Integrand size = 17, antiderivative size = 61 \[ \int \frac {\sqrt {1+x}}{(1-x)^{9/2}} \, dx=\frac {(1+x)^{3/2}}{7 (1-x)^{7/2}}+\frac {2 (1+x)^{3/2}}{35 (1-x)^{5/2}}+\frac {2 (1+x)^{3/2}}{105 (1-x)^{3/2}} \] Output:
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)
Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.49 \[ \int \frac {\sqrt {1+x}}{(1-x)^{9/2}} \, dx=\frac {(1+x)^{3/2} \left (23-10 x+2 x^2\right )}{105 (1-x)^{7/2}} \] Input:
Integrate[Sqrt[1 + x]/(1 - x)^(9/2),x]
Output:
((1 + x)^(3/2)*(23 - 10*x + 2*x^2))/(105*(1 - x)^(7/2))
Time = 0.15 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.08, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {55, 55, 48}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sqrt {x+1}}{(1-x)^{9/2}} \, dx\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {2}{7} \int \frac {\sqrt {x+1}}{(1-x)^{7/2}}dx+\frac {(x+1)^{3/2}}{7 (1-x)^{7/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {2}{7} \left (\frac {1}{5} \int \frac {\sqrt {x+1}}{(1-x)^{5/2}}dx+\frac {(x+1)^{3/2}}{5 (1-x)^{5/2}}\right )+\frac {(x+1)^{3/2}}{7 (1-x)^{7/2}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {(x+1)^{3/2}}{7 (1-x)^{7/2}}+\frac {2}{7} \left (\frac {(x+1)^{3/2}}{15 (1-x)^{3/2}}+\frac {(x+1)^{3/2}}{5 (1-x)^{5/2}}\right )\) |
Input:
Int[Sqrt[1 + x]/(1 - x)^(9/2),x]
Output:
(1 + x)^(3/2)/(7*(1 - x)^(7/2)) + (2*((1 + x)^(3/2)/(5*(1 - x)^(5/2)) + (1 + x)^(3/2)/(15*(1 - x)^(3/2))))/7
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] && EqQ[m + n + 2, 0] && NeQ[m, -1]
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] - Simp[d*(S implify[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] && 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] || !SumSimp lerQ[n, 1])
Time = 0.09 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.41
method | result | size |
gosper | \(\frac {\left (1+x \right )^{\frac {3}{2}} \left (2 x^{2}-10 x +23\right )}{105 \left (1-x \right )^{\frac {7}{2}}}\) | \(25\) |
orering | \(-\frac {\left (1+x \right )^{\frac {3}{2}} \left (-1+x \right ) \left (2 x^{2}-10 x +23\right )}{105 \left (1-x \right )^{\frac {9}{2}}}\) | \(28\) |
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 | \(-\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (2 x^{4}-6 x^{3}+5 x^{2}+36 x +23\right )}{105 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{3} \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\) | \(61\) |
Input:
int((1+x)^(1/2)/(1-x)^(9/2),x,method=_RETURNVERBOSE)
Output:
1/105/(1-x)^(7/2)*(1+x)^(3/2)*(2*x^2-10*x+23)
Time = 0.07 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.15 \[ \int \frac {\sqrt {1+x}}{(1-x)^{9/2}} \, dx=\frac {23 \, x^{4} - 92 \, x^{3} + 138 \, x^{2} + {\left (2 \, x^{3} - 8 \, x^{2} + 13 \, x + 23\right )} \sqrt {x + 1} \sqrt {-x + 1} - 92 \, x + 23}{105 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} \] Input:
integrate((1+x)^(1/2)/(1-x)^(9/2),x, algorithm="fricas")
Output:
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)
Result contains complex when optimal does not.
Time = 8.96 (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} \] Input:
integrate((1+x)**(1/2)/(1-x)**(9/2),x)
Output:
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) + 16 80*sqrt(x - 1)) - 18*I*(x + 1)**(7/2)/(105*sqrt(x - 1)*(x + 1)**4 - 840*sq rt(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*s qrt(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*sqrt(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))
Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (43) = 86\).
Time = 0.04 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.56 \[ \int \frac {\sqrt {1+x}}{(1-x)^{9/2}} \, dx=\frac {2 \, \sqrt {-x^{2} + 1}}{7 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{35 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac {2 \, \sqrt {-x^{2} + 1}}{105 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {2 \, \sqrt {-x^{2} + 1}}{105 \, {\left (x - 1\right )}} \] Input:
integrate((1+x)^(1/2)/(1-x)^(9/2),x, algorithm="maxima")
Output:
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)
Time = 0.13 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.48 \[ \int \frac {\sqrt {1+x}}{(1-x)^{9/2}} \, dx=\frac {{\left (2 \, {\left (x + 1\right )} {\left (x - 6\right )} + 35\right )} {\left (x + 1\right )}^{\frac {3}{2}} \sqrt {-x + 1}}{105 \, {\left (x - 1\right )}^{4}} \] Input:
integrate((1+x)^(1/2)/(1-x)^(9/2),x, algorithm="giac")
Output:
1/105*(2*(x + 1)*(x - 6) + 35)*(x + 1)^(3/2)*sqrt(-x + 1)/(x - 1)^4
Time = 0.17 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt {1+x}}{(1-x)^{9/2}} \, dx=\frac {\sqrt {1-x}\,\left (\frac {13\,x\,\sqrt {x+1}}{105}+\frac {23\,\sqrt {x+1}}{105}-\frac {8\,x^2\,\sqrt {x+1}}{105}+\frac {2\,x^3\,\sqrt {x+1}}{105}\right )}{x^4-4\,x^3+6\,x^2-4\,x+1} \] Input:
int((x + 1)^(1/2)/(1 - x)^(9/2),x)
Output:
((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)
Time = 0.15 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {1+x}}{(1-x)^{9/2}} \, dx=\frac {\sqrt {x +1}\, \left (-2 x^{3}+8 x^{2}-13 x -23\right )}{105 \sqrt {1-x}\, \left (x^{3}-3 x^{2}+3 x -1\right )} \] Input:
int((1+x)^(1/2)/(1-x)^(9/2),x)
Output:
(sqrt(x + 1)*( - 2*x**3 + 8*x**2 - 13*x - 23))/(105*sqrt( - x + 1)*(x**3 - 3*x**2 + 3*x - 1))