Integrand size = 20, antiderivative size = 44 \[ \int \frac {x^5}{\sqrt {1-x} \sqrt {1+x}} \, dx=-\sqrt {1-x^2}+\frac {2}{3} \left (1-x^2\right )^{3/2}-\frac {1}{5} \left (1-x^2\right )^{5/2} \] Output:
-(-x^2+1)^(1/2)+2/3*(-x^2+1)^(3/2)-1/5*(-x^2+1)^(5/2)
Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.61 \[ \int \frac {x^5}{\sqrt {1-x} \sqrt {1+x}} \, dx=\frac {1}{15} \sqrt {1-x^2} \left (-8-4 x^2-3 x^4\right ) \] Input:
Integrate[x^5/(Sqrt[1 - x]*Sqrt[1 + x]),x]
Output:
(Sqrt[1 - x^2]*(-8 - 4*x^2 - 3*x^4))/15
Time = 0.16 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.64, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {111, 27, 111, 27, 83}
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.
\(\displaystyle \int \frac {x^5}{\sqrt {1-x} \sqrt {x+1}} \, dx\) |
\(\Big \downarrow \) 111 |
\(\displaystyle -\frac {1}{5} \int -\frac {4 x^3}{\sqrt {1-x} \sqrt {x+1}}dx-\frac {1}{5} \sqrt {1-x} \sqrt {x+1} x^4\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {4}{5} \int \frac {x^3}{\sqrt {1-x} \sqrt {x+1}}dx-\frac {1}{5} \sqrt {1-x} x^4 \sqrt {x+1}\) |
\(\Big \downarrow \) 111 |
\(\displaystyle \frac {4}{5} \left (-\frac {1}{3} \int -\frac {2 x}{\sqrt {1-x} \sqrt {x+1}}dx-\frac {1}{3} \sqrt {1-x} \sqrt {x+1} x^2\right )-\frac {1}{5} \sqrt {1-x} x^4 \sqrt {x+1}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {4}{5} \left (\frac {2}{3} \int \frac {x}{\sqrt {1-x} \sqrt {x+1}}dx-\frac {1}{3} \sqrt {1-x} x^2 \sqrt {x+1}\right )-\frac {1}{5} \sqrt {1-x} x^4 \sqrt {x+1}\) |
\(\Big \downarrow \) 83 |
\(\displaystyle \frac {4}{5} \left (-\frac {1}{3} \sqrt {1-x} \sqrt {x+1} x^2-\frac {2}{3} \sqrt {1-x} \sqrt {x+1}\right )-\frac {1}{5} \sqrt {1-x} x^4 \sqrt {x+1}\) |
Input:
Int[x^5/(Sqrt[1 - x]*Sqrt[1 + x]),x]
Output:
-1/5*(Sqrt[1 - x]*x^4*Sqrt[1 + x]) + (4*((-2*Sqrt[1 - x]*Sqrt[1 + x])/3 - (Sqrt[1 - x]*x^2*Sqrt[1 + x])/3))/5
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && EqQ[a*d*f *(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1)) Int[(a + b*x) ^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & & GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
Time = 0.17 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.61
method | result | size |
gosper | \(-\frac {\sqrt {1-x}\, \sqrt {1+x}\, \left (3 x^{4}+4 x^{2}+8\right )}{15}\) | \(27\) |
default | \(-\frac {\sqrt {1-x}\, \sqrt {1+x}\, \left (3 x^{4}+4 x^{2}+8\right )}{15}\) | \(27\) |
orering | \(\frac {\sqrt {1+x}\, \left (-1+x \right ) \left (3 x^{4}+4 x^{2}+8\right )}{15 \sqrt {1-x}}\) | \(30\) |
risch | \(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \sqrt {1+x}\, \left (3 x^{4}+4 x^{2}+8\right ) \left (-1+x \right )}{15 \sqrt {1-x}\, \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\) | \(51\) |
Input:
int(x^5/(1-x)^(1/2)/(1+x)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/15*(1-x)^(1/2)*(1+x)^(1/2)*(3*x^4+4*x^2+8)
Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.59 \[ \int \frac {x^5}{\sqrt {1-x} \sqrt {1+x}} \, dx=-\frac {1}{15} \, {\left (3 \, x^{4} + 4 \, x^{2} + 8\right )} \sqrt {x + 1} \sqrt {-x + 1} \] Input:
integrate(x^5/(1-x)^(1/2)/(1+x)^(1/2),x, algorithm="fricas")
Output:
-1/15*(3*x^4 + 4*x^2 + 8)*sqrt(x + 1)*sqrt(-x + 1)
Result contains complex when optimal does not.
