Integrand size = 17, antiderivative size = 101 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{15/2}} \, dx=\frac {(1+x)^{5/2}}{13 (1-x)^{13/2}}+\frac {4 (1+x)^{5/2}}{143 (1-x)^{11/2}}+\frac {4 (1+x)^{5/2}}{429 (1-x)^{9/2}}+\frac {8 (1+x)^{5/2}}{3003 (1-x)^{7/2}}+\frac {8 (1+x)^{5/2}}{15015 (1-x)^{5/2}} \] Output:
1/13*(1+x)^(5/2)/(1-x)^(13/2)+4/143*(1+x)^(5/2)/(1-x)^(11/2)+4/429*(1+x)^( 5/2)/(1-x)^(9/2)+8/3003*(1+x)^(5/2)/(1-x)^(7/2)+8/15015*(1+x)^(5/2)/(1-x)^ (5/2)
Time = 0.05 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.40 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{15/2}} \, dx=\frac {(1+x)^{5/2} \left (1763-852 x+308 x^2-72 x^3+8 x^4\right )}{15015 (1-x)^{13/2}} \] Input:
Integrate[(1 + x)^(3/2)/(1 - x)^(15/2),x]
Output:
((1 + x)^(5/2)*(1763 - 852*x + 308*x^2 - 72*x^3 + 8*x^4))/(15015*(1 - x)^( 13/2))
Time = 0.17 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.15, 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, 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.
\(\displaystyle \int \frac {(x+1)^{3/2}}{(1-x)^{15/2}} \, dx\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {4}{13} \int \frac {(x+1)^{3/2}}{(1-x)^{13/2}}dx+\frac {(x+1)^{5/2}}{13 (1-x)^{13/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {4}{13} \left (\frac {3}{11} \int \frac {(x+1)^{3/2}}{(1-x)^{11/2}}dx+\frac {(x+1)^{5/2}}{11 (1-x)^{11/2}}\right )+\frac {(x+1)^{5/2}}{13 (1-x)^{13/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {4}{13} \left (\frac {3}{11} \left (\frac {2}{9} \int \frac {(x+1)^{3/2}}{(1-x)^{9/2}}dx+\frac {(x+1)^{5/2}}{9 (1-x)^{9/2}}\right )+\frac {(x+1)^{5/2}}{11 (1-x)^{11/2}}\right )+\frac {(x+1)^{5/2}}{13 (1-x)^{13/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {4}{13} \left (\frac {3}{11} \left (\frac {2}{9} \left (\frac {1}{7} \int \frac {(x+1)^{3/2}}{(1-x)^{7/2}}dx+\frac {(x+1)^{5/2}}{7 (1-x)^{7/2}}\right )+\frac {(x+1)^{5/2}}{9 (1-x)^{9/2}}\right )+\frac {(x+1)^{5/2}}{11 (1-x)^{11/2}}\right )+\frac {(x+1)^{5/2}}{13 (1-x)^{13/2}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {(x+1)^{5/2}}{13 (1-x)^{13/2}}+\frac {4}{13} \left (\frac {(x+1)^{5/2}}{11 (1-x)^{11/2}}+\frac {3}{11} \left (\frac {(x+1)^{5/2}}{9 (1-x)^{9/2}}+\frac {2}{9} \left (\frac {(x+1)^{5/2}}{35 (1-x)^{5/2}}+\frac {(x+1)^{5/2}}{7 (1-x)^{7/2}}\right )\right )\right )\) |
Input:
Int[(1 + x)^(3/2)/(1 - x)^(15/2),x]
Output:
(1 + x)^(5/2)/(13*(1 - x)^(13/2)) + (4*((1 + x)^(5/2)/(11*(1 - x)^(11/2)) + (3*((1 + x)^(5/2)/(9*(1 - x)^(9/2)) + (2*((1 + x)^(5/2)/(7*(1 - x)^(7/2) ) + (1 + x)^(5/2)/(35*(1 - x)^(5/2))))/9))/11))/13
