Integrand size = 17, antiderivative size = 110 \[ \int (1-x)^{9/2} (1+x)^{5/2} \, dx=\frac {45}{128} \sqrt {1-x} x \sqrt {1+x}+\frac {15}{64} (1-x)^{3/2} x (1+x)^{3/2}+\frac {3}{16} (1-x)^{5/2} x (1+x)^{5/2}+\frac {9}{56} (1-x)^{7/2} (1+x)^{7/2}+\frac {1}{8} (1-x)^{9/2} (1+x)^{7/2}+\frac {45 \arcsin (x)}{128} \]
15/64*(1-x)^(3/2)*x*(1+x)^(3/2)+3/16*(1-x)^(5/2)*x*(1+x)^(5/2)+9/56*(1-x)^ (7/2)*(1+x)^(7/2)+1/8*(1-x)^(9/2)*(1+x)^(7/2)+45/128*arcsin(x)+45/128*x*(1 -x)^(1/2)*(1+x)^(1/2)
Time = 0.19 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.65 \[ \int (1-x)^{9/2} (1+x)^{5/2} \, dx=\frac {1}{896} \left (\sqrt {1-x^2} \left (256+581 x-768 x^2-210 x^3+768 x^4-168 x^5-256 x^6+112 x^7\right )-630 \arctan \left (\frac {\sqrt {1-x^2}}{-1+x}\right )\right ) \]
(Sqrt[1 - x^2]*(256 + 581*x - 768*x^2 - 210*x^3 + 768*x^4 - 168*x^5 - 256* x^6 + 112*x^7) - 630*ArcTan[Sqrt[1 - x^2]/(-1 + x)])/896
Time = 0.17 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {59, 50, 211, 211, 211, 223}
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.
\(\displaystyle \int (1-x)^{9/2} (x+1)^{5/2} \, dx\) |
\(\Big \downarrow \) 59 |
\(\displaystyle \frac {9}{8} \int (1-x)^{7/2} (x+1)^{5/2}dx+\frac {1}{8} (x+1)^{7/2} (1-x)^{9/2}\) |
\(\Big \downarrow \) 50 |
\(\displaystyle \frac {9}{8} \left (\int \left (1-x^2\right )^{5/2}dx+\frac {1}{7} \left (1-x^2\right )^{7/2}\right )+\frac {1}{8} (x+1)^{7/2} (1-x)^{9/2}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {9}{8} \left (\frac {5}{6} \int \left (1-x^2\right )^{3/2}dx+\frac {1}{7} \left (1-x^2\right )^{7/2}+\frac {1}{6} x \left (1-x^2\right )^{5/2}\right )+\frac {1}{8} (x+1)^{7/2} (1-x)^{9/2}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {9}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {1-x^2}dx+\frac {1}{4} x \left (1-x^2\right )^{3/2}\right )+\frac {1}{7} \left (1-x^2\right )^{7/2}+\frac {1}{6} x \left (1-x^2\right )^{5/2}\right )+\frac {1}{8} (x+1)^{7/2} (1-x)^{9/2}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {9}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-x^2}}dx+\frac {1}{2} \sqrt {1-x^2} x\right )+\frac {1}{4} x \left (1-x^2\right )^{3/2}\right )+\frac {1}{7} \left (1-x^2\right )^{7/2}+\frac {1}{6} x \left (1-x^2\right )^{5/2}\right )+\frac {1}{8} (x+1)^{7/2} (1-x)^{9/2}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {9}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\arcsin (x)}{2}+\frac {1}{2} \sqrt {1-x^2} x\right )+\frac {1}{4} x \left (1-x^2\right )^{3/2}\right )+\frac {1}{7} \left (1-x^2\right )^{7/2}+\frac {1}{6} x \left (1-x^2\right )^{5/2}\right )+\frac {1}{8} (x+1)^{7/2} (1-x)^{9/2}\) |
((1 - x)^(9/2)*(1 + x)^(7/2))/8 + (9*((x*(1 - x^2)^(5/2))/6 + (1 - x^2)^(7 /2)/7 + (5*((x*(1 - x^2)^(3/2))/4 + (3*((x*Sqrt[1 - x^2])/2 + ArcSin[x]/2) )/4))/6))/8
3.11.89.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a *c + b*d*x^2)^m/(2*d*m), x] + Simp[a Int[(a*c + b*d*x^2)^n, x], x] /; Fre eQ[{a, b, c, d, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[m - n, 1] && GtQ[m, 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/(b*(m + n + 1))), x] + Simp[2*c*(n/(m + n + 1) ) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && IGtQ[m + 1/2, 0] && IGtQ[n + 1/2, 0] && LtQ[m, n]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Time = 0.32 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.93
method | result | size |
risch | \(-\frac {\left (112 x^{7}-256 x^{6}-168 x^{5}+768 x^{4}-210 x^{3}-768 x^{2}+581 x +256\right ) \left (-1+x \right ) \sqrt {1+x}\, \sqrt {\left (1+x \right ) \left (1-x \right )}}{896 \sqrt {-\left (-1+x \right ) \left (1+x \right )}\, \sqrt {1-x}}+\frac {45 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{128 \sqrt {1+x}\, \sqrt {1-x}}\) | \(102\) |
default | \(\frac {\left (1-x \right )^{\frac {9}{2}} \left (1+x \right )^{\frac {7}{2}}}{8}+\frac {9 \left (1-x \right )^{\frac {7}{2}} \left (1+x \right )^{\frac {7}{2}}}{56}+\frac {3 \left (1-x \right )^{\frac {5}{2}} \left (1+x \right )^{\frac {7}{2}}}{16}+\frac {3 \left (1-x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {7}{2}}}{16}+\frac {9 \sqrt {1-x}\, \left (1+x \right )^{\frac {7}{2}}}{64}-\frac {3 \sqrt {1-x}\, \left (1+x \right )^{\frac {5}{2}}}{64}-\frac {15 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}{128}-\frac {45 \sqrt {1-x}\, \sqrt {1+x}}{128}+\frac {45 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{128 \sqrt {1+x}\, \sqrt {1-x}}\) | \(141\) |
