Integrand size = 24, antiderivative size = 76 \[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{13/2} \sqrt [4]{1+x}} \, dx=-\frac {2 \left (1-x^2\right )^{3/4}}{11 e (e x)^{11/2}}-\frac {16 \left (1-x^2\right )^{3/4}}{77 e^3 (e x)^{7/2}}-\frac {64 \left (1-x^2\right )^{3/4}}{231 e^5 (e x)^{3/2}} \] Output:
-2/11*(-x^2+1)^(3/4)/e/(e*x)^(11/2)-16/77*(-x^2+1)^(3/4)/e^3/(e*x)^(7/2)-6 4/231*(-x^2+1)^(3/4)/e^5/(e*x)^(3/2)
Time = 0.14 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.46 \[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{13/2} \sqrt [4]{1+x}} \, dx=-\frac {2 x \left (1-x^2\right )^{3/4} \left (21+24 x^2+32 x^4\right )}{231 (e x)^{13/2}} \] Input:
Integrate[1/((1 - x)^(1/4)*(e*x)^(13/2)*(1 + x)^(1/4)),x]
Output:
(-2*x*(1 - x^2)^(3/4)*(21 + 24*x^2 + 32*x^4))/(231*(e*x)^(13/2))
Time = 0.19 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {135, 246, 246, 242}
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.
\(\displaystyle \int \frac {1}{\sqrt [4]{1-x} \sqrt [4]{x+1} (e x)^{13/2}} \, dx\) |
\(\Big \downarrow \) 135 |
\(\displaystyle \int \frac {1}{\sqrt [4]{1-x^2} (e x)^{13/2}}dx\) |
\(\Big \downarrow \) 246 |
\(\displaystyle -\frac {8}{3} \int \frac {\left (1-x^2\right )^{3/4}}{(e x)^{13/2}}dx-\frac {2 \left (1-x^2\right )^{3/4}}{3 e (e x)^{11/2}}\) |
\(\Big \downarrow \) 246 |
\(\displaystyle -\frac {8}{3} \left (-\frac {4}{7} \int \frac {\left (1-x^2\right )^{7/4}}{(e x)^{13/2}}dx-\frac {2 \left (1-x^2\right )^{7/4}}{7 e (e x)^{11/2}}\right )-\frac {2 \left (1-x^2\right )^{3/4}}{3 e (e x)^{11/2}}\) |
\(\Big \downarrow \) 242 |
\(\displaystyle -\frac {8}{3} \left (\frac {8 \left (1-x^2\right )^{11/4}}{77 e (e x)^{11/2}}-\frac {2 \left (1-x^2\right )^{7/4}}{7 e (e x)^{11/2}}\right )-\frac {2 \left (1-x^2\right )^{3/4}}{3 e (e x)^{11/2}}\) |
Input:
Int[1/((1 - x)^(1/4)*(e*x)^(13/2)*(1 + x)^(1/4)),x]
Output:
(-2*(1 - x^2)^(3/4))/(3*e*(e*x)^(11/2)) - (8*((-2*(1 - x^2)^(7/4))/(7*e*(e *x)^(11/2)) + (8*(1 - x^2)^(11/4))/(77*e*(e*x)^(11/2))))/3
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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x ] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x )^(m + 1))*((a + b*x^2)^(p + 1)/(a*c*2*(p + 1))), x] + Simp[(m + 2*p + 3)/( a*2*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m , p}, x] && ILtQ[Simplify[(m + 1)/2 + p + 1], 0] && NeQ[p, -1]
Time = 0.19 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.43
method | result | size |
gosper | \(-\frac {2 x \left (1-x \right )^{\frac {3}{4}} \left (1+x \right )^{\frac {3}{4}} \left (32 x^{4}+24 x^{2}+21\right )}{231 \left (e x \right )^{\frac {13}{2}}}\) | \(33\) |
orering | \(\frac {2 \left (-1+x \right ) x \left (1+x \right )^{\frac {3}{4}} \left (32 x^{4}+24 x^{2}+21\right )}{231 \left (1-x \right )^{\frac {1}{4}} \left (e x \right )^{\frac {13}{2}}}\) | \(36\) |
