Integrand size = 17, antiderivative size = 83 \[ \int \frac {1}{(1-x)^{9/2} (1+x)^{5/2}} \, dx=\frac {1}{7 (1-x)^{7/2} (1+x)^{3/2}}+\frac {1}{7 (1-x)^{5/2} (1+x)^{3/2}}+\frac {4 x}{21 (1-x)^{3/2} (1+x)^{3/2}}+\frac {8 x}{21 \sqrt {1-x} \sqrt {1+x}} \]
1/7/(1-x)^(7/2)/(1+x)^(3/2)+1/7/(1-x)^(5/2)/(1+x)^(3/2)+4/21*x/(1-x)^(3/2) /(1+x)^(3/2)+8/21*x/(1-x)^(1/2)/(1+x)^(1/2)
Time = 0.04 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.54 \[ \int \frac {1}{(1-x)^{9/2} (1+x)^{5/2}} \, dx=\frac {6+9 x-24 x^2+4 x^3+16 x^4-8 x^5}{21 (1-x)^{7/2} (1+x)^{3/2}} \]
Time = 0.16 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {55, 55, 39, 209, 208}
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.
\(\displaystyle \int \frac {1}{(1-x)^{9/2} (x+1)^{5/2}} \, dx\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {5}{7} \int \frac {1}{(1-x)^{7/2} (x+1)^{5/2}}dx+\frac {1}{7 (1-x)^{7/2} (x+1)^{3/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {5}{7} \left (\frac {4}{5} \int \frac {1}{(1-x)^{5/2} (x+1)^{5/2}}dx+\frac {1}{5 (1-x)^{5/2} (x+1)^{3/2}}\right )+\frac {1}{7 (1-x)^{7/2} (x+1)^{3/2}}\) |
\(\Big \downarrow \) 39 |
\(\displaystyle \frac {5}{7} \left (\frac {4}{5} \int \frac {1}{\left (1-x^2\right )^{5/2}}dx+\frac {1}{5 (1-x)^{5/2} (x+1)^{3/2}}\right )+\frac {1}{7 (1-x)^{7/2} (x+1)^{3/2}}\) |
\(\Big \downarrow \) 209 |
\(\displaystyle \frac {5}{7} \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {1}{\left (1-x^2\right )^{3/2}}dx+\frac {x}{3 \left (1-x^2\right )^{3/2}}\right )+\frac {1}{5 (1-x)^{5/2} (x+1)^{3/2}}\right )+\frac {1}{7 (1-x)^{7/2} (x+1)^{3/2}}\) |
\(\Big \downarrow \) 208 |
\(\displaystyle \frac {5}{7} \left (\frac {4}{5} \left (\frac {2 x}{3 \sqrt {1-x^2}}+\frac {x}{3 \left (1-x^2\right )^{3/2}}\right )+\frac {1}{5 (1-x)^{5/2} (x+1)^{3/2}}\right )+\frac {1}{7 (1-x)^{7/2} (x+1)^{3/2}}\) |
1/(7*(1 - x)^(7/2)*(1 + x)^(3/2)) + (5*(1/(5*(1 - x)^(5/2)*(1 + x)^(3/2)) + (4*(x/(3*(1 - x^2)^(3/2)) + (2*x)/(3*Sqrt[1 - x^2])))/5))/7
3.12.35.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[( a*c + b*d*x^2)^m, x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b*c + a*d, 0] && ( IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))
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])
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && ILtQ[p + 3/2, 0]
Time = 0.18 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.48
method | result | size |
gosper | \(-\frac {8 x^{5}-16 x^{4}-4 x^{3}+24 x^{2}-9 x -6}{21 \left (1-x \right )^{\frac {7}{2}} \left (1+x \right )^{\frac {3}{2}}}\) | \(40\) |
risch | \(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (8 x^{5}-16 x^{4}-4 x^{3}+24 x^{2}-9 x -6\right )}{21 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}} \left (-1+x \right )^{3} \sqrt {-\left (-1+x \right ) \left (1+x \right )}}\) | \(66\) |