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 = 0.10 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.35
method | result | size |
gosper | \(\frac {\left (1+x \right )^{\frac {5}{2}} \left (8 x^{4}-72 x^{3}+308 x^{2}-852 x +1763\right )}{15015 \left (1-x \right )^{\frac {13}{2}}}\) | \(35\) |
orering | \(-\frac {\left (1+x \right )^{\frac {5}{2}} \left (-1+x \right ) \left (8 x^{4}-72 x^{3}+308 x^{2}-852 x +1763\right )}{15015 \left (1-x \right )^{\frac {15}{2}}}\) | \(38\) |
risch | \(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (8 x^{7}-48 x^{6}+116 x^{5}-136 x^{4}+59 x^{3}+3041 x^{2}+4437 x +1763\right )}{15015 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{6} \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\) | \(76\) |
default | \(\frac {\left (1+x \right )^{\frac {3}{2}}}{5 \left (1-x \right )^{\frac {13}{2}}}-\frac {6 \sqrt {1+x}}{65 \left (1-x \right )^{\frac {13}{2}}}+\frac {3 \sqrt {1+x}}{715 \left (1-x \right )^{\frac {11}{2}}}+\frac {\sqrt {1+x}}{429 \left (1-x \right )^{\frac {9}{2}}}+\frac {4 \sqrt {1+x}}{3003 \left (1-x \right )^{\frac {7}{2}}}+\frac {4 \sqrt {1+x}}{5005 \left (1-x \right )^{\frac {5}{2}}}+\frac {8 \sqrt {1+x}}{15015 \left (1-x \right )^{\frac {3}{2}}}+\frac {8 \sqrt {1+x}}{15015 \sqrt {1-x}}\) | \(114\) |
Input:
int((1+x)^(3/2)/(1-x)^(15/2),x,method=_RETURNVERBOSE)
Output:
1/15015*(1+x)^(5/2)/(1-x)^(13/2)*(8*x^4-72*x^3+308*x^2-852*x+1763)
Time = 0.10 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.15 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{15/2}} \, dx=\frac {1763 \, x^{7} - 12341 \, x^{6} + 37023 \, x^{5} - 61705 \, x^{4} + 61705 \, x^{3} - 37023 \, x^{2} - {\left (8 \, x^{6} - 56 \, x^{5} + 172 \, x^{4} - 308 \, x^{3} + 367 \, x^{2} + 2674 \, x + 1763\right )} \sqrt {x + 1} \sqrt {-x + 1} + 12341 \, x - 1763}{15015 \, {\left (x^{7} - 7 \, x^{6} + 21 \, x^{5} - 35 \, x^{4} + 35 \, x^{3} - 21 \, x^{2} + 7 \, x - 1\right )}} \] Input:
integrate((1+x)^(3/2)/(1-x)^(15/2),x, algorithm="fricas")
Output:
1/15015*(1763*x^7 - 12341*x^6 + 37023*x^5 - 61705*x^4 + 61705*x^3 - 37023* x^2 - (8*x^6 - 56*x^5 + 172*x^4 - 308*x^3 + 367*x^2 + 2674*x + 1763)*sqrt( x + 1)*sqrt(-x + 1) + 12341*x - 1763)/(x^7 - 7*x^6 + 21*x^5 - 35*x^4 + 35* x^3 - 21*x^2 + 7*x - 1)
Timed out. \[ \int \frac {(1+x)^{3/2}}{(1-x)^{15/2}} \, dx=\text {Timed out} \] Input:
integrate((1+x)**(3/2)/(1-x)**(15/2),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (71) = 142\).
