Integrand size = 17, antiderivative size = 103 \[ \int \frac {(1-x)^{9/2}}{(1+x)^{5/2}} \, dx=-\frac {2 (1-x)^{9/2}}{3 (1+x)^{3/2}}+\frac {6 (1-x)^{7/2}}{\sqrt {1+x}}+\frac {105}{2} \sqrt {1-x} \sqrt {1+x}+\frac {35}{2} (1-x)^{3/2} \sqrt {1+x}+7 (1-x)^{5/2} \sqrt {1+x}+\frac {105 \arcsin (x)}{2} \]
-2/3*(1-x)^(9/2)/(1+x)^(3/2)+105/2*arcsin(x)+6*(1-x)^(7/2)/(1+x)^(1/2)+35/ 2*(1-x)^(3/2)*(1+x)^(1/2)+7*(1-x)^(5/2)*(1+x)^(1/2)+105/2*(1-x)^(1/2)*(1+x )^(1/2)
Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.59 \[ \int \frac {(1-x)^{9/2}}{(1+x)^{5/2}} \, dx=\frac {\sqrt {1-x} \left (494+679 x+102 x^2-17 x^3+2 x^4\right )}{6 (1+x)^{3/2}}-105 \arctan \left (\frac {\sqrt {1-x^2}}{-1+x}\right ) \]
(Sqrt[1 - x]*(494 + 679*x + 102*x^2 - 17*x^3 + 2*x^4))/(6*(1 + x)^(3/2)) - 105*ArcTan[Sqrt[1 - x^2]/(-1 + x)]
Time = 0.18 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {57, 57, 60, 60, 50, 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.
\(\displaystyle \int \frac {(1-x)^{9/2}}{(x+1)^{5/2}} \, dx\) |
\(\Big \downarrow \) 57 |
\(\displaystyle -3 \int \frac {(1-x)^{7/2}}{(x+1)^{3/2}}dx-\frac {2 (1-x)^{9/2}}{3 (x+1)^{3/2}}\) |
\(\Big \downarrow \) 57 |
\(\displaystyle -3 \left (-7 \int \frac {(1-x)^{5/2}}{\sqrt {x+1}}dx-\frac {2 (1-x)^{7/2}}{\sqrt {x+1}}\right )-\frac {2 (1-x)^{9/2}}{3 (x+1)^{3/2}}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle -3 \left (-7 \left (\frac {5}{3} \int \frac {(1-x)^{3/2}}{\sqrt {x+1}}dx+\frac {1}{3} \sqrt {x+1} (1-x)^{5/2}\right )-\frac {2 (1-x)^{7/2}}{\sqrt {x+1}}\right )-\frac {2 (1-x)^{9/2}}{3 (x+1)^{3/2}}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle -3 \left (-7 \left (\frac {5}{3} \left (\frac {3}{2} \int \frac {\sqrt {1-x}}{\sqrt {x+1}}dx+\frac {1}{2} \sqrt {x+1} (1-x)^{3/2}\right )+\frac {1}{3} \sqrt {x+1} (1-x)^{5/2}\right )-\frac {2 (1-x)^{7/2}}{\sqrt {x+1}}\right )-\frac {2 (1-x)^{9/2}}{3 (x+1)^{3/2}}\) |
\(\Big \downarrow \) 50 |