default | \(\frac {1}{7 \left (1-x \right )^{\frac {7}{2}} \left (1+x \right )^{\frac {3}{2}}}+\frac {1}{7 \left (1-x \right )^{\frac {5}{2}} \left (1+x \right )^{\frac {3}{2}}}+\frac {4}{21 \left (1-x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {3}{2}}}+\frac {4}{7 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}-\frac {8 \sqrt {1-x}}{21 \left (1+x \right )^{\frac {3}{2}}}-\frac {8 \sqrt {1-x}}{21 \sqrt {1+x}}\) | \(86\) |
Time = 0.23 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.22 \[ \int \frac {1}{(1-x)^{9/2} (1+x)^{5/2}} \, dx=\frac {6 \, x^{6} - 12 \, x^{5} - 6 \, x^{4} + 24 \, x^{3} - 6 \, x^{2} - {\left (8 \, x^{5} - 16 \, x^{4} - 4 \, x^{3} + 24 \, x^{2} - 9 \, x - 6\right )} \sqrt {x + 1} \sqrt {-x + 1} - 12 \, x + 6}{21 \, {\left (x^{6} - 2 \, x^{5} - x^{4} + 4 \, x^{3} - x^{2} - 2 \, x + 1\right )}} \]
1/21*(6*x^6 - 12*x^5 - 6*x^4 + 24*x^3 - 6*x^2 - (8*x^5 - 16*x^4 - 4*x^3 + 24*x^2 - 9*x - 6)*sqrt(x + 1)*sqrt(-x + 1) - 12*x + 6)/(x^6 - 2*x^5 - x^4 + 4*x^3 - x^2 - 2*x + 1)
Result contains complex when optimal does not.
Time = 61.34 (sec) , antiderivative size = 593, normalized size of antiderivative = 7.14 \[ \int \frac {1}{(1-x)^{9/2} (1+x)^{5/2}} \, dx=\begin {cases} - \frac {8 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{5}}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} + \frac {56 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{4}}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} - \frac {140 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{3}}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} + \frac {140 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2}}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} - \frac {35 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} - \frac {7 \sqrt {-1 + \frac {2}{x + 1}}}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\- \frac {8 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{5}}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} + \frac {56 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{4}}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} - \frac {140 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{3}}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} + \frac {140 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2}}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} - \frac {35 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} - \frac {7 i \sqrt {1 - \frac {2}{x + 1}}}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} & \text {otherwise} \end {cases} \]
Piecewise((-8*sqrt(-1 + 2/(x + 1))*(x + 1)**5/(336*x + 21*(x + 1)**5 - 168 *(x + 1)**4 + 504*(x + 1)**3 - 672*(x + 1)**2 + 336) + 56*sqrt(-1 + 2/(x + 1))*(x + 1)**4/(336*x + 21*(x + 1)**5 - 168*(x + 1)**4 + 504*(x + 1)**3 - 672*(x + 1)**2 + 336) - 140*sqrt(-1 + 2/(x + 1))*(x + 1)**3/(336*x + 21*( x + 1)**5 - 168*(x + 1)**4 + 504*(x + 1)**3 - 672*(x + 1)**2 + 336) + 140* sqrt(-1 + 2/(x + 1))*(x + 1)**2/(336*x + 21*(x + 1)**5 - 168*(x + 1)**4 + 504*(x + 1)**3 - 672*(x + 1)**2 + 336) - 35*sqrt(-1 + 2/(x + 1))*(x + 1)/( 336*x + 21*(x + 1)**5 - 168*(x + 