \, \sqrt {-x^{2} + 1}}{105 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {2 \, \sqrt {-x^{2} + 1}}{105 \, {\left (x - 1\right )}} \]

[In]

integrate((1+x)^(1/2)/(1-x)^(9/2),x, algorithm="maxima")

[Out]

2/7*sqrt(-x^2 + 1)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1) + 1/35*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x - 1) - 2/105*sqrt(
-x^2 + 1)/(x^2 - 2*x + 1) + 2/105*sqrt(-x^2 + 1)/(x - 1)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.48 \[ \int \frac {\sqrt {1+x}}{(1-x)^{9/2}} \, dx=\frac {{\left (2 \, {\left (x + 1\right )} {\left (x - 6\right )} + 35\right )} {\left (x + 1\right )}^{\frac {3}{2}} \sqrt {-x + 1}}{105 \, {\left (x - 1\right )}^{4}} \]

[In]

integrate((1+x)^(1/2)/(1-x)^(9/2),x, algorithm="giac")

[Out]

1/105*(2*(x + 1)*(x - 6) + 35)*(x + 1)^(3/2)*sqrt(-x + 1)/(x - 1)^4

Mupad [B] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt {1+x}}{(1-x)^{9/2}} \, dx=\frac {\sqrt {1-x}\,\left (\frac {13\,x\,\sqrt {x+1}}{105}+\frac {23\,\sqrt {x+1}}{105}-\frac {8\,x^2\,\sqrt {x+1}}{105}+\frac {2\,x^3\,\sqrt {x+1}}{105}\right )}{x^4-4\,x^3+6\,x^2-4\,x+1} \]

[In]

int((x + 1)^(1/2)/(1 - x)^(9/2),x)

[Out]

((1 - x)^(1/2)*((13*x*(x + 1)^(1/2))/105 + (23*(x + 1)^(1/2))/105 - (8*x^2*(x + 1)^(1/2))/105 + (2*x^3*(x + 1)
^(1/2))/105))/(6*x^2 - 4*x - 4*x^3 + x^4 + 1)