Integrand size = 17, antiderivative size = 82 \[ \int \frac {1}{(1-x)^{9/2} (1+x)^{3/2}} \, dx=\frac {1}{7 (1-x)^{7/2} \sqrt {1+x}}+\frac {4}{35 (1-x)^{5/2} \sqrt {1+x}}+\frac {4}{35 (1-x)^{3/2} \sqrt {1+x}}+\frac {8 x}{35 \sqrt {1-x} \sqrt {1+x}} \]
1/7/(1-x)^(7/2)/(1+x)^(1/2)+4/35/(1-x)^(5/2)/(1+x)^(1/2)+4/35/(1-x)^(3/2)/ (1+x)^(1/2)+8/35*x/(1-x)^(1/2)/(1+x)^(1/2)
Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.49 \[ \int \frac {1}{(1-x)^{9/2} (1+x)^{3/2}} \, dx=\frac {13-4 x-20 x^2+24 x^3-8 x^4}{35 (1-x)^{7/2} \sqrt {1+x}} \]
Time = 0.16 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {55, 55, 55, 39, 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.
\(\displaystyle \int \frac {1}{(1-x)^{9/2} (x+1)^{3/2}} \, dx\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {4}{7} \int \frac {1}{(1-x)^{7/2} (x+1)^{3/2}}dx+\frac {1}{7 (1-x)^{7/2} \sqrt {x+1}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {4}{7} \left (\frac {3}{5} \int \frac {1}{(1-x)^{5/2} (x+1)^{3/2}}dx+\frac {1}{5 (1-x)^{5/2} \sqrt {x+1}}\right )+\frac {1}{7 (1-x)^{7/2} \sqrt {x+1}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {4}{7} \left (\frac {3}{5} \left (\frac {2}{3} \int \frac {1}{(1-x)^{3/2} (x+1)^{3/2}}dx+\frac {1}{3 (1-x)^{3/2} \sqrt {x+1}}\right )+\frac {1}{5 (1-x)^{5/2} \sqrt {x+1}}\right )+\frac {1}{7 (1-x)^{7/2} \sqrt {x+1}}\) |
\(\Big \downarrow \) 39 |
\(\displaystyle \frac {4}{7} \left (\frac {3}{5} \left (\frac {2}{3} \int \frac {1}{\left (1-x^2\right )^{3/2}}dx+\frac {1}{3 (1-x)^{3/2} \sqrt {x+1}}\right )+\frac {1}{5 (1-x)^{5/2} \sqrt {x+1}}\right )+\frac {1}{7 (1-x)^{7/2} \sqrt {x+1}}\) |
\(\Big \downarrow \) 208 |
\(\displaystyle \frac {4}{7} \left (\frac {3}{5} \left (\frac {2 x}{3 \sqrt {1-x^2}}+\frac {1}{3 (1-x)^{3/2} \sqrt {x+1}}\right )+\frac {1}{5 (1-x)^{5/2} \sqrt {x+1}}\right )+\frac {1}{7 (1-x)^{7/2} \sqrt {x+1}}\) |
1/(7*(1 - x)^(7/2)*Sqrt[1 + x]) + (4*(1/(5*(1 - x)^(5/2)*Sqrt[1 + x]) + (3 *(1/(3*(1 - x)^(3/2)*Sqrt[1 + x]) + (2*x)/(3*Sqrt[1 - x^2])))/5))/7
3.12.24.3.1 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]
Time = 0.34 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.43
method | result | size |
gosper | \(-\frac {8 x^{4}-24 x^{3}+20 x^{2}+4 x -13}{35 \left (1-x \right )^{\frac {7}{2}} \sqrt {1+x}}\) | \(35\) |
risch | \(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (8 x^{4}-24 x^{3}+20 x^{2}+4 x -13\right )}{35 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{3} \sqrt {-\left (-1+x \right ) \left (1+x \right )}}\) | \(61\) |
default | \(\frac {1}{7 \left (1-x \right )^{\frac {7}{2}} \sqrt {1+x}}+\frac {4}{35 \left (1-x \right )^{\frac {5}{2}} \sqrt {1+x}}+\frac {4}{35 \left (1-x \right )^{\frac {3}{2}} \sqrt {1+x}}+\frac {8}{35 \sqrt {1-x}\, \sqrt {1+x}}-\frac {8 \sqrt {1-x}}{35 \sqrt {1+x}}\) | \(72\) |
