Integrand size = 17, antiderivative size = 61 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{13/2}} \, dx=\frac {(1+x)^{7/2}}{11 (1-x)^{11/2}}+\frac {2 (1+x)^{7/2}}{99 (1-x)^{9/2}}+\frac {2 (1+x)^{7/2}}{693 (1-x)^{7/2}} \]
Time = 0.04 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.49 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{13/2}} \, dx=\frac {(1+x)^{7/2} \left (79-18 x+2 x^2\right )}{693 (1-x)^{11/2}} \]
Time = 0.14 (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.
\(\displaystyle \int \frac {(x+1)^{5/2}}{(1-x)^{13/2}} \, dx\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {2}{11} \int \frac {(x+1)^{5/2}}{(1-x)^{11/2}}dx+\frac {(x+1)^{7/2}}{11 (1-x)^{11/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {2}{11} \left (\frac {1}{9} \int \frac {(x+1)^{5/2}}{(1-x)^{9/2}}dx+\frac {(x+1)^{7/2}}{9 (1-x)^{9/2}}\right )+\frac {(x+1)^{7/2}}{11 (1-x)^{11/2}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {(x+1)^{7/2}}{11 (1-x)^{11/2}}+\frac {2}{11} \left (\frac {(x+1)^{7/2}}{63 (1-x)^{7/2}}+\frac {(x+1)^{7/2}}{9 (1-x)^{9/2}}\right )\) |
(1 + x)^(7/2)/(11*(1 - x)^(11/2)) + (2*((1 + x)^(7/2)/(9*(1 - x)^(9/2)) + (1 + x)^(7/2)/(63*(1 - x)^(7/2))))/11
3.11.100.3.1 Defintions of rubi rules used
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 = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.41
method | result | size |
gosper | \(\frac {\left (1+x \right )^{\frac {7}{2}} \left (2 x^{2}-18 x +79\right )}{693 \left (1-x \right )^{\frac {11}{2}}}\) | \(25\) |
risch | \(-\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (2 x^{6}-10 x^{5}+19 x^{4}+216 x^{3}+404 x^{2}+298 x +79\right )}{693 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{5} \sqrt {-\left (-1+x \right ) \left (1+x \right )}}\) | \(71\) |
default | \(\frac {\left (1+x \right )^{\frac {5}{2}}}{3 \left (1-x \right )^{\frac {11}{2}}}-\frac {5 \left (1+x \right )^{\frac {3}{2}}}{12 \left (1-x \right )^{\frac {11}{2}}}+\frac {5 \sqrt {1+x}}{22 \left (1-x \right )^{\frac {11}{2}}}-\frac {5 \sqrt {1+x}}{396 \left (1-x \right )^{\frac {9}{2}}}-\frac {5 \sqrt {1+x}}{693 \left (1-x \right )^{\frac {7}{2}}}-\frac {\sqrt {1+x}}{231 \left (1-x \right )^{\frac {5}{2}}}-\frac {2 \sqrt {1+x}}{693 \left (1-x \right )^{\frac {3}{2}}}-\frac {2 \sqrt {1+x}}{693 \sqrt {1-x}}\) | \(114\) |
Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (43) = 86\).
Time = 0.22 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.64 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{13/2}} \, dx=\frac {79 \, x^{6} - 474 \, x^{5} + 1185 \, x^{4} - 1580 \, x^{3} + 1185 \, x^{2} + {\left (2 \, x^{5} - 12 \, x^{4} + 31 \, x^{3} + 185 \, x^{2} + 219 \, x + 79\right )} \sqrt {x + 1} \sqrt {-x + 1} - 474 \, x + 79}{693 \, {\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} \]
1/693*(79*x^6 - 474*x^5 + 1185*x^4 - 1580*x^3 + 1185*x^2 + (2*x^5 - 12*x^4 + 31*x^3 + 185*x^2 + 219*x + 79)*sqrt(x + 1)*sqrt(-x + 1) - 474*x + 79)/( x^6 - 6*x^5 + 15*x^4 - 20*x^3 + 15*x^2 - 6*x + 1)
Result contains complex when optimal does not.
