Integrand size = 24, antiderivative size = 42 \[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{3/2} \sqrt [4]{1+x}} \, dx=-\frac {2 \sqrt [4]{1-\frac {1}{x^2}} \sqrt {e x} E\left (\left .\frac {1}{2} \csc ^{-1}(x)\right |2\right )}{e^2 \sqrt [4]{1-x^2}} \] Output:
-2*(1-1/x^2)^(1/4)*(e*x)^(1/2)*EllipticE(sin(1/2*arccsc(x)),2^(1/2))/e^2/( -x^2+1)^(1/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.55 \[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{3/2} \sqrt [4]{1+x}} \, dx=-\frac {2 x \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{4},\frac {3}{4},x^2\right )}{(e x)^{3/2}} \] Input:
Integrate[1/((1 - x)^(1/4)*(e*x)^(3/2)*(1 + x)^(1/4)),x]
Output:
(-2*x*Hypergeometric2F1[-1/4, 1/4, 3/4, x^2])/(e*x)^(3/2)
Time = 0.17 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {135, 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.
\(\displaystyle \int \frac {1}{\sqrt [4]{1-x} \sqrt [4]{x+1} (e x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 135 |
\(\displaystyle \int \frac {1}{\sqrt [4]{1-x^2} (e x)^{3/2}}dx\) |
\(\Big \downarrow \) 258 |
\(\displaystyle \frac {\sqrt [4]{1-\frac {1}{x^2}} \sqrt {e x} \int \frac {1}{\sqrt [4]{1-\frac {1}{x^2}} x^2}dx}{e^2 \sqrt [4]{1-x^2}}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle -\frac {\sqrt [4]{1-\frac {1}{x^2}} \sqrt {e x} \int \frac {1}{\sqrt [4]{1-\frac {1}{x^2}}}d\frac {1}{x}}{e^2 \sqrt [4]{1-x^2}}\) |
\(\Big \downarrow \) 226 |
\(\displaystyle -\frac {2 \sqrt [4]{1-\frac {1}{x^2}} \sqrt {e x} E\left (\left .\frac {1}{2} \arcsin \left (\frac {1}{x}\right )\right |2\right )}{e^2 \sqrt [4]{1-x^2}}\) |
Input:
Int[1/((1 - x)^(1/4)*(e*x)^(3/2)*(1 + x)^(1/4)),x]
Output:
(-2*(1 - x^(-2))^(1/4)*Sqrt[e*x]*EllipticE[ArcSin[x^(-1)]/2, 2])/(e^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[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]
\[\int \frac {1}{\left (1-x \right )^{\frac {1}{4}} \left (e x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {1}{4}}}d x\]
Input:
int(1/(1-x)^(1/4)/(e*x)^(3/2)/(1+x)^(1/4),x)
Output:
int(1/(1-x)^(1/4)/(e*x)^(3/2)/(1+x)^(1/4),x)
\[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{3/2} \sqrt [4]{1+x}} \, dx=\int { \frac {1}{\left (e x\right )^{\frac {3}{2}} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate(1/(1-x)^(1/4)/(e*x)^(3/2)/(1+x)^(1/4),x, algorithm="fricas")
Output:
integral(-sqrt(e*x)*(x + 1)^(3/4)*(-x + 1)^(3/4)/(e^2*x^4 - e^2*x^2), x)
Result contains complex when optimal does not.
Time = 10.56 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.07 \[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{3/2} \sqrt [4]{1+x}} \, dx=- \frac {i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {7}{8}, \frac {11}{8}, 1 & 1, \frac {5}{4}, \frac {3}{2} \\\frac {1}{2}, \frac {7}{8}, 1, \frac {11}{8}, \frac {3}{2} & 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{x^{2}}} \right )} e^{\frac {i \pi }{4}}}{4 \pi e^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} - \frac {{G_{6, 6}^{2, 6}\left (\begin {matrix} \frac {1}{4}, \frac {3}{8}, \frac {3}{4}, \frac {7}{8}, \frac {5}{4}, 1 & \\\frac {3}{8}, \frac {7}{8} & \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 0 \end {matrix} \middle | {\frac {1}{x^{2}}} \right )}}{4 \pi e^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} \] Input:
integrate(1/(1-x)**(1/4)/(e*x)**(3/2)/(1+x)**(1/4),x)
Output:
-I*meijerg(((7/8, 11/8, 1), (1, 5/4, 3/2)), ((1/2, 7/8, 1, 11/8, 3/2), (0, )), exp_polar(-2*I*pi)/x**2)*exp(I*pi/4)/(4*pi*e**(3/2)*gamma(1/4)) - meij erg(((1/4, 3/8, 3/4, 7/8, 5/4, 1), ()), ((3/8, 7/8), (1/4, 1/2, 3/4, 0)), x**(-2))/(4*pi*e**(3/2)*gamma(1/4))
\[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{3/2} \sqrt [4]{1+x}} \, dx=\int { \frac {1}{\left (e x\right )^{\frac {3}{2}} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate(1/(1-x)^(1/4)/(e*x)^(3/2)/(1+x)^(1/4),x, algorithm="maxima")
Output:
integrate(1/((e*x)^(3/2)*(x + 1)^(1/4)*(-x + 1)^(1/4)), x)
\[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{3/2} \sqrt [4]{1+x}} \, dx=\int { \frac {1}{\left (e x\right )^{\frac {3}{2}} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate(1/(1-x)^(1/4)/(e*x)^(3/2)/(1+x)^(1/4),x, algorithm="giac")
Output:
integrate(1/((e*x)^(3/2)*(x + 1)^(1/4)*(-x + 1)^(1/4)), x)
Timed out. \[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{3/2} \sqrt [4]{1+x}} \, dx=\int \frac {1}{{\left (e\,x\right )}^{3/2}\,{\left (1-x\right )}^{1/4}\,{\left (x+1\right )}^{1/4}} \,d x \] Input:
int(1/((e*x)^(3/2)*(1 - x)^(1/4)*(x + 1)^(1/4)),x)
Output:
int(1/((e*x)^(3/2)*(1 - x)^(1/4)*(x + 1)^(1/4)), x)
\[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{3/2} \sqrt [4]{1+x}} \, dx=-\frac {\sqrt {e}\, \left (\int \frac {\sqrt {x}\, \left (x +1\right )^{\frac {3}{4}} \left (1-x \right )^{\frac {3}{4}}}{x^{4}-x^{2}}d x \right )}{e^{2}} \] Input:
int(1/(1-x)^(1/4)/(e*x)^(3/2)/(1+x)^(1/4),x)
Output:
( - sqrt(e)*int((sqrt(x)*(x + 1)**(3/4)*( - x + 1)**(3/4))/(x**4 - x**2),x ))/e**2