Integrand size = 17, antiderivative size = 63 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{5/2}} \, dx=-5 \sqrt {1-x} \sqrt {1+x}-\frac {10 (1+x)^{3/2}}{3 \sqrt {1-x}}+\frac {2 (1+x)^{5/2}}{3 (1-x)^{3/2}}+5 \arcsin (x) \]
2/3*(1+x)^(5/2)/(1-x)^(3/2)+5*arcsin(x)-10/3*(1+x)^(3/2)/(1-x)^(1/2)-5*(1- x)^(1/2)*(1+x)^(1/2)
Time = 0.09 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.81 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{5/2}} \, dx=-\frac {\sqrt {1-x^2} \left (23-34 x+3 x^2\right )}{3 (-1+x)^2}-10 \arctan \left (\frac {\sqrt {1-x^2}}{-1+x}\right ) \]
Time = 0.15 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {57, 57, 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.
\(\displaystyle \int \frac {(x+1)^{5/2}}{(1-x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 57 |
\(\displaystyle \frac {2 (x+1)^{5/2}}{3 (1-x)^{3/2}}-\frac {5}{3} \int \frac {(x+1)^{3/2}}{(1-x)^{3/2}}dx\) |
\(\Big \downarrow \) 57 |
\(\displaystyle \frac {2 (x+1)^{5/2}}{3 (1-x)^{3/2}}-\frac {5}{3} \left (\frac {2 (x+1)^{3/2}}{\sqrt {1-x}}-3 \int \frac {\sqrt {x+1}}{\sqrt {1-x}}dx\right )\) |
\(\Big \downarrow \) 50 |
\(\displaystyle \frac {2 (x+1)^{5/2}}{3 (1-x)^{3/2}}-\frac {5}{3} \left (\frac {2 (x+1)^{3/2}}{\sqrt {1-x}}-3 \left (\int \frac {1}{\sqrt {1-x^2}}dx-\sqrt {1-x^2}\right )\right )\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {2 (x+1)^{5/2}}{3 (1-x)^{3/2}}-\frac {5}{3} \left (\frac {2 (x+1)^{3/2}}{\sqrt {1-x}}-3 \left (\arcsin (x)-\sqrt {1-x^2}\right )\right )\) |
(2*(1 + x)^(5/2))/(3*(1 - x)^(3/2)) - (5*((2*(1 + x)^(3/2))/Sqrt[1 - x] - 3*(-Sqrt[1 - x^2] + ArcSin[x])))/3
3.11.96.3.1 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[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 = 0.17 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.33
method | result | size |
risch | \(\frac {\left (3 x^{3}-31 x^{2}-11 x +23\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{3 \left (-1+x \right ) \sqrt {-\left (-1+x \right ) \left (1+x \right )}\, \sqrt {1-x}\, \sqrt {1+x}}+\frac {5 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{\sqrt {1+x}\, \sqrt {1-x}}\) | \(84\) |
1/3*(3*x^3-31*x^2-11*x+23)/(-1+x)/(-(-1+x)*(1+x))^(1/2)*((1+x)*(1-x))^(1/2 )/(1-x)^(1/2)/(1+x)^(1/2)+5*((1+x)*(1-x))^(1/2)/(1+x)^(1/2)/(1-x)^(1/2)*ar csin(x)
Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.19 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{5/2}} \, dx=-\frac {23 \, x^{2} + {\left (3 \, x^{2} - 34 \, x + 23\right )} \sqrt {x + 1} \sqrt {-x + 1} + 30 \, {\left (x^{2} - 2 \, x + 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) - 46 \, x + 23}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} \]
-1/3*(23*x^2 + (3*x^2 - 34*x + 23)*sqrt(x + 1)*sqrt(-x + 1) + 30*(x^2 - 2* x + 1)*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) - 46*x + 23)/(x^2 - 2*x + 1)
Result contains complex when optimal does not.