Time = 0.23 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.05 \[ \int \frac {1}{(1-x)^{9/2} (1+x)^{3/2}} \, dx=\frac {13 \, x^{5} - 39 \, x^{4} + 26 \, x^{3} + 26 \, x^{2} - {\left (8 \, x^{4} - 24 \, x^{3} + 20 \, x^{2} + 4 \, x - 13\right )} \sqrt {x + 1} \sqrt {-x + 1} - 39 \, x + 13}{35 \, {\left (x^{5} - 3 \, x^{4} + 2 \, x^{3} + 2 \, x^{2} - 3 \, x + 1\right )}} \]
1/35*(13*x^5 - 39*x^4 + 26*x^3 + 26*x^2 - (8*x^4 - 24*x^3 + 20*x^2 + 4*x - 13)*sqrt(x + 1)*sqrt(-x + 1) - 39*x + 13)/(x^5 - 3*x^4 + 2*x^3 + 2*x^2 - 3*x + 1)
Result contains complex when optimal does not.
Time = 38.90 (sec) , antiderivative size = 425, normalized size of antiderivative = 5.18 \[ \int \frac {1}{(1-x)^{9/2} (1+x)^{3/2}} \, dx=\begin {cases} - \frac {8 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{4}}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} + \frac {56 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{3}}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} - \frac {140 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2}}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} + \frac {140 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} - \frac {35 \sqrt {-1 + \frac {2}{x + 1}}}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\- \frac {8 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{4}}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} + \frac {56 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{3}}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} - \frac {140 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2}}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} + \frac {140 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} - \frac {35 i \sqrt {1 - \frac {2}{x + 1}}}{- 1120 x + 35 \left (x + 1\right )^{4} - 280 \left (x + 1\right )^{3} + 840 \left (x + 1\right )^{2} - 560} & \text {otherwise} \end {cases} \]
Piecewise((-8*sqrt(-1 + 2/(x + 1))*(x + 1)**4/(-1120*x + 35*(x + 1)**4 - 2 80*(x + 1)**3 + 840*(x + 1)**2 - 560) + 56*sqrt(-1 + 2/(x + 1))*(x + 1)**3 /(-1120*x + 35*(x + 1)**4 - 280*(x + 1)**3 + 840*(x + 1)**2 - 560) - 140*s qrt(-1 + 2/(x + 1))*(x + 1)**2/(-1120*x + 35*(x + 1)**4 - 280*(x + 1)**3 + 840*(x + 1)**2 - 560) + 140*sqrt(-1 + 2/(x + 1))*(x + 1)/(-1120*x + 35*(x + 1)**4 - 280*(x + 1)**3 + 840*(x + 1)**2 - 560) - 35*sqrt(-1 + 2/(x + 1) )/(-1120*x + 35*(x + 1)**4 - 280*(x + 1)**3 + 840*(x + 1)**2 - 560), 1/Abs (x + 1) > 1/2), (-8*I*sqrt(1 - 2/(x + 1))*(x + 1)**4/(-1120*x + 35*(x + 1) **4 - 280*(x + 1)**3 + 840*(x + 1)**2 - 560) + 56*I*sqrt(1 - 2/(x + 1))*(x + 1)**3/(-1120*x + 35*(x + 1)**4 - 280*(x + 1)**3 + 840*(x + 1)**2 - 560) - 140*I*sqrt(1 - 2/(x + 1))*(x + 1)**2/(-1120*x + 35*(x + 1)**4 - 280*(x + 1)**3 + 840*(x + 1)**2 - 560) + 140*I*sqrt(1 - 2/(x + 1))*(x + 1)/(-1120 *x + 35*(x + 1)**4 - 280*(x + 1)**3 + 840*(x + 1)**2 - 560) - 35*I*sqrt(1 - 2/(x + 1))/(-1120*x + 35*(x + 1)**4 - 280*(x + 1)**3 + 840*(x + 1)**2 - 560), True))
Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (58) = 116\).
