Integrand size = 17, antiderivative size = 81 \[ \int \frac {1}{(1-x)^{9/2} \sqrt {1+x}} \, dx=\frac {\sqrt {1+x}}{7 (1-x)^{7/2}}+\frac {3 \sqrt {1+x}}{35 (1-x)^{5/2}}+\frac {2 \sqrt {1+x}}{35 (1-x)^{3/2}}+\frac {2 \sqrt {1+x}}{35 \sqrt {1-x}} \] Output:
1/7*(1+x)^(1/2)/(1-x)^(7/2)+3/35*(1+x)^(1/2)/(1-x)^(5/2)+2/35*(1+x)^(1/2)/ (1-x)^(3/2)+2/35*(1+x)^(1/2)/(1-x)^(1/2)
Time = 0.03 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.43 \[ \int \frac {1}{(1-x)^{9/2} \sqrt {1+x}} \, dx=\frac {\sqrt {1+x} \left (12-13 x+8 x^2-2 x^3\right )}{35 (1-x)^{7/2}} \] Input:
Integrate[1/((1 - x)^(9/2)*Sqrt[1 + x]),x]
Output:
(Sqrt[1 + x]*(12 - 13*x + 8*x^2 - 2*x^3))/(35*(1 - x)^(7/2))
Time = 0.16 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {55, 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.
\(\displaystyle \int \frac {1}{(1-x)^{9/2} \sqrt {x+1}} \, dx\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {3}{7} \int \frac {1}{(1-x)^{7/2} \sqrt {x+1}}dx+\frac {\sqrt {x+1}}{7 (1-x)^{7/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {3}{7} \left (\frac {2}{5} \int \frac {1}{(1-x)^{5/2} \sqrt {x+1}}dx+\frac {\sqrt {x+1}}{5 (1-x)^{5/2}}\right )+\frac {\sqrt {x+1}}{7 (1-x)^{7/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {3}{7} \left (\frac {2}{5} \left (\frac {1}{3} \int \frac {1}{(1-x)^{3/2} \sqrt {x+1}}dx+\frac {\sqrt {x+1}}{3 (1-x)^{3/2}}\right )+\frac {\sqrt {x+1}}{5 (1-x)^{5/2}}\right )+\frac {\sqrt {x+1}}{7 (1-x)^{7/2}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {3}{7} \left (\frac {2}{5} \left (\frac {\sqrt {x+1}}{3 \sqrt {1-x}}+\frac {\sqrt {x+1}}{3 (1-x)^{3/2}}\right )+\frac {\sqrt {x+1}}{5 (1-x)^{5/2}}\right )+\frac {\sqrt {x+1}}{7 (1-x)^{7/2}}\) |
Input:
Int[1/((1 - x)^(9/2)*Sqrt[1 + x]),x]
Output:
Sqrt[1 + x]/(7*(1 - x)^(7/2)) + (3*(Sqrt[1 + x]/(5*(1 - x)^(5/2)) + (2*(Sq rt[1 + x]/(3*(1 - x)^(3/2)) + Sqrt[1 + x]/(3*Sqrt[1 - x])))/5))/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 = 30, normalized size of antiderivative = 0.37
method | result | size |
gosper | \(-\frac {\sqrt {1+x}\, \left (2 x^{3}-8 x^{2}+13 x -12\right )}{35 \left (1-x \right )^{\frac {7}{2}}}\) | \(30\) |
orering | \(\frac {\sqrt {1+x}\, \left (-1+x \right ) \left (2 x^{3}-8 x^{2}+13 x -12\right )}{35 \left (1-x \right )^{\frac {9}{2}}}\) | \(33\) |
default | \(\frac {\sqrt {1+x}}{7 \left (1-x \right )^{\frac {7}{2}}}+\frac {3 \sqrt {1+x}}{35 \left (1-x \right )^{\frac {5}{2}}}+\frac {2 \sqrt {1+x}}{35 \left (1-x \right )^{\frac {3}{2}}}+\frac {2 \sqrt {1+x}}{35 \sqrt {1-x}}\) | \(58\) |
risch | \(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (2 x^{4}-6 x^{3}+5 x^{2}+x -12\right )}{35 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{3} \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\) | \(59\) |
Input:
int(1/(1-x)^(9/2)/(1+x)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/35/(1-x)^(7/2)*(1+x)^(1/2)*(2*x^3-8*x^2+13*x-12)
Time = 0.07 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(1-x)^{9/2} \sqrt {1+x}} \, dx=\frac {12 \, x^{4} - 48 \, x^{3} + 72 \, x^{2} - {\left (2 \, x^{3} - 8 \, x^{2} + 13 \, x - 12\right )} \sqrt {x + 1} \sqrt {-x + 1} - 48 \, x + 12}{35 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} \] Input:
integrate(1/(1-x)^(9/2)/(1+x)^(1/2),x, algorithm="fricas")
Output:
1/35*(12*x^4 - 48*x^3 + 72*x^2 - (2*x^3 - 8*x^2 + 13*x - 12)*sqrt(x + 1)*s qrt(-x + 1) - 48*x + 12)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1)
Result contains complex when optimal does not.
