\(\int \frac {\sqrt {1+x}}{(1-x)^{11/2}} \, dx\) [1073]
Optimal result
Integrand size = 17, antiderivative size = 81 \[
\int \frac {\sqrt {1+x}}{(1-x)^{11/2}} \, dx=\frac {(1+x)^{3/2}}{9 (1-x)^{9/2}}+\frac {(1+x)^{3/2}}{21 (1-x)^{7/2}}+\frac {2 (1+x)^{3/2}}{105 (1-x)^{5/2}}+\frac {2 (1+x)^{3/2}}{315 (1-x)^{3/2}}
\]
[Out]
1/9*(1+x)^(3/2)/(1-x)^(9/2)+1/21*(1+x)^(3/2)/(1-x)^(7/2)+2/105*(1+x)^(3/2)/(1-x)^(5/2)+2/315*(1+x)^(3/2)/(1-x)
^(3/2)
Rubi [A] (verified)
Time = 0.01 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00,
number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {47, 37}
\[
\int \frac {\sqrt {1+x}}{(1-x)^{11/2}} \, dx=\frac {2 (x+1)^{3/2}}{315 (1-x)^{3/2}}+\frac {2 (x+1)^{3/2}}{105 (1-x)^{5/2}}+\frac {(x+1)^{3/2}}{21 (1-x)^{7/2}}+\frac {(x+1)^{3/2}}{9 (1-x)^{9/2}}
\]
[In]
Int[Sqrt[1 + x]/(1 - x)^(11/2),x]
[Out]
(1 + x)^(3/2)/(9*(1 - x)^(9/2)) + (1 + x)^(3/2)/(21*(1 - x)^(7/2)) + (2*(1 + x)^(3/2))/(105*(1 - x)^(5/2)) + (
2*(1 + x)^(3/2))/(315*(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}}{9 (1-x)^{9/2}}+\frac {1}{3} \int \frac {\sqrt {1+x}}{(1-x)^{9/2}} \, dx \\ & = \frac {(1+x)^{3/2}}{9 (1-x)^{9/2}}+\frac {(1+x)^{3/2}}{21 (1-x)^{7/2}}+\frac {2}{21} \int \frac {\sqrt {1+x}}{(1-x)^{7/2}} \, dx \\ & = \frac {(1+x)^{3/2}}{9 (1-x)^{9/2}}+\frac {(1+x)^{3/2}}{21 (1-x)^{7/2}}+\frac {2 (1+x)^{3/2}}{105 (1-x)^{5/2}}+\frac {2}{105} \int \frac {\sqrt {1+x}}{(1-x)^{5/2}} \, dx \\ & = \frac {(1+x)^{3/2}}{9 (1-x)^{9/2}}+\frac {(1+x)^{3/2}}{21 (1-x)^{7/2}}+\frac {2 (1+x)^{3/2}}{105 (1-x)^{5/2}}+\frac {2 (1+x)^{3/2}}{315 (1-x)^{3/2}} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.43
\[
\int \frac {\sqrt {1+x}}{(1-x)^{11/2}} \, dx=\frac {(1+x)^{3/2} \left (58-33 x+12 x^2-2 x^3\right )}{315 (1-x)^{9/2}}
\]
[In]
Integrate[Sqrt[1 + x]/(1 - x)^(11/2),x]
[Out]
((1 + x)^(3/2)*(58 - 33*x + 12*x^2 - 2*x^3))/(315*(1 - x)^(9/2))
Maple [A] (verified)
Time = 0.17 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.37
| | |
| method | result | size |
| | |
| gosper |
\(-\frac {\left (1+x \right )^{\frac {3}{2}} \left (2 x^{3}-12 x^{2}+33 x -58\right )}{315 \left (1-x \right )^{\frac {9}{2}}}\) |
\(30\) |
| risch |
\(-\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (2 x^{5}-8 x^{4}+11 x^{3}-4 x^{2}-83 x -58\right )}{315 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{4} \sqrt {-\left (-1+x \right ) \left (1+x \right )}}\) |
\(66\) |
| default |
