\(\int \frac {1}{\sqrt [4]{1-x} (e x)^{11/2} \sqrt [4]{1+x}} \, dx\) [917]

Optimal result

Integrand size = 24, antiderivative size = 95 \[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{11/2} \sqrt [4]{1+x}} \, dx=-\frac {2 \left (1-x^2\right )^{3/4}}{9 e (e x)^{9/2}}-\frac {4 \left (1-x^2\right )^{3/4}}{15 e^3 (e x)^{5/2}}-\frac {8 \sqrt [4]{1-\frac {1}{x^2}} \sqrt {e x} E\left (\left .\frac {1}{2} \csc ^{-1}(x)\right |2\right )}{15 e^6 \sqrt [4]{1-x^2}} \]

[Out]

-2/9*(-x^2+1)^(3/4)/e/(e*x)^(9/2)-4/15*(-x^2+1)^(3/4)/e^3/(e*x)^(5/2)-8/15*(1-1/x^2)^(1/4)*(cos(1/2*arccsc(x))
^2)^(1/2)/cos(1/2*arccsc(x))*EllipticE(sin(1/2*arccsc(x)),2^(1/2))*(e*x)^(1/2)/e^6/(-x^2+1)^(1/4)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {126, 331, 323, 342, 234} \[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{11/2} \sqrt [4]{1+x}} \, dx=-\frac {8 \sqrt [4]{1-\frac {1}{x^2}} \sqrt {e x} E\left (\left .\frac {1}{2} \csc ^{-1}(x)\right |2\right )}{15 e^6 \sqrt [4]{1-x^2}}-\frac {4 \left (1-x^2\right )^{3/4}}{15 e^3 (e x)^{5/2}}-\frac {2 \left (1-x^2\right )^{3/4}}{9 e (e x)^{9/2}} \]

[In]

Int[1/((1 - x)^(1/4)*(e*x)^(11/2)*(1 + x)^(1/4)),x]

[Out]

(-2*(1 - x^2)^(3/4))/(9*e*(e*x)^(9/2)) - (4*(1 - x^2)^(3/4))/(15*e^3*(e*x)^(5/2)) - (8*(1 - x^(-2))^(1/4)*Sqrt
[e*x]*EllipticE[ArcCsc[x]/2, 2])/(15*e^6*(1 - x^2)^(1/4))

Rule 126

Int[((f_.)*(x_))^(p_.)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> 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]

Rule 234

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]

Rule 323

Int[1/(((c_.)*(x_))^(3/2)*((a_) + (b_.)*(x_)^2)^(1/4)), x_Symbol] :> Dist[Sqrt[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]

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 342

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] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(e x)^{11/2} \sqrt [4]{1-x^2}} \, dx \\ & = -\frac {2 \left (1-x^2\right )^{3/4}}{9 e (e x)^{9/2}}+\frac {2 \int \frac {1}{(e x)^{7/2} \sqrt [4]{1-x^2}} \, dx}{3 e^2} \\ & = -\frac {2 \left (1-x^2\right )^{3/4}}{9 e (e x)^{9/2}}-\frac {4 \left (1-x^2\right )^{3/4}}{15 e^3 (e x)^{5/2}}+\frac {4 \int \frac {1}{(e x)^{3/2} \sqrt [4]{1-x^2}} \, dx}{15 e^4} \\ & = -\frac {2 \left (1-x^2\right )^{3/4}}{9 e (e x)^{9/2}}-\frac {4 \left (1-x^2\right )^{3/4}}{15 e^3 (e x)^{5/2}}+\frac {\left (4 \sqrt [4]{1-\frac {1}{x^2}} \sqrt {e x}\right ) \int \frac {1}{\sqrt [4]{1-\frac {1}{x^2}} x^2} \, dx}{15 e^6 \sqrt [4]{1-x^2}} \\ & = -\frac {2 \left (1-x^2\right )^{3/4}}{9 e (e x)^{9/2}}-\frac {4 \left (1-x^2\right )^{3/4}}{15 e^3 (e x)^{5/2}}-\frac {\left (4 \sqrt [4]{1-\frac {1}{x^2}} \sqrt {e x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1-x^2}} \, dx,x,\frac {1}{x}\right )}{15 e^6 \sqrt [4]{1-x^2}} \\ & = -\frac {2 \left (1-x^2\right )^{3/4}}{9 e (e x)^{9/2}}-\frac {4 \left (1-x^2\right )^{3/4}}{15 e^3 (e x)^{5/2}}-\frac {8 \sqrt [4]{1-\frac {1}{x^2}} \sqrt {e x} E\left (\left .\frac {1}{2} \csc ^{-1}(x)\right |2\right )}{15 e^6 \sqrt [4]{1-x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.26 \[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{11/2} \sqrt [4]{1+x}} \, dx=-\frac {2 x \operatorname {Hypergeometric2F1}\left (-\frac {9}{4},\frac {1}{4},-\frac {5}{4},x^2\right )}{9 (e x)^{11/2}} \]

[In]

Integrate[1/((1 - x)^(1/4)*(e*x)^(11/2)*(1 + x)^(1/4)),x]

[Out]

(-2*x*Hypergeometric2F1[-9/4, 1/4, -5/4, x^2])/(9*(e*x)^(11/2))

Maple [F]

\[\int \frac {1}{\left (1-x \right )^{\frac {1}{4}} \left (e x \right )^{\frac {11}{2}} \left (1+x \right )^{\frac {1}{4}}}d x\]

[In]

int(1/(1-x)^(1/4)/(e*x)^(11/2)/(1+x)^(1/4),x)

[Out]

int(1/(1-x)^(1/4)/(e*x)^(11/2)/(1+x)^(1/4),x)

Fricas [F]

\[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{11/2} \sqrt [4]{1+x}} \, dx=\int { \frac {1}{\left (e x\right )^{\frac {11}{2}} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate(1/(1-x)^(1/4)/(e*x)^(11/2)/(1+x)^(1/4),x, algorithm="fricas")

[Out]

integral(-sqrt(e*x)*(x + 1)^(3/4)*(-x + 1)^(3/4)/(e^6*x^8 - e^6*x^6), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{11/2} \sqrt [4]{1+x}} \, dx=\text {Timed out} \]

[In]

integrate(1/(1-x)**(1/4)/(e*x)**(11/2)/(1+x)**(1/4),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{11/2} \sqrt [4]{1+x}} \, dx=\int { \frac {1}{\left (e x\right )^{\frac {11}{2}} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate(1/(1-x)^(1/4)/(e*x)^(11/2)/(1+x)^(1/4),x, algorithm="maxima")

[Out]

integrate(1/((e*x)^(11/2)*(x + 1)^(1/4)*(-x + 1)^(1/4)), x)

Giac [F]

\[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{11/2} \sqrt [4]{1+x}} \, dx=\int { \frac {1}{\left (e x\right )^{\frac {11}{2}} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate(1/(1-x)^(1/4)/(e*x)^(11/2)/(1+x)^(1/4),x, algorithm="giac")

[Out]

integrate(1/((e*x)^(11/2)*(x + 1)^(1/4)*(-x + 1)^(1/4)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{11/2} \sqrt [4]{1+x}} \, dx=\int \frac {1}{{\left (e\,x\right )}^{11/2}\,{\left (1-x\right )}^{1/4}\,{\left (x+1\right )}^{1/4}} \,d x \]

[In]

int(1/((e*x)^(11/2)*(1 - x)^(1/4)*(x + 1)^(1/4)),x)

[Out]

int(1/((e*x)^(11/2)*(1 - x)^(1/4)*(x + 1)^(1/4)), x)