Time = 10.53 (sec) , antiderivative size = 542, normalized size of antiderivative = 6.69 \[ \int \frac {1}{(1-x)^{9/2} \sqrt {1+x}} \, dx=\begin {cases} \frac {2 \left (x + 1\right )^{3}}{35 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{3} - 210 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2} + 420 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right ) - 280 \sqrt {-1 + \frac {2}{x + 1}}} - \frac {14 \left (x + 1\right )^{2}}{35 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{3} - 210 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2} + 420 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right ) - 280 \sqrt {-1 + \frac {2}{x + 1}}} + \frac {35 \left (x + 1\right )}{35 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{3} - 210 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2} + 420 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right ) - 280 \sqrt {-1 + \frac {2}{x + 1}}} - \frac {35}{35 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{3} - 210 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2} + 420 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right ) - 280 \sqrt {-1 + \frac {2}{x + 1}}} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\- \frac {2 i \left (x + 1\right )^{3}}{35 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{3} - 210 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2} + 420 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right ) - 280 \sqrt {1 - \frac {2}{x + 1}}} + \frac {14 i \left (x + 1\right )^{2}}{35 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{3} - 210 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2} + 420 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right ) - 280 \sqrt {1 - \frac {2}{x + 1}}} - \frac {35 i \left (x + 1\right )}{35 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{3} - 210 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2} + 420 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right ) - 280 \sqrt {1 - \frac {2}{x + 1}}} + \frac {35 i}{35 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{3} - 210 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2} + 420 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right ) - 280 \sqrt {1 - \frac {2}{x + 1}}} & \text {otherwise} \end {cases} \] Input:
integrate(1/(1-x)**(9/2)/(1+x)**(1/2),x)
Output:
Piecewise((2*(x + 1)**3/(35*sqrt(-1 + 2/(x + 1))*(x + 1)**3 - 210*sqrt(-1 + 2/(x + 1))*(x + 1)**2 + 420*sqrt(-1 + 2/(x + 1))*(x + 1) - 280*sqrt(-1 + 2/(x + 1))) - 14*(x + 1)**2/(35*sqrt(-1 + 2/(x + 1))*(x + 1)**3 - 210*sqr t(-1 + 2/(x + 1))*(x + 1)**2 + 420*sqrt(-1 + 2/(x + 1))*(x + 1) - 280*sqrt (-1 + 2/(x + 1))) + 35*(x + 1)/(35*sqrt(-1 + 2/(x + 1))*(x + 1)**3 - 210*s qrt(-1 + 2/(x + 1))*(x + 1)**2 + 420*sqrt(-1 + 2/(x + 1))*(x + 1) - 