\(\frac {2 \sqrt {1+x}}{9 \left (1-x \right )^{\frac {9}{2}}}-\frac {\sqrt {1+x}}{63 \left (1-x \right )^{\frac {7}{2}}}-\frac {\sqrt {1+x}}{105 \left (1-x \right )^{\frac {5}{2}}}-\frac {2 \sqrt {1+x}}{315 \left (1-x \right )^{\frac {3}{2}}}-\frac {2 \sqrt {1+x}}{315 \sqrt {1-x}}\) |
\(72\) |
| | |
|
|
|
|
[In]
int((1+x)^(1/2)/(1-x)^(11/2),x,method=_RETURNVERBOSE)
[Out]
-1/315*(1+x)^(3/2)/(1-x)^(9/2)*(2*x^3-12*x^2+33*x-58)
Fricas [A] (verification not implemented)
none
Time = 0.23 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.05
\[
\int \frac {\sqrt {1+x}}{(1-x)^{11/2}} \, dx=\frac {58 \, x^{5} - 290 \, x^{4} + 580 \, x^{3} - 580 \, x^{2} + {\left (2 \, x^{4} - 10 \, x^{3} + 21 \, x^{2} - 25 \, x - 58\right )} \sqrt {x + 1} \sqrt {-x + 1} + 290 \, x - 58}{315 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}}
\]
[In]
integrate((1+x)^(1/2)/(1-x)^(11/2),x, algorithm="fricas")
[Out]
1/315*(58*x^5 - 290*x^4 + 580*x^3 - 580*x^2 + (2*x^4 - 10*x^3 + 21*x^2 - 25*x - 58)*sqrt(x + 1)*sqrt(-x + 1) +
290*x - 58)/(x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 1)
Sympy [C] (verification not implemented)
Result contains complex when optimal does not.
Time = 70.62 (sec) , antiderivative size = 1561, normalized size of antiderivative = 19.27
\[
\int \frac {\sqrt {1+x}}{(1-x)^{11/2}} \, dx=\text {Too large to display}
\]
[In]
integrate((1+x)**(1/2)/(1-x)**(11/2),x)
[Out]
Piecewise((2*I*(x + 1)**(15/2)/(315*sqrt(x - 1)*(x + 1)**7 - 4410*sqrt(x - 1)*(x + 1)**6 + 26460*sqrt(x - 1)*(
x + 1)**5 - 88200*sqrt(x - 1)*(x + 1)**4 + 176400*sqrt(x - 1)*(x + 1)**3 - 211680*sqrt(x - 1)*(x + 1)**2 + 141
120*sqrt(x - 1)*(x + 1) - 40320*sqrt(x - 1)) - 30*I*(x + 1)**(13/2)/(315*sqrt(x - 1)*(x + 1)**7 - 4410*sqrt(x
- 1)*(x + 1)**6 + 26460*sqrt(x - 1)*(x + 1)**5 - 88200*sqrt(x - 1)*(x + 1)**4 + 176400*sqrt(x - 1)*(x + 1)**3
- 211680*sqrt(x - 1)*(x + 1)**2 + 141120*sqrt(x - 1)*(x + 1) - 40320*sqrt(x - 1)) + 195*I*(x + 1)**(11/2)/(315
*sqrt(x - 1)*(x + 1)**7 - 4410*sqrt(x - 1)*(x + 1)**6 + 26460*sqrt(x - 1)*(x + 1)**5 - 88200*sqrt(x - 1)*(x +
1)**4 + 176400*sqrt(x - 1)*(x + 1)**3 - 211680*sqrt(x - 1)*(x + 1)**2 + 141120*sqrt(x - 1)*(x + 1) - 40320*sqr
t(x - 1)) - 715*I*(x + 1)**(9/2)/(315*sqrt(x - 1)*(x + 1)**7 - 4410*sqrt(x - 1)*(x + 1)**6 + 26460*sqrt(x - 1)
*(x + 1)**5 - 88200*sqrt(x - 1)*(x + 1)**4 + 176400*sqrt(x - 1)*(x + 1)**3 - 211680*sqrt(x - 1)*(x + 1)**2 + 1
41120*sqrt(x - 1)*(x + 1) - 40320*sqrt(x - 1)) + 1530*I*(x + 1)**(7/2)/(315*sqrt(x - 1)*(x + 1)**7 - 4410*sqrt
(x - 1)*(x + 1)**6 + 26460*sqrt(x - 1)*(x + 1)**5 - 88200*sqrt(x - 1)*(x + 1)**4 + 176400*sqrt(x - 1)*(x + 1)*