\(\displaystyle -3 \left (-7 \left (\frac {5}{3} \left (\frac {3}{2} \left (\int \frac {1}{\sqrt {1-x^2}}dx+\sqrt {1-x^2}\right )+\frac {1}{2} \sqrt {x+1} (1-x)^{3/2}\right )+\frac {1}{3} \sqrt {x+1} (1-x)^{5/2}\right )-\frac {2 (1-x)^{7/2}}{\sqrt {x+1}}\right )-\frac {2 (1-x)^{9/2}}{3 (x+1)^{3/2}}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle -3 \left (-7 \left (\frac {5}{3} \left (\frac {3}{2} \left (\arcsin (x)+\sqrt {1-x^2}\right )+\frac {1}{2} \sqrt {x+1} (1-x)^{3/2}\right )+\frac {1}{3} \sqrt {x+1} (1-x)^{5/2}\right )-\frac {2 (1-x)^{7/2}}{\sqrt {x+1}}\right )-\frac {2 (1-x)^{9/2}}{3 (x+1)^{3/2}}\) |
(-2*(1 - x)^(9/2))/(3*(1 + x)^(3/2)) - 3*((-2*(1 - x)^(7/2))/Sqrt[1 + x] - 7*(((1 - x)^(5/2)*Sqrt[1 + x])/3 + (5*(((1 - x)^(3/2)*Sqrt[1 + x])/2 + (3 *(Sqrt[1 - x^2] + ArcSin[x]))/2))/3))
3.12.26.3.1 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 + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & & GtQ[n, 0] && LtQ[m, -1] && !(IntegerQ[n] && !IntegerQ[m]) && !(ILeQ[m + n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c , d, m, n, x]
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[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
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 = 0.34 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.86
method | result | size |
risch | \(-\frac {\left (2 x^{5}-19 x^{4}+119 x^{3}+577 x^{2}-185 x -494\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{6 \left (1+x \right )^{\frac {3}{2}} \sqrt {-\left (-1+x \right ) \left (1+x \right )}\, \sqrt {1-x}}+\frac {105 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{2 \sqrt {1+x}\, \sqrt {1-x}}\) | \(89\) |
-1/6*(2*x^5-19*x^4+119*x^3+577*x^2-185*x-494)/(1+x)^(3/2)/(-(-1+x)*(1+x))^ (1/2)*((1+x)*(1-x))^(1/2)/(1-x)^(1/2)+105/2*((1+x)*(1-x))^(1/2)/(1+x)^(1/2 )/(1-x)^(1/2)*arcsin(x)
Time = 0.23 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.83 \[ \int \frac {(1-x)^{9/2}}{(1+x)^{5/2}} \, dx=\frac {494 \, x^{2} + {\left (2 \, x^{4} - 17 \, x^{3} + 102 \, x^{2} + 679 \, x + 494\right )} \sqrt {x + 1} \sqrt {-x + 1} - 630 \, {\left (x^{2} + 2 \, x + 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) + 988 \, x + 494}{6 \, {\left (x^{2} + 2 \, x + 1\right )}} \]
1/6*(494*x^2 + (2*x^4 - 17*x^3 + 102*x^2 + 679*x + 494)*sqrt(x + 1)*sqrt(- x + 1) - 630*(x^2 + 2*x + 1)*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) + 98 8*x + 494)/(x^2 + 2*x + 1)
Result contains complex when optimal does not.