1)**4 + 504*(x + 1)**3 - 672*(x + 1)**2 + 336) - 7*sqrt(-1 + 2/(x + 1))/(336*x + 21*(x + 1)**5 - 168*(x + 1)**4 + 5 04*(x + 1)**3 - 672*(x + 1)**2 + 336), 1/Abs(x + 1) > 1/2), (-8*I*sqrt(1 - 2/(x + 1))*(x + 1)**5/(336*x + 21*(x + 1)**5 - 168*(x + 1)**4 + 504*(x + 1)**3 - 672*(x + 1)**2 + 336) + 56*I*sqrt(1 - 2/(x + 1))*(x + 1)**4/(336*x + 21*(x + 1)**5 - 168*(x + 1)**4 + 504*(x + 1)**3 - 672*(x + 1)**2 + 336) - 140*I*sqrt(1 - 2/(x + 1))*(x + 1)**3/(336*x + 21*(x + 1)**5 - 168*(x + 1)**4 + 504*(x + 1)**3 - 672*(x + 1)**2 + 336) + 140*I*sqrt(1 - 2/(x + 1)) *(x + 1)**2/(336*x + 21*(x + 1)**5 - 168*(x + 1)**4 + 504*(x + 1)**3 - 672 *(x + 1)**2 + 336) - 35*I*sqrt(1 - 2/(x + 1))*(x + 1)/(336*x + 21*(x + 1)* *5 - 168*(x + 1)**4 + 504*(x + 1)**3 - 672*(x + 1)**2 + 336) - 7*I*sqrt(1 - 2/(x + 1))/(336*x + 21*(x + 1)**5 - 168*(x + 1)**4 + 504*(x + 1)**3 - 67 2*(x + 1)**2 + 336), True))
Time = 0.20 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(1-x)^{9/2} (1+x)^{5/2}} \, dx=\frac {8 \, x}{21 \, \sqrt {-x^{2} + 1}} + \frac {4 \, x}{21 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {1}{7 \, {\left ({\left (-x^{2} + 1\right )}^{\frac {3}{2}} x^{2} - 2 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + {\left (-x^{2} + 1\right )}^{\frac {3}{2}}\right )}} - \frac {1}{7 \, {\left ({\left (-x^{2} + 1\right )}^{\frac {3}{2}} x - {\left (-x^{2} + 1\right )}^{\frac {3}{2}}\right )}} \]
8/21*x/sqrt(-x^2 + 1) + 4/21*x/(-x^2 + 1)^(3/2) + 1/7/((-x^2 + 1)^(3/2)*x^ 2 - 2*(-x^2 + 1)^(3/2)*x + (-x^2 + 1)^(3/2)) - 1/7/((-x^2 + 1)^(3/2)*x - ( -x^2 + 1)^(3/2))
Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (59) = 118\).
Time = 0.30 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.51 \[ \int \frac {1}{(1-x)^{9/2} (1+x)^{5/2}} \, dx=\frac {{\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}}{768 \, {\left (x + 1\right )}^{\frac {3}{2}}} + \frac {19 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}}{256 \, \sqrt {x + 1}} - \frac {{\left (x + 1\right )}^{\frac {3}{2}} {\left (\frac {57 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{2}}{x + 1} + 1\right )}}{768 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}} - \frac {{\left ({\left ({\left (79 \, x - 432\right )} {\left (x + 1\right )} + 1120\right )} {\left (x + 1\right )} - 840\right )} \sqrt {x + 1} \sqrt {-x + 1}}{336 \, {\left (x - 1\right )}^{4}} \]
1/768*(sqrt(2) - sqrt(-x + 1))^3/(x + 1)^(3/2) + 19/256*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - 1/768*(x + 1)^(3/2)*(57*(sqrt(2) - sqrt(-x + 1))^2/(x + 1) + 1)/(sqrt(2) - sqrt(-x + 1))^3 - 1/336*(((79*x - 432)*(x + 1) + 112 0)*(x + 1) - 840)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^4
Time = 0.45 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.04 \[ \int \frac {1}{(1-x)^{9/2} (1+x)^{5/2}} \, dx=\frac {9\,x\,\sqrt {1-x}+6\,\sqrt {1-x}-24\,x^2\,\sqrt {1-x}+4\,x^3\,\sqrt {1-x}+16\,x^4\,\sqrt {1-x}-8\,x^5\,\sqrt {1-x}}{\left (21\,x+21\right )\,{\left (x-1\right )}^4\,\sqrt {x+1}} \]