Time = 178.85 (sec) , antiderivative size = 784, normalized size of antiderivative = 12.85 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{13/2}} \, dx=\begin {cases} \frac {2 i \left (x + 1\right )^{\frac {13}{2}}}{693 \sqrt {x - 1} \left (x + 1\right )^{6} - 8316 \sqrt {x - 1} \left (x + 1\right )^{5} + 41580 \sqrt {x - 1} \left (x + 1\right )^{4} - 110880 \sqrt {x - 1} \left (x + 1\right )^{3} + 166320 \sqrt {x - 1} \left (x + 1\right )^{2} - 133056 \sqrt {x - 1} \left (x + 1\right ) + 44352 \sqrt {x - 1}} - \frac {26 i \left (x + 1\right )^{\frac {11}{2}}}{693 \sqrt {x - 1} \left (x + 1\right )^{6} - 8316 \sqrt {x - 1} \left (x + 1\right )^{5} + 41580 \sqrt {x - 1} \left (x + 1\right )^{4} - 110880 \sqrt {x - 1} \left (x + 1\right )^{3} + 166320 \sqrt {x - 1} \left (x + 1\right )^{2} - 133056 \sqrt {x - 1} \left (x + 1\right ) + 44352 \sqrt {x - 1}} + \frac {143 i \left (x + 1\right )^{\frac {9}{2}}}{693 \sqrt {x - 1} \left (x + 1\right )^{6} - 8316 \sqrt {x - 1} \left (x + 1\right )^{5} + 41580 \sqrt {x - 1} \left (x + 1\right )^{4} - 110880 \sqrt {x - 1} \left (x + 1\right )^{3} + 166320 \sqrt {x - 1} \left (x + 1\right )^{2} - 133056 \sqrt {x - 1} \left (x + 1\right ) + 44352 \sqrt {x - 1}} - \frac {198 i \left (x + 1\right )^{\frac {7}{2}}}{693 \sqrt {x - 1} \left (x + 1\right )^{6} - 8316 \sqrt {x - 1} \left (x + 1\right )^{5} + 41580 \sqrt {x - 1} \left (x + 1\right )^{4} - 110880 \sqrt {x - 1} \left (x + 1\right )^{3} + 166320 \sqrt {x - 1} \left (x + 1\right )^{2} - 133056 \sqrt {x - 1} \left (x + 1\right ) + 44352 \sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\- \frac {2 \left (x + 1\right )^{\frac {13}{2}}}{693 \sqrt {1 - x} \left (x + 1\right )^{6} - 8316 \sqrt {1 - x} \left (x + 1\right )^{5} + 41580 \sqrt {1 - x} \left (x + 1\right )^{4} - 110880 \sqrt {1 - x} \left (x + 1\right )^{3} + 166320 \sqrt {1 - x} \left (x + 1\right )^{2} - 133056 \sqrt {1 - x} \left (x + 1\right ) + 44352 \sqrt {1 - x}} + \frac {26 \left (x + 1\right )^{\frac {11}{2}}}{693 \sqrt {1 - x} \left (x + 1\right )^{6} - 8316 \sqrt {1 - x} \left (x + 1\right )^{5} + 41580 \sqrt {1 - x} \left (x + 1\right )^{4} - 110880 \sqrt {1 - x} \left (x + 1\right )^{3} + 166320 \sqrt {1 - x} \left (x + 1\right )^{2} - 133056 \sqrt {1 - x} \left (x + 1\right ) + 44352 \sqrt {1 - x}} - \frac {143 \left (x + 1\right )^{\frac {9}{2}}}{693 \sqrt {1 - x} \left (x + 1\right )^{6} - 8316 \sqrt {1 - x} \left (x + 1\right )^{5} + 41580 \sqrt {1 - x} \left (x + 1\right )^{4} - 110880 \sqrt {1 - x} \left (x + 1\right )^{3} + 166320 \sqrt {1 - x} \left (x + 1\right )^{2} - 133056 \sqrt {1 - x} \left (x + 1\right ) + 44352 \sqrt {1 - x}} + \frac {198 \left (x + 1\right )^{\frac {7}{2}}}{693 \sqrt {1 - x} \left (x + 1\right )^{6} - 8316 \sqrt {1 - x} \left (x + 1\right )^{5} + 41580 \sqrt {1 - x} \left (x + 1\right )^{4} - 110880 \sqrt {1 - x} \left (x + 1\right )^{3} + 166320 \sqrt {1 - x} \left (x + 1\right )^{2} - 133056 \sqrt {1 - x} \left (x + 1\right ) + 44352 \sqrt {1 - x}} & \text {otherwise} \end {cases} \]
