Integrand size = 17, antiderivative size = 121 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{19/2}} \, dx=\frac {(1+x)^{7/2}}{17 (1-x)^{17/2}}+\frac {(1+x)^{7/2}}{51 (1-x)^{15/2}}+\frac {4 (1+x)^{7/2}}{663 (1-x)^{13/2}}+\frac {4 (1+x)^{7/2}}{2431 (1-x)^{11/2}}+\frac {8 (1+x)^{7/2}}{21879 (1-x)^{9/2}}+\frac {8 (1+x)^{7/2}}{153153 (1-x)^{7/2}} \]
1/17*(1+x)^(7/2)/(1-x)^(17/2)+1/51*(1+x)^(7/2)/(1-x)^(15/2)+4/663*(1+x)^(7 /2)/(1-x)^(13/2)+4/2431*(1+x)^(7/2)/(1-x)^(11/2)+8/21879*(1+x)^(7/2)/(1-x) ^(9/2)+8/153153*(1+x)^(7/2)/(1-x)^(7/2)
Time = 0.06 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.37 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{19/2}} \, dx=\frac {(1+x)^{7/2} \left (13252-5871 x+2096 x^2-556 x^3+96 x^4-8 x^5\right )}{153153 (1-x)^{17/2}} \]
((1 + x)^(7/2)*(13252 - 5871*x + 2096*x^2 - 556*x^3 + 96*x^4 - 8*x^5))/(15 3153*(1 - x)^(17/2))
Time = 0.18 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.17, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {55, 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.
\(\displaystyle \int \frac {(x+1)^{5/2}}{(1-x)^{19/2}} \, dx\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {5}{17} \int \frac {(x+1)^{5/2}}{(1-x)^{17/2}}dx+\frac {(x+1)^{7/2}}{17 (1-x)^{17/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {5}{17} \left (\frac {4}{15} \int \frac {(x+1)^{5/2}}{(1-x)^{15/2}}dx+\frac {(x+1)^{7/2}}{15 (1-x)^{15/2}}\right )+\frac {(x+1)^{7/2}}{17 (1-x)^{17/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {5}{17} \left (\frac {4}{15} \left (\frac {3}{13} \int \frac {(x+1)^{5/2}}{(1-x)^{13/2}}dx+\frac {(x+1)^{7/2}}{13 (1-x)^{13/2}}\right )+\frac {(x+1)^{7/2}}{15 (1-x)^{15/2}}\right )+\frac {(x+1)^{7/2}}{17 (1-x)^{17/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {5}{17} \left (\frac {4}{15} \left (\frac {3}{13} \left (\frac {2}{11} \int \frac {(x+1)^{5/2}}{(1-x)^{11/2}}dx+\frac {(x+1)^{7/2}}{11 (1-x)^{11/2}}\right )+\frac {(x+1)^{7/2}}{13 (1-x)^{13/2}}\right )+\frac {(x+1)^{7/2}}{15 (1-x)^{15/2}}\right )+\frac {(x+1)^{7/2}}{17 (1-x)^{17/2}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {5}{17} \left (\frac {4}{15} \left (\frac {3}{13} \left (\frac {2}{11} \left (\frac {1}{9} \int \frac {(x+1)^{5/2}}{(1-x)^{9/2}}dx+\frac {(x+1)^{7/2}}{9 (1-x)^{9/2}}\right )+\frac {(x+1)^{7/2}}{11 (1-x)^{11/2}}\right )+\frac {(x+1)^{7/2}}{13 (1-x)^{13/2}}\right )+\frac {(x+1)^{7/2}}{15 (1-x)^{15/2}}\right )+\frac {(x+1)^{7/2}}{17 (1-x)^{17/2}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {(x+1)^{7/2}}{17 (1-x)^{17/2}}+\frac {5}{17} \left (\frac {(x+1)^{7/2}}{15 (1-x)^{15/2}}+\frac {4}{15} \left (\frac {(x+1)^{7/2}}{13 (1-x)^{13/2}}+\frac {3}{13} \left (\frac {(x+1)^{7/2}}{11 (1-x)^{11/2}}+\frac {2}{11} \left (\frac {(x+1)^{7/2}}{63 (1-x)^{7/2}}+\frac {(x+1)^{7/2}}{9 (1-x)^{9/2}}\right )\right )\right )\right )\) |