280*sq rt(-1 + 2/(x + 1))) - 35/(35*sqrt(-1 + 2/(x + 1))*(x + 1)**3 - 210*sqrt(-1 + 2/(x + 1))*(x + 1)**2 + 420*sqrt(-1 + 2/(x + 1))*(x + 1) - 280*sqrt(-1 + 2/(x + 1))), 1/Abs(x + 1) > 1/2), (-2*I*(x + 1)**3/(35*sqrt(1 - 2/(x + 1 ))*(x + 1)**3 - 210*sqrt(1 - 2/(x + 1))*(x + 1)**2 + 420*sqrt(1 - 2/(x + 1 ))*(x + 1) - 280*sqrt(1 - 2/(x + 1))) + 14*I*(x + 1)**2/(35*sqrt(1 - 2/(x + 1))*(x + 1)**3 - 210*sqrt(1 - 2/(x + 1))*(x + 1)**2 + 420*sqrt(1 - 2/(x + 1))*(x + 1) - 280*sqrt(1 - 2/(x + 1))) - 35*I*(x + 1)/(35*sqrt(1 - 2/(x + 1))*(x + 1)**3 - 210*sqrt(1 - 2/(x + 1))*(x + 1)**2 + 420*sqrt(1 - 2/(x + 1))*(x + 1) - 280*sqrt(1 - 2/(x + 1))) + 35*I/(35*sqrt(1 - 2/(x + 1))*(x + 1)**3 - 210*sqrt(1 - 2/(x + 1))*(x + 1)**2 + 420*sqrt(1 - 2/(x + 1))*(x + 1) - 280*sqrt(1 - 2/(x + 1))), True))
Time = 0.11 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.17 \[ \int \frac {1}{(1-x)^{9/2} \sqrt {1+x}} \, dx=\frac {\sqrt {-x^{2} + 1}}{7 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} - \frac {3 \, \sqrt {-x^{2} + 1}}{35 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac {2 \, \sqrt {-x^{2} + 1}}{35 \, {\left (x^{2} - 2 \, x + 1\right )}} - \frac {2 \, \sqrt {-x^{2} + 1}}{35 \, {\left (x - 1\right )}} \] Input:
integrate(1/(1-x)^(9/2)/(1+x)^(1/2),x, algorithm="maxima")
Output:
1/7*sqrt(-x^2 + 1)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1) - 3/35*sqrt(-x^2 + 1)/( x^3 - 3*x^2 + 3*x - 1) + 2/35*sqrt(-x^2 + 1)/(x^2 - 2*x + 1) - 2/35*sqrt(- x^2 + 1)/(x - 1)
Time = 0.13 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.43 \[ \int \frac {1}{(1-x)^{9/2} \sqrt {1+x}} \, dx=-\frac {{\left ({\left (2 \, {\left (x + 1\right )} {\left (x - 6\right )} + 35\right )} {\left (x + 1\right )} - 35\right )} \sqrt {x + 1} \sqrt {-x + 1}}{35 \, {\left (x - 1\right )}^{4}} \] Input:
integrate(1/(1-x)^(9/2)/(1+x)^(1/2),x, algorithm="giac")
Output:
-1/35*((2*(x + 1)*(x - 6) + 35)*(x + 1) - 35)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^4
Time = 0.21 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.83 \[ \int \frac {1}{(1-x)^{9/2} \sqrt {1+x}} \, dx=-\frac {x\,\sqrt {1-x}-12\,\sqrt {1-x}+5\,x^2\,\sqrt {1-x}-6\,x^3\,\sqrt {1-x}+2\,x^4\,\sqrt {1-x}}{35\,{\left (x-1\right )}^4\,\sqrt {x+1}} \] Input:
int(1/((1 - x)^(9/2)*(x + 1)^(1/2)),x)
Output:
-(x*(1 - x)^(1/2) - 12*(1 - x)^(1/2) + 5*x^2*(1 - x)^(1/2) - 6*x^3*(1 - x) ^(1/2) + 2*x^4*(1 - x)^(1/2))/(35*(x - 1)^4*(x + 1)^(1/2))
Time = 0.16 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.54 \[ \int \frac {1}{(1-x)^{9/2} \sqrt {1+x}} \, dx=\frac {\sqrt {x +1}\, \left (2 x^{3}-8 x^{2}+13 x -12\right )}{35 \sqrt {1-x}\, \left (x^{3}-3 x^{2}+3 x -1\right )} \] Input:
int(1/(1-x)^(9/2)/(1+x)^(1/2),x)
Output:
(sqrt(x + 1)*(2*x**3 - 8*x**2 + 13*x - 12))/(35*sqrt( - x + 1)*(x**3 - 3*x **2 + 3*x - 1))