Integrand size = 17, antiderivative size = 87 \[ \int \frac {(1-x)^{7/2}}{(1+x)^{5/2}} \, dx=-\frac {2 (1-x)^{7/2}}{3 (1+x)^{3/2}}+\frac {14 (1-x)^{5/2}}{3 \sqrt {1+x}}+\frac {35}{2} \sqrt {1-x} \sqrt {1+x}+\frac {35}{6} (1-x)^{3/2} \sqrt {1+x}+\frac {35 \arcsin (x)}{2} \]
-2/3*(1-x)^(7/2)/(1+x)^(3/2)+35/2*arcsin(x)+14/3*(1-x)^(5/2)/(1+x)^(1/2)+3 5/6*(1-x)^(3/2)*(1+x)^(1/2)+35/2*(1-x)^(1/2)*(1+x)^(1/2)
Time = 0.07 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.64 \[ \int \frac {(1-x)^{7/2}}{(1+x)^{5/2}} \, dx=\frac {\sqrt {1-x} \left (164+229 x+30 x^2-3 x^3\right )}{6 (1+x)^{3/2}}-35 \arctan \left (\frac {\sqrt {1-x^2}}{-1+x}\right ) \]
(Sqrt[1 - x]*(164 + 229*x + 30*x^2 - 3*x^3))/(6*(1 + x)^(3/2)) - 35*ArcTan [Sqrt[1 - x^2]/(-1 + x)]
Time = 0.16 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {57, 57, 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.
\(\displaystyle \int \frac {(1-x)^{7/2}}{(x+1)^{5/2}} \, dx\) |
\(\Big \downarrow \) 57 |
\(\displaystyle -\frac {7}{3} \int \frac {(1-x)^{5/2}}{(x+1)^{3/2}}dx-\frac {2 (1-x)^{7/2}}{3 (x+1)^{3/2}}\) |
\(\Big \downarrow \) 57 |
\(\displaystyle -\frac {7}{3} \left (-5 \int \frac {(1-x)^{3/2}}{\sqrt {x+1}}dx-\frac {2 (1-x)^{5/2}}{\sqrt {x+1}}\right )-\frac {2 (1-x)^{7/2}}{3 (x+1)^{3/2}}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle -\frac {7}{3} \left (-5 \left (\frac {3}{2} \int \frac {\sqrt {1-x}}{\sqrt {x+1}}dx+\frac {1}{2} \sqrt {x+1} (1-x)^{3/2}\right )-\frac {2 (1-x)^{5/2}}{\sqrt {x+1}}\right )-\frac {2 (1-x)^{7/2}}{3 (x+1)^{3/2}}\) |
\(\Big \downarrow \) 50 |
\(\displaystyle -\frac {7}{3} \left (-5 \left (\frac {3}{2} \left (\int \frac {1}{\sqrt {1-x^2}}dx+\sqrt {1-x^2}\right )+\frac {1}{2} \sqrt {x+1} (1-x)^{3/2}\right )-\frac {2 (1-x)^{5/2}}{\sqrt {x+1}}\right )-\frac {2 (1-x)^{7/2}}{3 (x+1)^{3/2}}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle -\frac {7}{3} \left (-5 \left (\frac {3}{2} \left (\arcsin (x)+\sqrt {1-x^2}\right )+\frac {1}{2} \sqrt {x+1} (1-x)^{3/2}\right )-\frac {2 (1-x)^{5/2}}{\sqrt {x+1}}\right )-\frac {2 (1-x)^{7/2}}{3 (x+1)^{3/2}}\) |
(-2*(1 - x)^(7/2))/(3*(1 + x)^(3/2)) - (7*((-2*(1 - x)^(5/2))/Sqrt[1 + x] - 5*(((1 - x)^(3/2)*Sqrt[1 + x])/2 + (3*(Sqrt[1 - x^2] + ArcSin[x]))/2)))/ 3
3.12.27.3.1 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 = 0.18 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.97
method | result | size |
risch | \(\frac {\left (3 x^{4}-33 x^{3}-199 x^{2}+65 x +164\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{6 \left (1+x \right )^{\frac {3}{2}} \sqrt {-\left (-1+x \right ) \left (1+x \right )}\, \sqrt {1-x}}+\frac {35 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{2 \sqrt {1+x}\, \sqrt {1-x}}\) | \(84\) |
1/6*(3*x^4-33*x^3-199*x^2+65*x+164)/(1+x)^(3/2)/(-(-1+x)*(1+x))^(1/2)*((1+ x)*(1-x))^(1/2)/(1-x)^(1/2)+35/2*((1+x)*(1-x))^(1/2)/(1+x)^(1/2)/(1-x)^(1/ 2)*arcsin(x)
Time = 0.23 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.93 \[ \int \frac {(1-x)^{7/2}}{(1+x)^{5/2}} \, dx=\frac {164 \, x^{2} - {\left (3 \, x^{3} - 30 \, x^{2} - 229 \, x - 164\right )} \sqrt {x + 1} \sqrt {-x + 1} - 210 \, {\left (x^{2} + 2 \, x + 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) + 328 \, x + 164}{6 \, {\left (x^{2} + 2 \, x + 1\right )}} \]
1/6*(164*x^2 - (3*x^3 - 30*x^2 - 229*x - 164)*sqrt(x + 1)*sqrt(-x + 1) - 2 10*(x^2 + 2*x + 1)*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) + 328*x + 164) /(x^2 + 2*x + 1)
Result contains complex when optimal does not.