Piecewise((2*I*(x + 1)**(13/2)/(693*sqrt(x - 1)*(x + 1)**6 - 8316*sqrt(x - 1)*(x + 1)**5 + 41580*sqrt(x - 1)*(x + 1)**4 - 110880*sqrt(x - 1)*(x + 1) **3 + 166320*sqrt(x - 1)*(x + 1)**2 - 133056*sqrt(x - 1)*(x + 1) + 44352*s qrt(x - 1)) - 26*I*(x + 1)**(11/2)/(693*sqrt(x - 1)*(x + 1)**6 - 8316*sqrt (x - 1)*(x + 1)**5 + 41580*sqrt(x - 1)*(x + 1)**4 - 110880*sqrt(x - 1)*(x + 1)**3 + 166320*sqrt(x - 1)*(x + 1)**2 - 133056*sqrt(x - 1)*(x + 1) + 443 52*sqrt(x - 1)) + 143*I*(x + 1)**(9/2)/(693*sqrt(x - 1)*(x + 1)**6 - 8316* sqrt(x - 1)*(x + 1)**5 + 41580*sqrt(x - 1)*(x + 1)**4 - 110880*sqrt(x - 1) *(x + 1)**3 + 166320*sqrt(x - 1)*(x + 1)**2 - 133056*sqrt(x - 1)*(x + 1) + 44352*sqrt(x - 1)) - 198*I*(x + 1)**(7/2)/(693*sqrt(x - 1)*(x + 1)**6 - 8 316*sqrt(x - 1)*(x + 1)**5 + 41580*sqrt(x - 1)*(x + 1)**4 - 110880*sqrt(x - 1)*(x + 1)**3 + 166320*sqrt(x - 1)*(x + 1)**2 - 133056*sqrt(x - 1)*(x + 1) + 44352*sqrt(x - 1)), Abs(x + 1) > 2), (-2*(x + 1)**(13/2)/(693*sqrt(1 - x)*(x + 1)**6 - 8316*sqrt(1 - x)*(x + 1)**5 + 41580*sqrt(1 - x)*(x + 1)* *4 - 110880*sqrt(1 - x)*(x + 1)**3 + 166320*sqrt(1 - x)*(x + 1)**2 - 13305 6*sqrt(1 - x)*(x + 1) + 44352*sqrt(1 - x)) + 26*(x + 1)**(11/2)/(693*sqrt( 1 - x)*(x + 1)**6 - 8316*sqrt(1 - x)*(x + 1)**5 + 41580*sqrt(1 - x)*(x + 1 )**4 - 110880*sqrt(1 - x)*(x + 1)**3 + 166320*sqrt(1 - x)*(x + 1)**2 - 133 056*sqrt(1 - x)*(x + 1) + 44352*sqrt(1 - x)) - 143*(x + 1)**(9/2)/(693*sqr t(1 - x)*(x + 1)**6 - 8316*sqrt(1 - x)*(x + 1)**5 + 41580*sqrt(1 - x)*(...
Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (43) = 86\).
Time = 0.24 (sec) , antiderivative size = 269, normalized size of antiderivative = 4.41 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{13/2}} \, dx=\frac {{\left (-x^{2} + 1\right )}^{\frac {5}{2}}}{3 \, {\left (x^{8} - 8 \, x^{7} + 28 \, x^{6} - 56 \, x^{5} + 70 \, x^{4} - 56 \, x^{3} + 28 \, x^{2} - 8 \, x + 1\right )}} + \frac {5 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{12 \, {\left (x^{7} - 7 \, x^{6} + 21 \, x^{5} - 35 \, x^{4} + 35 \, x^{3} - 21 \, x^{2} + 7 \, x - 1\right )}} + \frac {5 \, \sqrt {-x^{2} + 1}}{22 \, {\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} + \frac {5 \, \sqrt {-x^{2} + 1}}{396 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} - \frac {5 \, \sqrt {-x^{2} + 1}}{693 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{231 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac {2 \, \sqrt {-x^{2} + 1}}{693 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {2 \, \sqrt {-x^{2} + 1}}{693 \, {\left (x - 1\right )}} \]
1/3*(-x^2 + 1)^(5/2)/(x^8 - 8*x^7 + 28*x^6 - 56*x^5 + 70*x^4 - 56*x^3 + 28 *x^2 - 8*x + 1) + 5/12*(-x^2 + 1)^(3/2)/(x^7 - 7*x^6 + 21*x^5 - 35*x^4 + 3 5*x^3 - 21*x^2 + 7*x - 1) + 5/22*sqrt(-x^2 + 1)/(x^6 - 6*x^5 + 15*x^4 - 20 *x^3 + 15*x^2 - 6*x + 1) + 5/396*sqrt(-x^2 + 1)/(x^5 - 5*x^4 + 10*x^3 - 10 *x^2 + 5*x - 1) - 5/693*sqrt(-x^2 + 1)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1) + 1 /231*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x - 1) - 2/693*sqrt(-x^2 + 1)/(x^2 - 2*x + 1) + 2/693*sqrt(-x^2 + 1)/(x - 1)
Time = 0.33 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.48 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{13/2}} \, dx=\frac {{\left (2 \, {\left (x + 1\right )} {\left (x - 10\right )} + 99\right )} {\left (x + 1\right )}^{\frac {7}{2}} \sqrt {-x + 1}}{693 \, {\left (x - 1\right )}^{6}} \]
Time = 0.33 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.54 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{13/2}} \, dx=\frac {\sqrt {1-x}\,\left (\frac {73\,x\,\sqrt {x+1}}{231}+\frac {79\,\sqrt {x+1}}{693}+\frac {185\,x^2\,\sqrt {x+1}}{693}+\frac {31\,x^3\,\sqrt {x+1}}{693}-\frac {4\,x^4\,\sqrt {x+1}}{231}+\frac {2\,x^5\,\sqrt {x+1}}{693}\right )}{x^6-6\,x^5+15\,x^4-20\,x^3+15\,x^2-6\,x+1} \]