Integrand size = 24, antiderivative size = 60 \[ \int \frac {\sqrt {e x}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx=-\frac {e \left (1-x^2\right )^{3/4}}{\sqrt {e x}}+\frac {\sqrt [4]{1-\frac {1}{x^2}} \sqrt {e x} E\left (\left .\frac {1}{2} \csc ^{-1}(x)\right |2\right )}{\sqrt [4]{1-x^2}} \] Output:
-e*(-x^2+1)^(3/4)/(e*x)^(1/2)+(1-1/x^2)^(1/4)*(e*x)^(1/2)*EllipticE(sin(1/ 2*arccsc(x)),2^(1/2))/(-x^2+1)^(1/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 9.42 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.42 \[ \int \frac {\sqrt {e x}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx=\frac {2}{3} x \sqrt {e x} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{4},\frac {7}{4},x^2\right ) \] Input:
Integrate[Sqrt[e*x]/((1 - x)^(1/4)*(1 + x)^(1/4)),x]
Output:
(2*x*Sqrt[e*x]*Hypergeometric2F1[1/4, 3/4, 7/4, x^2])/3
Time = 0.20 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {135, 256, 258, 858, 226}
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.
\(\displaystyle \int \frac {\sqrt {e x}}{\sqrt [4]{1-x} \sqrt [4]{x+1}} \, dx\) |
\(\Big \downarrow \) 135 |
\(\displaystyle \int \frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}dx\) |
\(\Big \downarrow \) 256 |
\(\displaystyle -\frac {1}{2} e^2 \int \frac {1}{(e x)^{3/2} \sqrt [4]{1-x^2}}dx-\frac {e \left (1-x^2\right )^{3/4}}{\sqrt {e x}}\) |
\(\Big \downarrow \) 258 |
\(\displaystyle -\frac {\sqrt [4]{1-\frac {1}{x^2}} \sqrt {e x} \int \frac {1}{\sqrt [4]{1-\frac {1}{x^2}} x^2}dx}{2 \sqrt [4]{1-x^2}}-\frac {e \left (1-x^2\right )^{3/4}}{\sqrt {e x}}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle \frac {\sqrt [4]{1-\frac {1}{x^2}} \sqrt {e x} \int \frac {1}{\sqrt [4]{1-\frac {1}{x^2}}}d\frac {1}{x}}{2 \sqrt [4]{1-x^2}}-\frac {e \left (1-x^2\right )^{3/4}}{\sqrt {e x}}\) |
\(\Big \downarrow \) 226 |
\(\displaystyle \frac {\sqrt [4]{1-\frac {1}{x^2}} \sqrt {e x} E\left (\left .\frac {1}{2} \arcsin \left (\frac {1}{x}\right )\right |2\right )}{\sqrt [4]{1-x^2}}-\frac {e \left (1-x^2\right )^{3/4}}{\sqrt {e x}}\) |
Input:
Int[Sqrt[e*x]/((1 - x)^(1/4)*(1 + x)^(1/4)),x]
Output:
-((e*(1 - x^2)^(3/4))/Sqrt[e*x]) + ((1 - x^(-2))^(1/4)*Sqrt[e*x]*EllipticE [ArcSin[x^(-1)]/2, 2])/(1 - x^2)^(1/4)
Int[((f_.)*(x_))^(p_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_] :> Int[(a*c + b*d*x^2)^m*(f*x)^p, x] /; FreeQ[{a, b, c, d, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && GtQ[a, 0] && GtQ[c, 0]
Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[(2/(a^(1/4)*Rt[-b/a, 2] ))*EllipticE[(1/2)*ArcSin[Rt[-b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ [a, 0] && NegQ[b/a]
Int[Sqrt[(c_)*(x_)]/((a_) + (b_.)*(x_)^2)^(1/4), x_Symbol] :> Simp[c*((a + b*x^2)^(3/4)/(b*Sqrt[c*x])), x] + Simp[a*(c^2/(2*b)) Int[1/((c*x)^(3/2)*( a + b*x^2)^(1/4)), x], x] /; FreeQ[{a, b, c}, x] && NegQ[b/a]
Int[1/(((c_.)*(x_))^(3/2)*((a_) + (b_.)*(x_)^2)^(1/4)), x_Symbol] :> Simp[S qrt[c*x]*((1 + a/(b*x^2))^(1/4)/(c^2*(a + b*x^2)^(1/4))) Int[1/(x^2*(1 + a/(b*x^2))^(1/4)), x], x] /; FreeQ[{a, b, c}, x] && NegQ[b/a]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && Int egerQ[m]
\[\int \frac {\sqrt {e x}}{\left (1-x \right )^{\frac {1}{4}} \left (1+x \right )^{\frac {1}{4}}}d x\]
Input:
int((e*x)^(1/2)/(1-x)^(1/4)/(1+x)^(1/4),x)
Output:
int((e*x)^(1/2)/(1-x)^(1/4)/(1+x)^(1/4),x)
\[ \int \frac {\sqrt {e x}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx=\int { \frac {\sqrt {e x}}{{\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate((e*x)^(1/2)/(1-x)^(1/4)/(1+x)^(1/4),x, algorithm="fricas")
Output:
integral(-sqrt(e*x)*(x + 1)^(3/4)*(-x + 1)^(3/4)/(x^2 - 1), x)
Result contains complex when optimal does not.