Time = 57.29 (sec) , antiderivative size = 248, normalized size of antiderivative = 2.41 \[ \int \frac {(1-x)^{9/2}}{(1+x)^{5/2}} \, dx=\begin {cases} - 105 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} + \frac {i \left (x + 1\right )^{\frac {7}{2}}}{3 \sqrt {x - 1}} - \frac {29 i \left (x + 1\right )^{\frac {5}{2}}}{6 \sqrt {x - 1}} + \frac {215 i \left (x + 1\right )^{\frac {3}{2}}}{6 \sqrt {x - 1}} + \frac {43 i \sqrt {x + 1}}{3 \sqrt {x - 1}} - \frac {448 i}{3 \sqrt {x - 1} \sqrt {x + 1}} + \frac {64 i}{3 \sqrt {x - 1} \left (x + 1\right )^{\frac {3}{2}}} & \text {for}\: \left |{x + 1}\right | > 2 \\105 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {\left (x + 1\right )^{\frac {7}{2}}}{3 \sqrt {1 - x}} + \frac {29 \left (x + 1\right )^{\frac {5}{2}}}{6 \sqrt {1 - x}} - \frac {215 \left (x + 1\right )^{\frac {3}{2}}}{6 \sqrt {1 - x}} - \frac {43 \sqrt {x + 1}}{3 \sqrt {1 - x}} + \frac {448}{3 \sqrt {1 - x} \sqrt {x + 1}} - \frac {64}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
Piecewise((-105*I*acosh(sqrt(2)*sqrt(x + 1)/2) + I*(x + 1)**(7/2)/(3*sqrt( x - 1)) - 29*I*(x + 1)**(5/2)/(6*sqrt(x - 1)) + 215*I*(x + 1)**(3/2)/(6*sq rt(x - 1)) + 43*I*sqrt(x + 1)/(3*sqrt(x - 1)) - 448*I/(3*sqrt(x - 1)*sqrt( x + 1)) + 64*I/(3*sqrt(x - 1)*(x + 1)**(3/2)), Abs(x + 1) > 2), (105*asin( sqrt(2)*sqrt(x + 1)/2) - (x + 1)**(7/2)/(3*sqrt(1 - x)) + 29*(x + 1)**(5/2 )/(6*sqrt(1 - x)) - 215*(x + 1)**(3/2)/(6*sqrt(1 - x)) - 43*sqrt(x + 1)/(3 *sqrt(1 - x)) + 448/(3*sqrt(1 - x)*sqrt(x + 1)) - 64/(3*sqrt(1 - x)*(x + 1 )**(3/2)), True))
Time = 0.30 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.21 \[ \int \frac {(1-x)^{9/2}}{(1+x)^{5/2}} \, dx=\frac {x^{6}}{3 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} - \frac {7 \, x^{5}}{2 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {23 \, x^{4}}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {35}{2} \, x {\left (\frac {3 \, x^{2}}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}}} - \frac {2}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}\right )} - \frac {143 \, x}{6 \, \sqrt {-x^{2} + 1}} - \frac {127 \, x^{2}}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {22 \, x}{3 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {247}{3 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {105}{2} \, \arcsin \left (x\right ) \]
1/3*x^6/(-x^2 + 1)^(3/2) - 7/2*x^5/(-x^2 + 1)^(3/2) + 23*x^4/(-x^2 + 1)^(3 /2) + 35/2*x*(3*x^2/(-x^2 + 1)^(3/2) - 2/(-x^2 + 1)^(3/2)) - 143/6*x/sqrt( -x^2 + 1) - 127*x^2/(-x^2 + 1)^(3/2) + 22/3*x/(-x^2 + 1)^(3/2) + 247/3/(-x ^2 + 1)^(3/2) + 105/2*arcsin(x)
Time = 0.35 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.23 \[ \int \frac {(1-x)^{9/2}}{(1+x)^{5/2}} \, dx=\frac {1}{6} \, {\left ({\left (2 \, x - 23\right )} {\left (x + 1\right )} + 165\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {2 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}}{3 \, {\left (x + 1\right )}^{\frac {3}{2}}} - \frac {34 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}}{\sqrt {x + 1}} + \frac {2 \, {\left (x + 1\right )}^{\frac {3}{2}} {\left (\frac {51 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{2}}{x + 1} - 1\right )}}{3 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}} + 105 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
1/6*((2*x - 23)*(x + 1) + 165)*sqrt(x + 1)*sqrt(-x + 1) + 2/3*(sqrt(2) - s qrt(-x + 1))^3/(x + 1)^(3/2) - 34*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) + 2 /3*(x + 1)^(3/2)*(51*(sqrt(2) - sqrt(-x + 1))^2/(x + 1) - 1)/(sqrt(2) - sq rt(-x + 1))^3 + 105*arcsin(1/2*sqrt(2)*sqrt(x + 1))
Timed out. \[ \int \frac {(1-x)^{9/2}}{(1+x)^{5/2}} \, dx=\int \frac {{\left (1-x\right )}^{9/2}}{{\left (x+1\right )}^{5/2}} \,d x \]