-1/896*(112*x^7-256*x^6-168*x^5+768*x^4-210*x^3-768*x^2+581*x+256)*(-1+x)* (1+x)^(1/2)/(-(-1+x)*(1+x))^(1/2)*((1+x)*(1-x))^(1/2)/(1-x)^(1/2)+45/128*( (1+x)*(1-x))^(1/2)/(1+x)^(1/2)/(1-x)^(1/2)*arcsin(x)
Time = 0.22 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.65 \[ \int (1-x)^{9/2} (1+x)^{5/2} \, dx=\frac {1}{896} \, {\left (112 \, x^{7} - 256 \, x^{6} - 168 \, x^{5} + 768 \, x^{4} - 210 \, x^{3} - 768 \, x^{2} + 581 \, x + 256\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {45}{64} \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \]
1/896*(112*x^7 - 256*x^6 - 168*x^5 + 768*x^4 - 210*x^3 - 768*x^2 + 581*x + 256)*sqrt(x + 1)*sqrt(-x + 1) - 45/64*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x)
Timed out. \[ \int (1-x)^{9/2} (1+x)^{5/2} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.58 \[ \int (1-x)^{9/2} (1+x)^{5/2} \, dx=-\frac {1}{8} \, {\left (-x^{2} + 1\right )}^{\frac {7}{2}} x + \frac {2}{7} \, {\left (-x^{2} + 1\right )}^{\frac {7}{2}} + \frac {3}{16} \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}} x + \frac {15}{64} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + \frac {45}{128} \, \sqrt {-x^{2} + 1} x + \frac {45}{128} \, \arcsin \left (x\right ) \]
-1/8*(-x^2 + 1)^(7/2)*x + 2/7*(-x^2 + 1)^(7/2) + 3/16*(-x^2 + 1)^(5/2)*x + 15/64*(-x^2 + 1)^(3/2)*x + 45/128*sqrt(-x^2 + 1)*x + 45/128*arcsin(x)
Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (78) = 156\).
Time = 0.37 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.69 \[ \int (1-x)^{9/2} (1+x)^{5/2} \, dx=\frac {1}{13440} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (6 \, {\left (7 \, x - 50\right )} {\left (x + 1\right )} + 1219\right )} {\left (x + 1\right )} - 12463\right )} {\left (x + 1\right )} + 64233\right )} {\left (x + 1\right )} - 53963\right )} {\left (x + 1\right )} + 59465\right )} {\left (x + 1\right )} - 23205\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{1680} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (6 \, x - 37\right )} {\left (x + 1\right )} + 661\right )} {\left (x + 1\right )} - 4551\right )} {\left (x + 1\right )} + 4781\right )} {\left (x + 1\right )} - 6335\right )} {\left (x + 1\right )} + 2835\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{80} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, x - 26\right )} {\left (x + 1\right )} + 321\right )} {\left (x + 1\right )} - 451\right )} {\left (x + 1\right )} + 745\right )} {\left (x + 1\right )} - 405\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{40} \, {\left ({\left (2 \, {\left (3 \, {\left (4 \, x - 17\right )} {\left (x + 1\right )} + 133\right )} {\left (x + 1\right )} - 295\right )} {\left (x + 1\right )} + 195\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{8} \, {\left ({\left (2 \, {\left (3 \, x - 10\right )} {\left (x + 1\right )} + 43\right )} {\left (x + 1\right )} - 39\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{2} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{2} \, \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} + \sqrt {x + 1} \sqrt {-x + 1} + \frac {45}{64} \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
1/13440*((2*((4*(5*(6*(7*x - 50)*(x + 1) + 1219)*(x + 1) - 12463)*(x + 1) + 64233)*(x + 1) - 53963)*(x + 1) + 59465)*(x + 1) - 23205)*sqrt(x + 1)*sq rt(-x + 1) - 1/1680*((2*((4*(5*(6*x - 37)*(x + 1) + 661)*(x + 1) - 4551)*( x + 1) + 4781)*(x + 1) - 6335)*(x + 1) + 2835)*sqrt(x + 1)*sqrt(-x + 1) - 1/80*((2*((4*(5*x - 26)*(x + 1) + 321)*(x + 1) - 451)*(x + 1) + 745)*(x + 1) - 405)*sqrt(x + 1)*sqrt(-x + 1) + 1/40*((2*(3*(4*x - 17)*(x + 1) + 133) *(x + 1) - 295)*(x + 1) + 195)*sqrt(x + 1)*sqrt(-x + 1) + 1/8*((2*(3*x - 1 0)*(x + 1) + 43)*(x + 1) - 39)*sqrt(x + 1)*sqrt(-x + 1) - 1/2*((2*x - 5)*( x + 1) + 9)*sqrt(x + 1)*sqrt(-x + 1) - 1/2*sqrt(x + 1)*(x - 2)*sqrt(-x + 1 ) + sqrt(x + 1)*sqrt(-x + 1) + 45/64*arcsin(1/2*sqrt(2)*sqrt(x + 1))
Timed out. \[ \int (1-x)^{9/2} (1+x)^{5/2} \, dx=\int {\left (1-x\right )}^{9/2}\,{\left (x+1\right )}^{5/2} \,d x \]