*3 - 211680*sqrt(x - 1)*(x + 1)**2 + 141120*sqrt(x - 1)*(x + 1) - 40320*sqrt(x - 1)) - 1764*I*(x + 1)**(5/2)/(
315*sqrt(x - 1)*(x + 1)**7 - 4410*sqrt(x - 1)*(x + 1)**6 + 26460*sqrt(x - 1)*(x + 1)**5 - 88200*sqrt(x - 1)*(x
+ 1)**4 + 176400*sqrt(x - 1)*(x + 1)**3 - 211680*sqrt(x - 1)*(x + 1)**2 + 141120*sqrt(x - 1)*(x + 1) - 40320*
sqrt(x - 1)) + 840*I*(x + 1)**(3/2)/(315*sqrt(x - 1)*(x + 1)**7 - 4410*sqrt(x - 1)*(x + 1)**6 + 26460*sqrt(x -
1)*(x + 1)**5 - 88200*sqrt(x - 1)*(x + 1)**4 + 176400*sqrt(x - 1)*(x + 1)**3 - 211680*sqrt(x - 1)*(x + 1)**2
+ 141120*sqrt(x - 1)*(x + 1) - 40320*sqrt(x - 1)), Abs(x + 1) > 2), (-2*(x + 1)**(15/2)/(315*sqrt(1 - x)*(x +
1)**7 - 4410*sqrt(1 - x)*(x + 1)**6 + 26460*sqrt(1 - x)*(x + 1)**5 - 88200*sqrt(1 - x)*(x + 1)**4 + 176400*sqr
t(1 - x)*(x + 1)**3 - 211680*sqrt(1 - x)*(x + 1)**2 + 141120*sqrt(1 - x)*(x + 1) - 40320*sqrt(1 - x)) + 30*(x
+ 1)**(13/2)/(315*sqrt(1 - x)*(x + 1)**7 - 4410*sqrt(1 - x)*(x + 1)**6 + 26460*sqrt(1 - x)*(x + 1)**5 - 88200*
sqrt(1 - x)*(x + 1)**4 + 176400*sqrt(1 - x)*(x + 1)**3 - 211680*sqrt(1 - x)*(x + 1)**2 + 141120*sqrt(1 - x)*(x
+ 1) - 40320*sqrt(1 - x)) - 195*(x + 1)**(11/2)/(315*sqrt(1 - x)*(x + 1)**7 - 4410*sqrt(1 - x)*(x + 1)**6 + 2
6460*sqrt(1 - x)*(x + 1)**5 - 88200*sqrt(1 - x)*(x + 1)**4 + 176400*sqrt(1 - x)*(x + 1)**3 - 211680*sqrt(1 - x
)*(x + 1)**2 + 141120*sqrt(1 - x)*(x + 1) - 40320*sqrt(1 - x)) + 715*(x + 1)**(9/2)/(315*sqrt(1 - x)*(x + 1)**
7 - 4410*sqrt(1 - x)*(x + 1)**6 + 26460*sqrt(1 - x)*(x + 1)**5 - 88200*sqrt(1 - x)*(x + 1)**4 + 176400*sqrt(1
- x)*(x + 1)**3 - 211680*sqrt(1 - x)*(x + 1)**2 + 141120*sqrt(1 - x)*(x + 1) - 40320*sqrt(1 - x)) - 1530*(x +
1)**(7/2)/(315*sqrt(1 - x)*(x + 1)**7 - 4410*sqrt(1 - x)*(x + 1)**6 + 26460*sqrt(1 - x)*(x + 1)**5 - 88200*sqr
t(1 - x)*(x + 1)**4 + 176400*sqrt(1 - x)*(x + 1)**3 - 211680*sqrt(1 - x)*(x + 1)**2 + 141120*sqrt(1 - x)*(x +
1) - 40320*sqrt(1 - x)) + 1764*(x + 1)**(5/2)/(315*sqrt(1 - x)*(x + 1)**7 - 4410*sqrt(1 - x)*(x + 1)**6 + 2646
0*sqrt(1 - x)*(x + 1)**5 - 88200*sqrt(1 - x)*(x + 1)**4 + 176400*sqrt(1 - x)*(x + 1)**3 - 211680*sqrt(1 - x)*(
x + 1)**2 + 141120*sqrt(1 - x)*(x + 1) - 40320*sqrt(1 - x)) - 840*(x + 1)**(3/2)/(315*sqrt(1 - x)*(x + 1)**7 -
4410*sqrt(1 - x)*(x + 1)**6 + 26460*sqrt(1 - x)*(x + 1)**5 - 88200*sqrt(1 - x)*(x + 1)**4 + 176400*sqrt(1 - x
)*(x + 1)**3 - 211680*sqrt(1 - x)*(x + 1)**2 + 141120*sqrt(1 - x)*(x + 1) - 40320*sqrt(1 - x)), True))
Maxima [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (57) = 114\).