Time = 5.62 (sec) , antiderivative size = 575, normalized size of antiderivative = 9.13 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{5/2}} \, dx=\begin {cases} - \frac {30 i \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}} \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{3 \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}} - 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}}} + \frac {15 \pi \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}}}{3 \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}} - 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}}} + \frac {60 i \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}} \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{3 \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}} - 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}}} - \frac {30 \pi \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}}}{3 \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}} - 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}}} - \frac {3 i \left (x + 1\right )^{15}}{3 \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}} - 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}}} + \frac {40 i \left (x + 1\right )^{14}}{3 \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}} - 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}}} - \frac {60 i \left (x + 1\right )^{13}}{3 \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}} - 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}}} & \text {for}\: \left |{x + 1}\right | > 2 \\\frac {30 \sqrt {1 - x} \left (x + 1\right )^{\frac {27}{2}} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {27}{2}} - 6 \sqrt {1 - x} \left (x + 1\right )^{\frac {25}{2}}} - \frac {60 \sqrt {1 - x} \left (x + 1\right )^{\frac {25}{2}} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {27}{2}} - 6 \sqrt {1 - x} \left (x + 1\right )^{\frac {25}{2}}} + \frac {3 \left (x + 1\right )^{15}}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {27}{2}} - 6 \sqrt {1 - x} \left (x + 1\right )^{\frac {25}{2}}} - \frac {40 \left (x + 1\right )^{14}}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {27}{2}} - 6 \sqrt {1 - x} \left (x + 1\right )^{\frac {25}{2}}} + \frac {60 \left (x + 1\right )^{13}}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {27}{2}} - 6 \sqrt {1 - x} \left (x + 1\right )^{\frac {25}{2}}} & \text {otherwise} \end {cases} \]
Piecewise((-30*I*sqrt(x - 1)*(x + 1)**(27/2)*acosh(sqrt(2)*sqrt(x + 1)/2)/ (3*sqrt(x - 1)*(x + 1)**(27/2) - 6*sqrt(x - 1)*(x + 1)**(25/2)) + 15*pi*sq rt(x - 1)*(x + 1)**(27/2)/(3*sqrt(x - 1)*(x + 1)**(27/2) - 6*sqrt(x - 1)*( x + 1)**(25/2)) + 60*I*sqrt(x - 1)*(x + 1)**(25/2)*acosh(sqrt(2)*sqrt(x + 1)/2)/(3*sqrt(x - 1)*(x + 1)**(27/2) - 6*sqrt(x - 1)*(x + 1)**(25/2)) - 30 *pi*sqrt(x - 1)*(x + 1)**(25/2)/(3*sqrt(x - 1)*(x + 1)**(27/2) - 6*sqrt(x - 1)*(x + 1)**(25/2)) - 3*I*(x + 1)**15/(3*sqrt(x - 1)*(x + 1)**(27/2) - 6 *sqrt(x - 1)*(x + 1)**(25/2)) + 40*I*(x + 1)**14/(3*sqrt(x - 1)*(x + 1)**( 27/2) - 6*sqrt(x - 1)*(x + 1)**(25/2)) - 60*I*(x + 1)**13/(3*sqrt(x - 1)*( x + 1)**(27/2) - 6*sqrt(x - 1)*(x + 1)**(25/2)), Abs(x + 1) > 2), (30*sqrt (1 - x)*(x + 1)**(27/2)*asin(sqrt(2)*sqrt(x + 1)/2)/(3*sqrt(1 - x)*(x + 1) **(27/2) - 6*sqrt(1 - x)*(x + 1)**(25/2)) - 60*sqrt(1 - x)*(x + 1)**(25/2) *asin(sqrt(2)*sqrt(x + 1)/2)/(3*sqrt(1 - x)*(x + 1)**(27/2) - 6*sqrt(1 - x )*(x + 1)**(25/2)) + 3*(x + 1)**15/(3*sqrt(1 - x)*(x + 1)**(27/2) - 6*sqrt (1 - x)*(x + 1)**(25/2)) - 40*(x + 1)**14/(3*sqrt(1 - x)*(x + 1)**(27/2) - 6*sqrt(1 - x)*(x + 1)**(25/2)) + 60*(x + 1)**13/(3*sqrt(1 - x)*(x + 1)**( 27/2) - 6*sqrt(1 - x)*(x + 1)**(25/2)), True))
Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (47) = 94\).
Time = 0.31 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.57 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{5/2}} \, dx=-\frac {{\left (-x^{2} + 1\right )}^{\frac {5}{2}}}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1} - \frac {5 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{3 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac {10 \, \sqrt {-x^{2} + 1}}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {35 \, \sqrt {-x^{2} + 1}}{3 \, {\left (x - 1\right )}} + 5 \, \arcsin \left (x\right ) \]
-(-x^2 + 1)^(5/2)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1) - 5/3*(-x^2 + 1)^(3/2)/( x^3 - 3*x^2 + 3*x - 1) + 10/3*sqrt(-x^2 + 1)/(x^2 - 2*x + 1) + 35/3*sqrt(- x^2 + 1)/(x - 1) + 5*arcsin(x)
Time = 0.31 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.70 \[ \int \frac {(1+x)^{5/2}}{(1-x)^{5/2}} \, dx=-\frac {{\left ({\left (3 \, x - 37\right )} {\left (x + 1\right )} + 60\right )} \sqrt {x + 1} \sqrt {-x + 1}}{3 \, {\left (x - 1\right )}^{2}} + 10 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
-1/3*((3*x - 37)*(x + 1) + 60)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^2 + 10*arc sin(1/2*sqrt(2)*sqrt(x + 1))
Timed out. \[ \int \frac {(1+x)^{5/2}}{(1-x)^{5/2}} \, dx=\int \frac {{\left (x+1\right )}^{5/2}}{{\left (1-x\right )}^{5/2}} \,d x \]