\(\int \frac {1}{\sqrt [4]{1-x} (e x)^{9/2} \sqrt [4]{1+x}} \, dx\) [911]

Optimal result

Integrand size = 24, antiderivative size = 51 \[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{9/2} \sqrt [4]{1+x}} \, dx=-\frac {2 \left (1-x^2\right )^{3/4}}{3 e (e x)^{7/2}}+\frac {8 \left (1-x^2\right )^{7/4}}{21 e (e x)^{7/2}} \]

[Out]

-2/3*(-x^2+1)^(3/4)/e/(e*x)^(7/2)+8/21*(-x^2+1)^(7/4)/e/(e*x)^(7/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {126, 279, 270} \[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{9/2} \sqrt [4]{1+x}} \, dx=\frac {8 \left (1-x^2\right )^{7/4}}{21 e (e x)^{7/2}}-\frac {2 \left (1-x^2\right )^{3/4}}{3 e (e x)^{7/2}} \]

[In]

Int[1/((1 - x)^(1/4)*(e*x)^(9/2)*(1 + x)^(1/4)),x]

[Out]

(-2*(1 - x^2)^(3/4))/(3*e*(e*x)^(7/2)) + (8*(1 - x^2)^(7/4))/(21*e*(e*x)^(7/2))

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 270

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] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 279

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(e x)^{9/2} \sqrt [4]{1-x^2}} \, dx \\ & = -\frac {2 \left (1-x^2\right )^{3/4}}{3 e (e x)^{7/2}}-\frac {4}{3} \int \frac {\left (1-x^2\right )^{3/4}}{(e x)^{9/2}} \, dx \\ & = -\frac {2 \left (1-x^2\right )^{3/4}}{3 e (e x)^{7/2}}+\frac {8 \left (1-x^2\right )^{7/4}}{21 e (e x)^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.59 \[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{9/2} \sqrt [4]{1+x}} \, dx=-\frac {2 x \left (1-x^2\right )^{3/4} \left (3+4 x^2\right )}{21 (e x)^{9/2}} \]

[In]

Integrate[1/((1 - x)^(1/4)*(e*x)^(9/2)*(1 + x)^(1/4)),x]

[Out]

(-2*x*(1 - x^2)^(3/4)*(3 + 4*x^2))/(21*(e*x)^(9/2))

Maple [A] (verified)

Time = 1.85 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.55

method	result	size
gosper	\(-\frac {2 x \left (4 x^{2}+3\right ) \left (1-x \right )^{\frac {3}{4}} \left (1+x \right )^{\frac {3}{4}}}{21 \left (e x \right )^{\frac {9}{2}}}\)	\(28\)
risch	\(\frac {2 \left (e^{2} x^{2} \left (1-x \right ) \left (1+x \right )\right )^{\frac {1}{4}} \left (1+x \right )^{\frac {3}{4}} \left (-1+x \right ) \left (4 x^{2}+3\right )}{21 \sqrt {e x}\, \left (1-x \right )^{\frac {1}{4}} e^{4} x^{3} \left (-e^{2} x^{2} \left (-1+x \right ) \left (1+x \right )\right )^{\frac {1}{4}}}\)	\(69\)

[In]

int(1/(1-x)^(1/4)/(e*x)^(9/2)/(1+x)^(1/4),x,method=_RETURNVERBOSE)

[Out]

-2/21*x*(4*x^2+3)/(e*x)^(9/2)*(1-x)^(3/4)*(1+x)^(3/4)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.63 \[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{9/2} \sqrt [4]{1+x}} \, dx=-\frac {2 \, \sqrt {e x} {\left (4 \, x^{2} + 3\right )} {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}}}{21 \, e^{5} x^{4}} \]

[In]

integrate(1/(1-x)^(1/4)/(e*x)^(9/2)/(1+x)^(1/4),x, algorithm="fricas")

[Out]

-2/21*sqrt(e*x)*(4*x^2 + 3)*(x + 1)^(3/4)*(-x + 1)^(3/4)/(e^5*x^4)

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{9/2} \sqrt [4]{1+x}} \, dx=\text {Timed out} \]

[In]

integrate(1/(1-x)**(1/4)/(e*x)**(9/2)/(1+x)**(1/4),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{9/2} \sqrt [4]{1+x}} \, dx=\int { \frac {1}{\left (e x\right )^{\frac {9}{2}} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate(1/(1-x)^(1/4)/(e*x)^(9/2)/(1+x)^(1/4),x, algorithm="maxima")

[Out]

integrate(1/((e*x)^(9/2)*(x + 1)^(1/4)*(-x + 1)^(1/4)), x)

Giac [F]

\[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{9/2} \sqrt [4]{1+x}} \, dx=\int { \frac {1}{\left (e x\right )^{\frac {9}{2}} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate(1/(1-x)^(1/4)/(e*x)^(9/2)/(1+x)^(1/4),x, algorithm="giac")

[Out]

integrate(1/((e*x)^(9/2)*(x + 1)^(1/4)*(-x + 1)^(1/4)), x)

Mupad [B] (verification not implemented)

Time = 1.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{9/2} \sqrt [4]{1+x}} \, dx=-\frac {\sqrt {e\,x}\,\left (\frac {2}{7\,e^5}+\frac {2\,x^2}{21\,e^5}-\frac {8\,x^4}{21\,e^5}\right )}{x^4\,{\left (1-x\right )}^{1/4}\,{\left (x+1\right )}^{1/4}} \]

[In]

int(1/((e*x)^(9/2)*(1 - x)^(1/4)*(x + 1)^(1/4)),x)

[Out]

-((e*x)^(1/2)*(2/(7*e^5) + (2*x^2)/(21*e^5) - (8*x^4)/(21*e^5)))/(x^4*(1 - x)^(1/4)*(x + 1)^(1/4))