Integrand size = 17, antiderivative size = 89 \[ \int (1-x)^{7/2} (1+x)^{3/2} \, dx=\frac {7}{16} \sqrt {1-x} x \sqrt {1+x}+\frac {7}{24} (1-x)^{3/2} x (1+x)^{3/2}+\frac {7}{30} (1-x)^{5/2} (1+x)^{5/2}+\frac {1}{6} (1-x)^{7/2} (1+x)^{5/2}+\frac {7 \arcsin (x)}{16} \]
7/24*(1-x)^(3/2)*x*(1+x)^(3/2)+7/30*(1-x)^(5/2)*(1+x)^(5/2)+1/6*(1-x)^(7/2 )*(1+x)^(5/2)+7/16*arcsin(x)+7/16*x*(1-x)^(1/2)*(1+x)^(1/2)
Time = 0.15 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.71 \[ \int (1-x)^{7/2} (1+x)^{3/2} \, dx=\frac {1}{240} \sqrt {1-x^2} \left (96+135 x-192 x^2+10 x^3+96 x^4-40 x^5\right )-\frac {7}{8} \arctan \left (\frac {\sqrt {1-x^2}}{-1+x}\right ) \]
(Sqrt[1 - x^2]*(96 + 135*x - 192*x^2 + 10*x^3 + 96*x^4 - 40*x^5))/240 - (7 *ArcTan[Sqrt[1 - x^2]/(-1 + x)])/8
Time = 0.17 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {59, 50, 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.
\(\displaystyle \int (1-x)^{7/2} (x+1)^{3/2} \, dx\) |
\(\Big \downarrow \) 59 |
\(\displaystyle \frac {7}{6} \int (1-x)^{5/2} (x+1)^{3/2}dx+\frac {1}{6} (x+1)^{5/2} (1-x)^{7/2}\) |
\(\Big \downarrow \) 50 |
\(\displaystyle \frac {7}{6} \left (\int \left (1-x^2\right )^{3/2}dx+\frac {1}{5} \left (1-x^2\right )^{5/2}\right )+\frac {1}{6} (x+1)^{5/2} (1-x)^{7/2}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {7}{6} \left (\frac {3}{4} \int \sqrt {1-x^2}dx+\frac {1}{5} \left (1-x^2\right )^{5/2}+\frac {1}{4} x \left (1-x^2\right )^{3/2}\right )+\frac {1}{6} (x+1)^{5/2} (1-x)^{7/2}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {7}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-x^2}}dx+\frac {1}{2} \sqrt {1-x^2} x\right )+\frac {1}{5} \left (1-x^2\right )^{5/2}+\frac {1}{4} x \left (1-x^2\right )^{3/2}\right )+\frac {1}{6} (x+1)^{5/2} (1-x)^{7/2}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {7}{6} \left (\frac {3}{4} \left (\frac {\arcsin (x)}{2}+\frac {1}{2} \sqrt {1-x^2} x\right )+\frac {1}{5} \left (1-x^2\right )^{5/2}+\frac {1}{4} x \left (1-x^2\right )^{3/2}\right )+\frac {1}{6} (x+1)^{5/2} (1-x)^{7/2}\) |
((1 - x)^(7/2)*(1 + x)^(5/2))/6 + (7*((x*(1 - x^2)^(3/2))/4 + (1 - x^2)^(5 /2)/5 + (3*((x*Sqrt[1 - x^2])/2 + ArcSin[x]/2))/4))/6
3.11.76.3.1 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 = 92, normalized size of antiderivative = 1.03
method | result | size |
risch | \(\frac {\left (40 x^{5}-96 x^{4}-10 x^{3}+192 x^{2}-135 x -96\right ) \left (-1+x \right ) \sqrt {1+x}\, \sqrt {\left (1+x \right ) \left (1-x \right )}}{240 \sqrt {-\left (-1+x \right ) \left (1+x \right )}\, \sqrt {1-x}}+\frac {7 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{16 \sqrt {1+x}\, \sqrt {1-x}}\) | \(92\) |
default | \(\frac {\left (1-x \right )^{\frac {7}{2}} \left (1+x \right )^{\frac {5}{2}}}{6}+\frac {7 \left (1-x \right )^{\frac {5}{2}} \left (1+x \right )^{\frac {5}{2}}}{30}+\frac {7 \left (1-x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {5}{2}}}{24}+\frac {7 \sqrt {1-x}\, \left (1+x \right )^{\frac {5}{2}}}{24}-\frac {7 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}{48}-\frac {7 \sqrt {1-x}\, \sqrt {1+x}}{16}+\frac {7 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{16 \sqrt {1+x}\, \sqrt {1-x}}\) | \(113\) |
1/240*(40*x^5-96*x^4-10*x^3+192*x^2-135*x-96)*(-1+x)*(1+x)^(1/2)/(-(-1+x)* (1+x))^(1/2)*((1+x)*(1-x))^(1/2)/(1-x)^(1/2)+7/16*((1+x)*(1-x))^(1/2)/(1+x )^(1/2)/(1-x)^(1/2)*arcsin(x)
Time = 0.22 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.70 \[ \int (1-x)^{7/2} (1+x)^{3/2} \, dx=-\frac {1}{240} \, {\left (40 \, x^{5} - 96 \, x^{4} - 10 \, x^{3} + 192 \, x^{2} - 135 \, x - 96\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {7}{8} \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \]
-1/240*(40*x^5 - 96*x^4 - 10*x^3 + 192*x^2 - 135*x - 96)*sqrt(x + 1)*sqrt( -x + 1) - 7/8*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x)
Result contains complex when optimal does not.