Time = 0.21 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.63 \[ \int \frac {1}{(1-x)^{9/2} (1+x)^{3/2}} \, dx=\frac {8 \, x}{35 \, \sqrt {-x^{2} + 1}} - \frac {1}{7 \, {\left (\sqrt {-x^{2} + 1} x^{3} - 3 \, \sqrt {-x^{2} + 1} x^{2} + 3 \, \sqrt {-x^{2} + 1} x - \sqrt {-x^{2} + 1}\right )}} + \frac {4}{35 \, {\left (\sqrt {-x^{2} + 1} x^{2} - 2 \, \sqrt {-x^{2} + 1} x + \sqrt {-x^{2} + 1}\right )}} - \frac {4}{35 \, {\left (\sqrt {-x^{2} + 1} x - \sqrt {-x^{2} + 1}\right )}} \]
8/35*x/sqrt(-x^2 + 1) - 1/7/(sqrt(-x^2 + 1)*x^3 - 3*sqrt(-x^2 + 1)*x^2 + 3 *sqrt(-x^2 + 1)*x - sqrt(-x^2 + 1)) + 4/35/(sqrt(-x^2 + 1)*x^2 - 2*sqrt(-x ^2 + 1)*x + sqrt(-x^2 + 1)) - 4/35/(sqrt(-x^2 + 1)*x - sqrt(-x^2 + 1))
Time = 0.34 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.96 \[ \int \frac {1}{(1-x)^{9/2} (1+x)^{3/2}} \, dx=\frac {\sqrt {2} - \sqrt {-x + 1}}{32 \, \sqrt {x + 1}} - \frac {\sqrt {x + 1}}{32 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}} - \frac {{\left ({\left ({\left (93 \, x - 523\right )} {\left (x + 1\right )} + 1400\right )} {\left (x + 1\right )} - 1120\right )} \sqrt {x + 1} \sqrt {-x + 1}}{560 \, {\left (x - 1\right )}^{4}} \]
1/32*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - 1/32*sqrt(x + 1)/(sqrt(2) - sq rt(-x + 1)) - 1/560*(((93*x - 523)*(x + 1) + 1400)*(x + 1) - 1120)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^4
Time = 0.39 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.83 \[ \int \frac {1}{(1-x)^{9/2} (1+x)^{3/2}} \, dx=-\frac {4\,x\,\sqrt {1-x}-13\,\sqrt {1-x}+20\,x^2\,\sqrt {1-x}-24\,x^3\,\sqrt {1-x}+8\,x^4\,\sqrt {1-x}}{35\,{\left (x-1\right )}^4\,\sqrt {x+1}} \]
-(4*x*(1 - x)^(1/2) - 13*(1 - x)^(1/2) + 20*x^2*(1 - x)^(1/2) - 24*x^3*(1 - x)^(1/2) + 8*x^4*(1 - x)^(1/2))/(35*(x - 1)^4*(x + 1)^(1/2))
Time = 0.00 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.62 \[ \int \frac {1}{(1-x)^{9/2} (1+x)^{3/2}} \, dx=\frac {8 x^{4}-24 x^{3}+20 x^{2}+4 x -13}{35 \sqrt {x +1}\, \sqrt {1-x}\, \left (x^{3}-3 x^{2}+3 x -1\right )} \]