Time = 0.21 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.62
\[
\int \frac {\sqrt {1+x}}{(1-x)^{11/2}} \, dx=-\frac {2 \, \sqrt {-x^{2} + 1}}{9 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} - \frac {\sqrt {-x^{2} + 1}}{63 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{105 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac {2 \, \sqrt {-x^{2} + 1}}{315 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {2 \, \sqrt {-x^{2} + 1}}{315 \, {\left (x - 1\right )}}
\]
[In]
integrate((1+x)^(1/2)/(1-x)^(11/2),x, algorithm="maxima")
[Out]
-2/9*sqrt(-x^2 + 1)/(x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 1) - 1/63*sqrt(-x^2 + 1)/(x^4 - 4*x^3 + 6*x^2 - 4*x
+ 1) + 1/105*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x - 1) - 2/315*sqrt(-x^2 + 1)/(x^2 - 2*x + 1) + 2/315*sqrt(-x^2
+ 1)/(x - 1)
Giac [A] (verification not implemented)
none
Time = 0.31 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.43
\[
\int \frac {\sqrt {1+x}}{(1-x)^{11/2}} \, dx=\frac {{\left ({\left (2 \, {\left (x + 1\right )} {\left (x - 8\right )} + 63\right )} {\left (x + 1\right )} - 105\right )} {\left (x + 1\right )}^{\frac {3}{2}} \sqrt {-x + 1}}{315 \, {\left (x - 1\right )}^{5}}
\]
[In]
integrate((1+x)^(1/2)/(1-x)^(11/2),x, algorithm="giac")
[Out]
1/315*((2*(x + 1)*(x - 8) + 63)*(x + 1) - 105)*(x + 1)^(3/2)*sqrt(-x + 1)/(x - 1)^5
Mupad [B] (verification not implemented)
Time = 0.47 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.99
\[
\int \frac {\sqrt {1+x}}{(1-x)^{11/2}} \, dx=-\frac {\sqrt {1-x}\,\left (\frac {5\,x\,\sqrt {x+1}}{63}+\frac {58\,\sqrt {x+1}}{315}-\frac {x^2\,\sqrt {x+1}}{15}+\frac {2\,x^3\,\sqrt {x+1}}{63}-\frac {2\,x^4\,\sqrt {x+1}}{315}\right )}{x^5-5\,x^4+10\,x^3-10\,x^2+5\,x-1}
\]
[In]
int((x + 1)^(1/2)/(1 - x)^(11/2),x)
[Out]
-((1 - x)^(1/2)*((5*x*(x + 1)^(1/2))/63 + (58*(x + 1)^(1/2))/315 - (x^2*(x + 1)^(1/2))/15 + (2*x^3*(x + 1)^(1/
2))/63 - (2*x^4*(x + 1)^(1/2))/315))/(5*x - 10*x^2 + 10*x^3 - 5*x^4 + x^5 - 1)