risch | \(\frac {2 \left (e^{2} x^{2} \left (1-x \right ) \left (1+x \right )\right )^{\frac {1}{4}} \left (1+x \right )^{\frac {3}{4}} \left (-1+x \right ) \left (32 x^{4}+24 x^{2}+21\right )}{231 \sqrt {e x}\, \left (1-x \right )^{\frac {1}{4}} e^{6} x^{5} \left (-e^{2} x^{2} \left (-1+x \right ) \left (1+x \right )\right )^{\frac {1}{4}}}\) | \(74\) |
Input:
int(1/(1-x)^(1/4)/(e*x)^(13/2)/(1+x)^(1/4),x,method=_RETURNVERBOSE)
Output:
-2/231*x*(1-x)^(3/4)*(1+x)^(3/4)/(e*x)^(13/2)*(32*x^4+24*x^2+21)
Time = 0.09 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.49 \[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{13/2} \sqrt [4]{1+x}} \, dx=-\frac {2 \, {\left (32 \, x^{4} + 24 \, x^{2} + 21\right )} \sqrt {e x} {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}}}{231 \, e^{7} x^{6}} \] Input:
integrate(1/(1-x)^(1/4)/(e*x)^(13/2)/(1+x)^(1/4),x, algorithm="fricas")
Output:
-2/231*(32*x^4 + 24*x^2 + 21)*sqrt(e*x)*(x + 1)^(3/4)*(-x + 1)^(3/4)/(e^7* x^6)
Timed out. \[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{13/2} \sqrt [4]{1+x}} \, dx=\text {Timed out} \] Input:
integrate(1/(1-x)**(1/4)/(e*x)**(13/2)/(1+x)**(1/4),x)
Output:
Timed out
\[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{13/2} \sqrt [4]{1+x}} \, dx=\int { \frac {1}{\left (e x\right )^{\frac {13}{2}} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate(1/(1-x)^(1/4)/(e*x)^(13/2)/(1+x)^(1/4),x, algorithm="maxima")
Output:
integrate(1/((e*x)^(13/2)*(x + 1)^(1/4)*(-x + 1)^(1/4)), x)
\[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{13/2} \sqrt [4]{1+x}} \, dx=\int { \frac {1}{\left (e x\right )^{\frac {13}{2}} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate(1/(1-x)^(1/4)/(e*x)^(13/2)/(1+x)^(1/4),x, algorithm="giac")
Output:
integrate(1/((e*x)^(13/2)*(x + 1)^(1/4)*(-x + 1)^(1/4)), x)
Time = 1.08 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.68 \[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{13/2} \sqrt [4]{1+x}} \, dx=-\frac {\sqrt {e\,x}\,\left (\frac {2}{11\,e^7}+\frac {2\,x^2}{77\,e^7}+\frac {16\,x^4}{231\,e^7}-\frac {64\,x^6}{231\,e^7}\right )}{x^6\,{\left (1-x\right )}^{1/4}\,{\left (x+1\right )}^{1/4}} \] Input:
int(1/((e*x)^(13/2)*(1 - x)^(1/4)*(x + 1)^(1/4)),x)
Output:
-((e*x)^(1/2)*(2/(11*e^7) + (2*x^2)/(77*e^7) + (16*x^4)/(231*e^7) - (64*x^ 6)/(231*e^7)))/(x^6*(1 - x)^(1/4)*(x + 1)^(1/4))
Time = 0.60 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{13/2} \sqrt [4]{1+x}} \, dx=\frac {\sqrt {e}\, \left (-128 \left (x +1\right ) x^{6}+32 \left (x +1\right ) x^{4}+12 \left (x +1\right ) x^{2}-70 x -70\right )}{385 \left (x +1\right )^{\frac {3}{4}} \sqrt {x}\, \sqrt {x +1}\, \left (1-x \right )^{\frac {1}{4}} e^{7} x^{5}} \] Input:
int(1/(1-x)^(1/4)/(e*x)^(13/2)/(1+x)^(1/4),x)
Output:
(sqrt(e)*(x + 1)**(1/4)*( - 128*(x + 1)*x**6 + 32*(x + 1)*x**4 + 12*(x + 1 )*x**2 + 7*(x + 1) - 77*x - 77))/(385*sqrt(x)*sqrt(x + 1)*( - x + 1)**(1/4 )*e**7*x**5*(x + 1))