Time = 164.06 (sec) , antiderivative size = 287, normalized size of antiderivative = 3.22 \[ \int (1-x)^{7/2} (1+x)^{3/2} \, dx=\begin {cases} - \frac {7 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{8} - \frac {i \left (x + 1\right )^{\frac {13}{2}}}{6 \sqrt {x - 1}} + \frac {47 i \left (x + 1\right )^{\frac {11}{2}}}{30 \sqrt {x - 1}} - \frac {683 i \left (x + 1\right )^{\frac {9}{2}}}{120 \sqrt {x - 1}} + \frac {1151 i \left (x + 1\right )^{\frac {7}{2}}}{120 \sqrt {x - 1}} - \frac {1543 i \left (x + 1\right )^{\frac {5}{2}}}{240 \sqrt {x - 1}} - \frac {7 i \left (x + 1\right )^{\frac {3}{2}}}{48 \sqrt {x - 1}} + \frac {7 i \sqrt {x + 1}}{8 \sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\\frac {7 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{8} + \frac {\left (x + 1\right )^{\frac {13}{2}}}{6 \sqrt {1 - x}} - \frac {47 \left (x + 1\right )^{\frac {11}{2}}}{30 \sqrt {1 - x}} + \frac {683 \left (x + 1\right )^{\frac {9}{2}}}{120 \sqrt {1 - x}} - \frac {1151 \left (x + 1\right )^{\frac {7}{2}}}{120 \sqrt {1 - x}} + \frac {1543 \left (x + 1\right )^{\frac {5}{2}}}{240 \sqrt {1 - x}} + \frac {7 \left (x + 1\right )^{\frac {3}{2}}}{48 \sqrt {1 - x}} - \frac {7 \sqrt {x + 1}}{8 \sqrt {1 - x}} & \text {otherwise} \end {cases} \]
Piecewise((-7*I*acosh(sqrt(2)*sqrt(x + 1)/2)/8 - I*(x + 1)**(13/2)/(6*sqrt (x - 1)) + 47*I*(x + 1)**(11/2)/(30*sqrt(x - 1)) - 683*I*(x + 1)**(9/2)/(1 20*sqrt(x - 1)) + 1151*I*(x + 1)**(7/2)/(120*sqrt(x - 1)) - 1543*I*(x + 1) **(5/2)/(240*sqrt(x - 1)) - 7*I*(x + 1)**(3/2)/(48*sqrt(x - 1)) + 7*I*sqrt (x + 1)/(8*sqrt(x - 1)), Abs(x + 1) > 2), (7*asin(sqrt(2)*sqrt(x + 1)/2)/8 + (x + 1)**(13/2)/(6*sqrt(1 - x)) - 47*(x + 1)**(11/2)/(30*sqrt(1 - x)) + 683*(x + 1)**(9/2)/(120*sqrt(1 - x)) - 1151*(x + 1)**(7/2)/(120*sqrt(1 - x)) + 1543*(x + 1)**(5/2)/(240*sqrt(1 - x)) + 7*(x + 1)**(3/2)/(48*sqrt(1 - x)) - 7*sqrt(x + 1)/(8*sqrt(1 - x)), True))
Time = 0.30 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.58 \[ \int (1-x)^{7/2} (1+x)^{3/2} \, dx=-\frac {1}{6} \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}} x + \frac {2}{5} \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}} + \frac {7}{24} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + \frac {7}{16} \, \sqrt {-x^{2} + 1} x + \frac {7}{16} \, \arcsin \left (x\right ) \]
-1/6*(-x^2 + 1)^(5/2)*x + 2/5*(-x^2 + 1)^(5/2) + 7/24*(-x^2 + 1)^(3/2)*x + 7/16*sqrt(-x^2 + 1)*x + 7/16*arcsin(x)
Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (63) = 126\).
Time = 0.34 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.08 \[ \int (1-x)^{7/2} (1+x)^{3/2} \, dx=-\frac {1}{240} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, x - 26\right )} {\left (x + 1\right )} + 321\right )} {\left (x + 1\right )} - 451\right )} {\left (x + 1\right )} + 745\right )} {\left (x + 1\right )} - 405\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{120} \, {\left ({\left (2 \, {\left (3 \, {\left (4 \, x - 17\right )} {\left (x + 1\right )} + 133\right )} {\left (x + 1\right )} - 295\right )} {\left (x + 1\right )} + 195\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{12} \, {\left ({\left (2 \, {\left (3 \, x - 10\right )} {\left (x + 1\right )} + 43\right )} {\left (x + 1\right )} - 39\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{3} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{2} \, \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} + \sqrt {x + 1} \sqrt {-x + 1} + \frac {7}{8} \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
-1/240*((2*((4*(5*x - 26)*(x + 1) + 321)*(x + 1) - 451)*(x + 1) + 745)*(x + 1) - 405)*sqrt(x + 1)*sqrt(-x + 1) + 1/120*((2*(3*(4*x - 17)*(x + 1) + 1 33)*(x + 1) - 295)*(x + 1) + 195)*sqrt(x + 1)*sqrt(-x + 1) + 1/12*((2*(3*x - 10)*(x + 1) + 43)*(x + 1) - 39)*sqrt(x + 1)*sqrt(-x + 1) - 1/3*((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) + 7/8*arcsin(1/2*sqrt(2)*sqrt(x + 1))
Timed out. \[ \int (1-x)^{7/2} (1+x)^{3/2} \, dx=\int {\left (1-x\right )}^{7/2}\,{\left (x+1\right )}^{3/2} \,d x \]