Time = 0.03 (sec) , antiderivative size = 269, normalized size of antiderivative = 2.66 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{15/2}} \, dx=\frac {{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{5 \, {\left (x^{8} - 8 \, x^{7} + 28 \, x^{6} - 56 \, x^{5} + 70 \, x^{4} - 56 \, x^{3} + 28 \, x^{2} - 8 \, x + 1\right )}} + \frac {6 \, \sqrt {-x^{2} + 1}}{65 \, {\left (x^{7} - 7 \, x^{6} + 21 \, x^{5} - 35 \, x^{4} + 35 \, x^{3} - 21 \, x^{2} + 7 \, x - 1\right )}} + \frac {3 \, \sqrt {-x^{2} + 1}}{715 \, {\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} - \frac {\sqrt {-x^{2} + 1}}{429 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} + \frac {4 \, \sqrt {-x^{2} + 1}}{3003 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} - \frac {4 \, \sqrt {-x^{2} + 1}}{5005 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac {8 \, \sqrt {-x^{2} + 1}}{15015 \, {\left (x^{2} - 2 \, x + 1\right )}} - \frac {8 \, \sqrt {-x^{2} + 1}}{15015 \, {\left (x - 1\right )}} \] Input:
integrate((1+x)^(3/2)/(1-x)^(15/2),x, algorithm="maxima")
Output:
1/5*(-x^2 + 1)^(3/2)/(x^8 - 8*x^7 + 28*x^6 - 56*x^5 + 70*x^4 - 56*x^3 + 28 *x^2 - 8*x + 1) + 6/65*sqrt(-x^2 + 1)/(x^7 - 7*x^6 + 21*x^5 - 35*x^4 + 35* x^3 - 21*x^2 + 7*x - 1) + 3/715*sqrt(-x^2 + 1)/(x^6 - 6*x^5 + 15*x^4 - 20* x^3 + 15*x^2 - 6*x + 1) - 1/429*sqrt(-x^2 + 1)/(x^5 - 5*x^4 + 10*x^3 - 10* x^2 + 5*x - 1) + 4/3003*sqrt(-x^2 + 1)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1) - 4 /5005*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x - 1) + 8/15015*sqrt(-x^2 + 1)/(x^2 - 2*x + 1) - 8/15015*sqrt(-x^2 + 1)/(x - 1)
Time = 0.16 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.42 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{15/2}} \, dx=-\frac {{\left (4 \, {\left ({\left (2 \, {\left (x + 1\right )} {\left (x - 12\right )} + 143\right )} {\left (x + 1\right )} - 429\right )} {\left (x + 1\right )} + 3003\right )} {\left (x + 1\right )}^{\frac {5}{2}} \sqrt {-x + 1}}{15015 \, {\left (x - 1\right )}^{7}} \] Input:
integrate((1+x)^(3/2)/(1-x)^(15/2),x, algorithm="giac")
Output:
-1/15015*(4*((2*(x + 1)*(x - 12) + 143)*(x + 1) - 429)*(x + 1) + 3003)*(x + 1)^(5/2)*sqrt(-x + 1)/(x - 1)^7
Time = 0.19 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.09 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{15/2}} \, dx=-\frac {\sqrt {1-x}\,\left (\frac {382\,x\,\sqrt {x+1}}{2145}+\frac {1763\,\sqrt {x+1}}{15015}+\frac {367\,x^2\,\sqrt {x+1}}{15015}-\frac {4\,x^3\,\sqrt {x+1}}{195}+\frac {172\,x^4\,\sqrt {x+1}}{15015}-\frac {8\,x^5\,\sqrt {x+1}}{2145}+\frac {8\,x^6\,\sqrt {x+1}}{15015}\right )}{x^7-7\,x^6+21\,x^5-35\,x^4+35\,x^3-21\,x^2+7\,x-1} \] Input:
int((x + 1)^(3/2)/(1 - x)^(15/2),x)
Output:
-((1 - x)^(1/2)*((382*x*(x + 1)^(1/2))/2145 + (1763*(x + 1)^(1/2))/15015 + (367*x^2*(x + 1)^(1/2))/15015 - (4*x^3*(x + 1)^(1/2))/195 + (172*x^4*(x + 1)^(1/2))/15015 - (8*x^5*(x + 1)^(1/2))/2145 + (8*x^6*(x + 1)^(1/2))/1501 5))/(7*x - 21*x^2 + 35*x^3 - 35*x^4 + 21*x^5 - 7*x^6 + x^7 - 1)
Time = 0.15 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.73 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{15/2}} \, dx=\frac {\sqrt {x +1}\, \left (8 x^{6}-56 x^{5}+172 x^{4}-308 x^{3}+367 x^{2}+2674 x +1763\right )}{15015 \sqrt {1-x}\, \left (x^{6}-6 x^{5}+15 x^{4}-20 x^{3}+15 x^{2}-6 x +1\right )} \] Input:
int((1+x)^(3/2)/(1-x)^(15/2),x)
Output:
(sqrt(x + 1)*(8*x**6 - 56*x**5 + 172*x**4 - 308*x**3 + 367*x**2 + 2674*x + 1763))/(15015*sqrt( - x + 1)*(x**6 - 6*x**5 + 15*x**4 - 20*x**3 + 15*x**2 - 6*x + 1))