Time = 2.34 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.14 \[ \int \frac {x^5}{\sqrt {1-x} \sqrt {1+x}} \, dx=- \frac {i {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {9}{4}, - \frac {7}{4} & -2, -2, - \frac {3}{2}, 1 \\- \frac {5}{2}, - \frac {9}{4}, -2, - \frac {7}{4}, - \frac {3}{2}, 0 & \end {matrix} \middle | {\frac {1}{x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} - \frac {{G_{6, 6}^{2, 6}\left (\begin {matrix} -3, - \frac {11}{4}, - \frac {5}{2}, - \frac {9}{4}, -2, 1 & \\- \frac {11}{4}, - \frac {9}{4} & -3, - \frac {5}{2}, - \frac {5}{2}, 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} \] Input:
integrate(x**5/(1-x)**(1/2)/(1+x)**(1/2),x)
Output:
-I*meijerg(((-9/4, -7/4), (-2, -2, -3/2, 1)), ((-5/2, -9/4, -2, -7/4, -3/2 , 0), ()), x**(-2))/(4*pi**(3/2)) - meijerg(((-3, -11/4, -5/2, -9/4, -2, 1 ), ()), ((-11/4, -9/4), (-3, -5/2, -5/2, 0)), exp_polar(-2*I*pi)/x**2)/(4* pi**(3/2))
Time = 0.11 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.91 \[ \int \frac {x^5}{\sqrt {1-x} \sqrt {1+x}} \, dx=-\frac {1}{5} \, \sqrt {-x^{2} + 1} x^{4} - \frac {4}{15} \, \sqrt {-x^{2} + 1} x^{2} - \frac {8}{15} \, \sqrt {-x^{2} + 1} \] Input:
integrate(x^5/(1-x)^(1/2)/(1+x)^(1/2),x, algorithm="maxima")
Output:
-1/5*sqrt(-x^2 + 1)*x^4 - 4/15*sqrt(-x^2 + 1)*x^2 - 8/15*sqrt(-x^2 + 1)
Time = 0.13 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.82 \[ \int \frac {x^5}{\sqrt {1-x} \sqrt {1+x}} \, dx=-\frac {1}{15} \, {\left ({\left ({\left (3 \, {\left (x + 1\right )} {\left (x - 3\right )} + 22\right )} {\left (x + 1\right )} - 20\right )} {\left (x + 1\right )} + 15\right )} \sqrt {x + 1} \sqrt {-x + 1} \] Input:
integrate(x^5/(1-x)^(1/2)/(1+x)^(1/2),x, algorithm="giac")
Output:
-1/15*(((3*(x + 1)*(x - 3) + 22)*(x + 1) - 20)*(x + 1) + 15)*sqrt(x + 1)*s qrt(-x + 1)
Time = 1.86 (sec) , antiderivative size = 317, normalized size of antiderivative = 7.20 \[ \int \frac {x^5}{\sqrt {1-x} \sqrt {1+x}} \, dx=\frac {{\left (\sqrt {x+1}-1\right )}^6\,{\left (\sqrt {1-x}-1\right )}^6\,\left (\frac {296\,x\,\sqrt {x+1}}{5}-\frac {752\,x\,{\left (x+1\right )}^{3/2}}{15}+8\,x\,{\left (x+1\right )}^{5/2}-\frac {32\,\sqrt {1-x}\,\sqrt {x+1}}{5}+\frac {224\,\sqrt {1-x}\,{\left (x+1\right )}^{3/2}}{15}+\frac {224\,{\left (1-x\right )}^{3/2}\,\sqrt {x+1}}{15}+16\,\sqrt {1-x}\,{\left (x+1\right )}^{5/2}-\frac {192\,{\left (1-x\right )}^{3/2}\,{\left (x+1\right )}^{3/2}}{5}+16\,{\left (1-x\right )}^{5/2}\,\sqrt {x+1}-\frac {96\,\sqrt {x+1}}{5}+\frac {304\,{\left (x+1\right )}^{3/2}}{5}+\frac {64\,{\left (x+1\right )}^{5/2}}{3}+\frac {16\,{\left (x+1\right )}^{7/2}}{3}-\frac {296\,x\,\sqrt {1-x}}{5}+\frac {752\,x\,{\left (1-x\right )}^{3/2}}{15}-8\,x\,{\left (1-x\right )}^{5/2}-\frac {192\,x^2\,\sqrt {x+1}}{5}+\frac {8\,x^3\,\sqrt {x+1}}{3}+\frac {48\,x^2\,{\left (x+1\right )}^{3/2}}{5}-\frac {96\,\sqrt {1-x}}{5}+\frac {304\,{\left (1-x\right )}^{3/2}}{5}+\frac {64\,{\left (1-x\right )}^{5/2}}{3}+\frac {16\,{\left (1-x\right )}^{7/2}}{3}-\frac {192\,x^2\,\sqrt {1-x}}{5}-\frac {8\,x^3\,\sqrt {1-x}}{3}+\frac {48\,x^2\,{\left (1-x\right )}^{3/2}}{5}-\frac {1952\,x^2}{15}-\frac {32\,x^4}{5}-\frac {768}{5}\right )}{{\left (\sqrt {x+1}+\sqrt {1-x}-2\right )}^{10}} \] Input:
int(x^5/((1 - x)^(1/2)*(x + 1)^(1/2)),x)
Output:
(((x + 1)^(1/2) - 1)^6*((1 - x)^(1/2) - 1)^6*((296*x*(x + 1)^(1/2))/5 - (7 52*x*(x + 1)^(3/2))/15 + 8*x*(x + 1)^(5/2) - (32*(1 - x)^(1/2)*(x + 1)^(1/ 2))/5 + (224*(1 - x)^(1/2)*(x + 1)^(3/2))/15 + (224*(1 - x)^(3/2)*(x + 1)^ (1/2))/15 + 16*(1 - x)^(1/2)*(x + 1)^(5/2) - (192*(1 - x)^(3/2)*(x + 1)^(3 /2))/5 + 16*(1 - x)^(5/2)*(x + 1)^(1/2) - (96*(x + 1)^(1/2))/5 + (304*(x + 1)^(3/2))/5 + (64*(x + 1)^(5/2))/3 + (16*(x + 1)^(7/2))/3 - (296*x*(1 - x )^(1/2))/5 + (752*x*(1 - x)^(3/2))/15 - 8*x*(1 - x)^(5/2) - (192*x^2*(x + 1)^(1/2))/5 + (8*x^3*(x + 1)^(1/2))/3 + (48*x^2*(x + 1)^(3/2))/5 - (96*(1 - x)^(1/2))/5 + (304*(1 - x)^(3/2))/5 + (64*(1 - x)^(5/2))/3 + (16*(1 - x) ^(7/2))/3 - (192*x^2*(1 - x)^(1/2))/5 - (8*x^3*(1 - x)^(1/2))/3 + (48*x^2* (1 - x)^(3/2))/5 - (1952*x^2)/15 - (32*x^4)/5 - 768/5))/((x + 1)^(1/2) + ( 1 - x)^(1/2) - 2)^10
Time = 0.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.55 \[ \int \frac {x^5}{\sqrt {1-x} \sqrt {1+x}} \, dx=\frac {\sqrt {x +1}\, \sqrt {1-x}\, \left (-3 x^{4}-4 x^{2}-8\right )}{15} \] Input:
int(x^5/(1-x)^(1/2)/(1+x)^(1/2),x)
Output:
(sqrt(x + 1)*sqrt( - x + 1)*( - 3*x**4 - 4*x**2 - 8))/15