(1 + x)^(7/2)/(17*(1 - x)^(17/2)) + (5*((1 + x)^(7/2)/(15*(1 - x)^(15/2)) + (4*((1 + x)^(7/2)/(13*(1 - x)^(13/2)) + (3*((1 + x)^(7/2)/(11*(1 - x)^(1 1/2)) + (2*((1 + x)^(7/2)/(9*(1 - x)^(9/2)) + (1 + x)^(7/2)/(63*(1 - x)^(7 /2))))/11))/13))/15))/17
3.12.3.3.1 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 = 0.33 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.33
method | result | size |
gosper | \(-\frac {\left (1+x \right )^{\frac {7}{2}} \left (8 x^{5}-96 x^{4}+556 x^{3}-2096 x^{2}+5871 x -13252\right )}{153153 \left (1-x \right )^{\frac {17}{2}}}\) | \(40\) |
risch | \(-\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (8 x^{9}-64 x^{8}+220 x^{7}-416 x^{6}+447 x^{5}-216 x^{4}-25610 x^{3}-58124 x^{2}-47137 x -13252\right )}{153153 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{8} \sqrt {-\left (-1+x \right ) \left (1+x \right )}}\) | \(86\) |
default | \(\frac {\left (1+x \right )^{\frac {5}{2}}}{6 \left (1-x \right )^{\frac {17}{2}}}-\frac {5 \left (1+x \right )^{\frac {3}{2}}}{42 \left (1-x \right )^{\frac {17}{2}}}+\frac {5 \sqrt {1+x}}{119 \left (1-x \right )^{\frac {17}{2}}}-\frac {\sqrt {1+x}}{714 \left (1-x \right )^{\frac {15}{2}}}-\frac {\sqrt {1+x}}{1326 \left (1-x \right )^{\frac {13}{2}}}-\frac {\sqrt {1+x}}{2431 \left (1-x \right )^{\frac {11}{2}}}-\frac {5 \sqrt {1+x}}{21879 \left (1-x \right )^{\frac {9}{2}}}-\frac {20 \sqrt {1+x}}{153153 \left (1-x \right )^{\frac {7}{2}}}-\frac {4 \sqrt {1+x}}{51051 \left (1-x \right )^{\frac {5}{2}}}-\frac {8 \sqrt {1+x}}{153153 \left (1-x \right )^{\frac {3}{2}}}-\frac {8 \sqrt {1+x}}{153153 \sqrt {1-x}}\) | \(156\) |
Time = 0.23 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.20 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{19/2}} \, dx=\frac {13252 \, x^{9} - 119268 \, x^{8} + 477072 \, x^{7} - 1113168 \, x^{6} + 1669752 \, x^{5} - 1669752 \, x^{4} + 1113168 \, x^{3} - 477072 \, x^{2} + {\left (8 \, x^{8} - 72 \, x^{7} + 292 \, x^{6} - 708 \, x^{5} + 1155 \, x^{4} - 1371 \, x^{3} - 24239 \, x^{2} - 33885 \, x - 13252\right )} \sqrt {x + 1} \sqrt {-x + 1} + 119268 \, x - 13252}{153153 \, {\left (x^{9} - 9 \, x^{8} + 36 \, x^{7} - 84 \, x^{6} + 126 \, x^{5} - 126 \, x^{4} + 84 \, x^{3} - 36 \, x^{2} + 9 \, x - 1\right )}} \]
1/153153*(13252*x^9 - 119268*x^8 + 477072*x^7 - 1113168*x^6 + 1669752*x^5 - 1669752*x^4 + 1113168*x^3 - 477072*x^2 + (8*x^8 - 72*x^7 + 292*x^6 - 708 *x^5 + 1155*x^4 - 1371*x^3 - 24239*x^2 - 33885*x - 13252)*sqrt(x + 1)*sqrt (-x + 1) + 119268*x - 13252)/(x^9 - 9*x^8 + 36*x^7 - 84*x^6 + 126*x^5 - 12 6*x^4 + 84*x^3 - 36*x^2 + 9*x - 1)
Timed out. \[ \int \frac {(1+x)^{5/2}}{(1-x)^{19/2}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 452 vs. \(2 (85) = 170\).