Time = 16.83 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.44 \[ \int \frac {(1-x)^{7/2}}{(1+x)^{5/2}} \, dx=\begin {cases} - 35 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {i \left (x + 1\right )^{\frac {5}{2}}}{2 \sqrt {x - 1}} + \frac {15 i \left (x + 1\right )^{\frac {3}{2}}}{2 \sqrt {x - 1}} + \frac {41 i \sqrt {x + 1}}{3 \sqrt {x - 1}} - \frac {176 i}{3 \sqrt {x - 1} \sqrt {x + 1}} + \frac {32 i}{3 \sqrt {x - 1} \left (x + 1\right )^{\frac {3}{2}}} & \text {for}\: \left |{x + 1}\right | > 2 \\35 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} + \frac {\left (x + 1\right )^{\frac {5}{2}}}{2 \sqrt {1 - x}} - \frac {15 \left (x + 1\right )^{\frac {3}{2}}}{2 \sqrt {1 - x}} - \frac {41 \sqrt {x + 1}}{3 \sqrt {1 - x}} + \frac {176}{3 \sqrt {1 - x} \sqrt {x + 1}} - \frac {32}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
Piecewise((-35*I*acosh(sqrt(2)*sqrt(x + 1)/2) - I*(x + 1)**(5/2)/(2*sqrt(x - 1)) + 15*I*(x + 1)**(3/2)/(2*sqrt(x - 1)) + 41*I*sqrt(x + 1)/(3*sqrt(x - 1)) - 176*I/(3*sqrt(x - 1)*sqrt(x + 1)) + 32*I/(3*sqrt(x - 1)*(x + 1)**( 3/2)), Abs(x + 1) > 2), (35*asin(sqrt(2)*sqrt(x + 1)/2) + (x + 1)**(5/2)/( 2*sqrt(1 - x)) - 15*(x + 1)**(3/2)/(2*sqrt(1 - x)) - 41*sqrt(x + 1)/(3*sqr t(1 - x)) + 176/(3*sqrt(1 - x)*sqrt(x + 1)) - 32/(3*sqrt(1 - x)*(x + 1)**( 3/2)), True))
Time = 0.29 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.28 \[ \int \frac {(1-x)^{7/2}}{(1+x)^{5/2}} \, dx=-\frac {x^{5}}{2 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {6 \, x^{4}}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {35}{6} \, x {\left (\frac {3 \, x^{2}}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}}} - \frac {2}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}\right )} - \frac {61 \, x}{6 \, \sqrt {-x^{2} + 1}} - \frac {44 \, x^{2}}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {16 \, x}{3 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {82}{3 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {35}{2} \, \arcsin \left (x\right ) \]
-1/2*x^5/(-x^2 + 1)^(3/2) + 6*x^4/(-x^2 + 1)^(3/2) + 35/6*x*(3*x^2/(-x^2 + 1)^(3/2) - 2/(-x^2 + 1)^(3/2)) - 61/6*x/sqrt(-x^2 + 1) - 44*x^2/(-x^2 + 1 )^(3/2) + 16/3*x/(-x^2 + 1)^(3/2) + 82/3/(-x^2 + 1)^(3/2) + 35/2*arcsin(x)
Time = 0.35 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.37 \[ \int \frac {(1-x)^{7/2}}{(1+x)^{5/2}} \, dx=-\frac {1}{2} \, \sqrt {x + 1} {\left (x - 12\right )} \sqrt {-x + 1} + \frac {{\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}}{3 \, {\left (x + 1\right )}^{\frac {3}{2}}} - \frac {13 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}}{\sqrt {x + 1}} + \frac {{\left (x + 1\right )}^{\frac {3}{2}} {\left (\frac {39 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{2}}{x + 1} - 1\right )}}{3 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}} + 35 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
-1/2*sqrt(x + 1)*(x - 12)*sqrt(-x + 1) + 1/3*(sqrt(2) - sqrt(-x + 1))^3/(x + 1)^(3/2) - 13*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) + 1/3*(x + 1)^(3/2)* (39*(sqrt(2) - sqrt(-x + 1))^2/(x + 1) - 1)/(sqrt(2) - sqrt(-x + 1))^3 + 3 5*arcsin(1/2*sqrt(2)*sqrt(x + 1))
Timed out. \[ \int \frac {(1-x)^{7/2}}{(1+x)^{5/2}} \, dx=\int \frac {{\left (1-x\right )}^{7/2}}{{\left (x+1\right )}^{5/2}} \,d x \]
Time = 0.00 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.03 \[ \int \frac {(1-x)^{7/2}}{(1+x)^{5/2}} \, dx=\frac {-210 \sqrt {x +1}\, \mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right ) x -210 \sqrt {x +1}\, \mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )-3 \sqrt {1-x}\, x^{3}+30 \sqrt {1-x}\, x^{2}+229 \sqrt {1-x}\, x +164 \sqrt {1-x}}{6 \sqrt {x +1}\, \left (x +1\right )} \]