Time = 2.05 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.75 \[ \int \frac {\sqrt {e x}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx=\frac {i \sqrt {e} {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {1}{8}, \frac {3}{8} & 0, \frac {1}{4}, \frac {1}{2}, 1 \\- \frac {1}{2}, - \frac {1}{8}, 0, \frac {3}{8}, \frac {1}{2}, 0 & \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{x^{2}}} \right )} e^{\frac {i \pi }{4}}}{4 \pi \Gamma \left (\frac {1}{4}\right )} - \frac {\sqrt {e} {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {3}{4}, - \frac {5}{8}, - \frac {1}{4}, - \frac {1}{8}, \frac {1}{4}, 1 & \\- \frac {5}{8}, - \frac {1}{8} & - \frac {3}{4}, - \frac {1}{2}, - \frac {1}{4}, 0 \end {matrix} \middle | {\frac {1}{x^{2}}} \right )}}{4 \pi \Gamma \left (\frac {1}{4}\right )} \] Input:
integrate((e*x)**(1/2)/(1-x)**(1/4)/(1+x)**(1/4),x)
Output:
I*sqrt(e)*meijerg(((-1/8, 3/8), (0, 1/4, 1/2, 1)), ((-1/2, -1/8, 0, 3/8, 1 /2, 0), ()), exp_polar(-2*I*pi)/x**2)*exp(I*pi/4)/(4*pi*gamma(1/4)) - sqrt (e)*meijerg(((-3/4, -5/8, -1/4, -1/8, 1/4, 1), ()), ((-5/8, -1/8), (-3/4, -1/2, -1/4, 0)), x**(-2))/(4*pi*gamma(1/4))
\[ \int \frac {\sqrt {e x}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx=\int { \frac {\sqrt {e x}}{{\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate((e*x)^(1/2)/(1-x)^(1/4)/(1+x)^(1/4),x, algorithm="maxima")
Output:
integrate(sqrt(e*x)/((x + 1)^(1/4)*(-x + 1)^(1/4)), x)
\[ \int \frac {\sqrt {e x}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx=\int { \frac {\sqrt {e x}}{{\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate((e*x)^(1/2)/(1-x)^(1/4)/(1+x)^(1/4),x, algorithm="giac")
Output:
integrate(sqrt(e*x)/((x + 1)^(1/4)*(-x + 1)^(1/4)), x)
Timed out. \[ \int \frac {\sqrt {e x}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx=\int \frac {\sqrt {e\,x}}{{\left (1-x\right )}^{1/4}\,{\left (x+1\right )}^{1/4}} \,d x \] Input:
int((e*x)^(1/2)/((1 - x)^(1/4)*(x + 1)^(1/4)),x)
Output:
int((e*x)^(1/2)/((1 - x)^(1/4)*(x + 1)^(1/4)), x)
\[ \int \frac {\sqrt {e x}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx=\sqrt {e}\, \left (\int \frac {\sqrt {x}}{\left (x +1\right )^{\frac {1}{4}} \left (1-x \right )^{\frac {1}{4}}}d x \right ) \] Input:
int((e*x)^(1/2)/(1-x)^(1/4)/(1+x)^(1/4),x)
Output:
sqrt(e)*int(sqrt(x)/((x + 1)**(1/4)*( - x + 1)**(1/4)),x)