Time = 0.21 (sec) , antiderivative size = 452, normalized size of antiderivative = 3.74 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{19/2}} \, dx=-\frac {{\left (-x^{2} + 1\right )}^{\frac {5}{2}}}{6 \, {\left (x^{11} - 11 \, x^{10} + 55 \, x^{9} - 165 \, x^{8} + 330 \, x^{7} - 462 \, x^{6} + 462 \, x^{5} - 330 \, x^{4} + 165 \, x^{3} - 55 \, x^{2} + 11 \, x - 1\right )}} - \frac {5 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{42 \, {\left (x^{10} - 10 \, x^{9} + 45 \, x^{8} - 120 \, x^{7} + 210 \, x^{6} - 252 \, x^{5} + 210 \, x^{4} - 120 \, x^{3} + 45 \, x^{2} - 10 \, x + 1\right )}} - \frac {5 \, \sqrt {-x^{2} + 1}}{119 \, {\left (x^{9} - 9 \, x^{8} + 36 \, x^{7} - 84 \, x^{6} + 126 \, x^{5} - 126 \, x^{4} + 84 \, x^{3} - 36 \, x^{2} + 9 \, x - 1\right )}} - \frac {\sqrt {-x^{2} + 1}}{714 \, {\left (x^{8} - 8 \, x^{7} + 28 \, x^{6} - 56 \, x^{5} + 70 \, x^{4} - 56 \, x^{3} + 28 \, x^{2} - 8 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{1326 \, {\left (x^{7} - 7 \, x^{6} + 21 \, x^{5} - 35 \, x^{4} + 35 \, x^{3} - 21 \, x^{2} + 7 \, x - 1\right )}} - \frac {\sqrt {-x^{2} + 1}}{2431 \, {\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} + \frac {5 \, \sqrt {-x^{2} + 1}}{21879 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} - \frac {20 \, \sqrt {-x^{2} + 1}}{153153 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac {4 \, \sqrt {-x^{2} + 1}}{51051 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac {8 \, \sqrt {-x^{2} + 1}}{153153 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {8 \, \sqrt {-x^{2} + 1}}{153153 \, {\left (x - 1\right )}} \]
-1/6*(-x^2 + 1)^(5/2)/(x^11 - 11*x^10 + 55*x^9 - 165*x^8 + 330*x^7 - 462*x ^6 + 462*x^5 - 330*x^4 + 165*x^3 - 55*x^2 + 11*x - 1) - 5/42*(-x^2 + 1)^(3 /2)/(x^10 - 10*x^9 + 45*x^8 - 120*x^7 + 210*x^6 - 252*x^5 + 210*x^4 - 120* x^3 + 45*x^2 - 10*x + 1) - 5/119*sqrt(-x^2 + 1)/(x^9 - 9*x^8 + 36*x^7 - 84 *x^6 + 126*x^5 - 126*x^4 + 84*x^3 - 36*x^2 + 9*x - 1) - 1/714*sqrt(-x^2 + 1)/(x^8 - 8*x^7 + 28*x^6 - 56*x^5 + 70*x^4 - 56*x^3 + 28*x^2 - 8*x + 1) + 1/1326*sqrt(-x^2 + 1)/(x^7 - 7*x^6 + 21*x^5 - 35*x^4 + 35*x^3 - 21*x^2 + 7 *x - 1) - 1/2431*sqrt(-x^2 + 1)/(x^6 - 6*x^5 + 15*x^4 - 20*x^3 + 15*x^2 - 6*x + 1) + 5/21879*sqrt(-x^2 + 1)/(x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 1 ) - 20/153153*sqrt(-x^2 + 1)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1) + 4/51051*sqr t(-x^2 + 1)/(x^3 - 3*x^2 + 3*x - 1) - 8/153153*sqrt(-x^2 + 1)/(x^2 - 2*x + 1) + 8/153153*sqrt(-x^2 + 1)/(x - 1)
Time = 0.35 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.40 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{19/2}} \, dx=\frac {{\left ({\left (4 \, {\left ({\left (2 \, {\left (x + 1\right )} {\left (x - 16\right )} + 255\right )} {\left (x + 1\right )} - 1105\right )} {\left (x + 1\right )} + 12155\right )} {\left (x + 1\right )} - 21879\right )} {\left (x + 1\right )}^{\frac {7}{2}} \sqrt {-x + 1}}{153153 \, {\left (x - 1\right )}^{9}} \]
1/153153*((4*((2*(x + 1)*(x - 16) + 255)*(x + 1) - 1105)*(x + 1) + 12155)* (x + 1) - 21879)*(x + 1)^(7/2)*sqrt(-x + 1)/(x - 1)^9
Time = 0.55 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.16 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{19/2}} \, dx=-\frac {\sqrt {1-x}\,\left (\frac {3765\,x\,\sqrt {x+1}}{17017}+\frac {13252\,\sqrt {x+1}}{153153}+\frac {24239\,x^2\,\sqrt {x+1}}{153153}+\frac {457\,x^3\,\sqrt {x+1}}{51051}-\frac {5\,x^4\,\sqrt {x+1}}{663}+\frac {236\,x^5\,\sqrt {x+1}}{51051}-\frac {292\,x^6\,\sqrt {x+1}}{153153}+\frac {8\,x^7\,\sqrt {x+1}}{17017}-\frac {8\,x^8\,\sqrt {x+1}}{153153}\right )}{x^9-9\,x^8+36\,x^7-84\,x^6+126\,x^5-126\,x^4+84\,x^3-36\,x^2+9\,x-1} \]
-((1 - x)^(1/2)*((3765*x*(x + 1)^(1/2))/17017 + (13252*(x + 1)^(1/2))/1531 53 + (24239*x^2*(x + 1)^(1/2))/153153 + (457*x^3*(x + 1)^(1/2))/51051 - (5 *x^4*(x + 1)^(1/2))/663 + (236*x^5*(x + 1)^(1/2))/51051 - (292*x^6*(x + 1) ^(1/2))/153153 + (8*x^7*(x + 1)^(1/2))/17017 - (8*x^8*(x + 1)^(1/2))/15315 3))/(9*x - 36*x^2 + 84*x^3 - 126*x^4 + 126*x^5 - 84*x^6 + 36*x^7 - 9*x^8 + x^9 - 1)