Integrand size = 24, antiderivative size = 244 \[ \int \frac {(e x)^{3/2}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx=-\frac {1}{2} e \sqrt {e x} \left (1-x^2\right )^{3/4}-\frac {e^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}\right )}{4 \sqrt {2}}+\frac {e^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}\right )}{4 \sqrt {2}}-\frac {e^{3/2} \log \left (\sqrt {e}+\frac {\sqrt {e} x}{\sqrt {1-x^2}}-\frac {\sqrt {2} \sqrt {e x}}{\sqrt [4]{1-x^2}}\right )}{8 \sqrt {2}}+\frac {e^{3/2} \log \left (\sqrt {e}+\frac {\sqrt {e} x}{\sqrt {1-x^2}}+\frac {\sqrt {2} \sqrt {e x}}{\sqrt [4]{1-x^2}}\right )}{8 \sqrt {2}} \]
-1/8*e^(3/2)*arctan(1-2^(1/2)*(e*x)^(1/2)/(-x^2+1)^(1/4)/e^(1/2))*2^(1/2)+ 1/8*e^(3/2)*arctan(1+2^(1/2)*(e*x)^(1/2)/(-x^2+1)^(1/4)/e^(1/2))*2^(1/2)-1 /16*e^(3/2)*ln(e^(1/2)-2^(1/2)*(e*x)^(1/2)/(-x^2+1)^(1/4)+x*e^(1/2)/(-x^2+ 1)^(1/2))*2^(1/2)+1/16*e^(3/2)*ln(e^(1/2)+2^(1/2)*(e*x)^(1/2)/(-x^2+1)^(1/ 4)+x*e^(1/2)/(-x^2+1)^(1/2))*2^(1/2)-1/2*e*(-x^2+1)^(3/4)*(e*x)^(1/2)
Time = 0.29 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.51 \[ \int \frac {(e x)^{3/2}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx=\frac {(e x)^{3/2} \left (-4 \sqrt {x} \left (1-x^2\right )^{3/4}+\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {x} \sqrt [4]{1-x^2}}{-x+\sqrt {1-x^2}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {x} \sqrt [4]{1-x^2}}{x+\sqrt {1-x^2}}\right )\right )}{8 x^{3/2}} \]
((e*x)^(3/2)*(-4*Sqrt[x]*(1 - x^2)^(3/4) + Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[x] *(1 - x^2)^(1/4))/(-x + Sqrt[1 - x^2])] + Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[x] *(1 - x^2)^(1/4))/(x + Sqrt[1 - x^2])]))/(8*x^(3/2))
Time = 0.38 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.94, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {135, 262, 266, 770, 755, 27, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int \frac {(e x)^{3/2}}{\sqrt [4]{1-x} \sqrt [4]{x+1}} \, dx\) |
| \(\Big \downarrow \) 135 |
| \(\displaystyle \int \frac {(e x)^{3/2}}{\sqrt [4]{1-x^2}}dx\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {1}{4} e^2 \int \frac {1}{\sqrt {e x} \sqrt [4]{1-x^2}}dx-\frac {1}{2} e \left (1-x^2\right )^{3/4} \sqrt {e x}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {1}{2} e \int \frac {1}{\sqrt [4]{1-x^2}}d\sqrt {e x}-\frac {1}{2} e \left (1-x^2\right )^{3/4} \sqrt {e x}\) |
| \(\Big \downarrow \) 770 |
| \(\displaystyle \frac {1}{2} e \int \frac {1}{x^2+1}d\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}-\frac {1}{2} e \left (1-x^2\right )^{3/4} \sqrt {e x}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {1}{2} e \left (\frac {\int \frac {e^2 (e-e x)}{x^2 e^2+e^2}d\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}}{2 e}+\frac {\int \frac {e^2 (x e+e)}{x^2 e^2+e^2}d\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}}{2 e}\right )-\frac {1}{2} e \left (1-x^2\right )^{3/4} \sqrt {e x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} e \left (\frac {1}{2} e \int \frac {e-e x}{x^2 e^2+e^2}d\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}+\frac {1}{2} e \int \frac {x e+e}{x^2 e^2+e^2}d\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}\right )-\frac {1}{2} e \left (1-x^2\right )^{3/4} \sqrt {e x}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {1}{2} e \left (\frac {1}{2} e \int \frac {e-e x}{x^2 e^2+e^2}d\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}+\frac {1}{2} e \left (\frac {1}{2} \int \frac {1}{x e+e-\frac {\sqrt {2} \sqrt {e x} \sqrt {e}}{\sqrt [4]{1-x^2}}}d\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}+\frac {1}{2} \int \frac {1}{x e+e+\frac {\sqrt {2} \sqrt {e x} \sqrt {e}}{\sqrt [4]{1-x^2}}}d\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}\right )\right )-\frac {1}{2} e \left (1-x^2\right )^{3/4} \sqrt {e x}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{2} e \left (\frac {1}{2} e \int \frac {e-e x}{x^2 e^2+e^2}d\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}+\frac {1}{2} e \left (\frac {\int \frac {1}{-e x-1}d\left (1-\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}\right )}{\sqrt {2} \sqrt {e}}-\frac {\int \frac {1}{-e x-1}d\left (\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}+1\right )}{\sqrt {2} \sqrt {e}}\right )\right )-\frac {1}{2} e \left (1-x^2\right )^{3/4} \sqrt {e x}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{2} e \left (\frac {1}{2} e \int \frac {e-e x}{x^2 e^2+e^2}d\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}+\frac {1}{2} e \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )-\frac {1}{2} e \left (1-x^2\right )^{3/4} \sqrt {e x}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {1}{2} e \left (\frac {1}{2} e \left (-\frac {\int -\frac {\sqrt {2} \sqrt {e}-\frac {2 \sqrt {e x}}{\sqrt [4]{1-x^2}}}{x e+e-\frac {\sqrt {2} \sqrt {e x} \sqrt {e}}{\sqrt [4]{1-x^2}}}d\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}}{2 \sqrt {2} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {e}+\frac {\sqrt {2} \sqrt {e x}}{\sqrt [4]{1-x^2}}\right )}{x e+e+\frac {\sqrt {2} \sqrt {e x} \sqrt {e}}{\sqrt [4]{1-x^2}}}d\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}}{2 \sqrt {2} \sqrt {e}}\right )+\frac {1}{2} e \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )-\frac {1}{2} e \left (1-x^2\right )^{3/4} \sqrt {e x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} e \left (\frac {1}{2} e \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-\frac {2 \sqrt {e x}}{\sqrt [4]{1-x^2}}}{x e+e-\frac {\sqrt {2} \sqrt {e x} \sqrt {e}}{\sqrt [4]{1-x^2}}}d\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {e}+\frac {\sqrt {2} \sqrt {e x}}{\sqrt [4]{1-x^2}}\right )}{x e+e+\frac {\sqrt {2} \sqrt {e x} \sqrt {e}}{\sqrt [4]{1-x^2}}}d\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}}{2 \sqrt {2} \sqrt {e}}\right )+\frac {1}{2} e \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )-\frac {1}{2} e \left (1-x^2\right )^{3/4} \sqrt {e x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} e \left (\frac {1}{2} e \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-\frac {2 \sqrt {e x}}{\sqrt [4]{1-x^2}}}{x e+e-\frac {\sqrt {2} \sqrt {e x} \sqrt {e}}{\sqrt [4]{1-x^2}}}d\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {e}+\frac {\sqrt {2} \sqrt {e x}}{\sqrt [4]{1-x^2}}}{x e+e+\frac {\sqrt {2} \sqrt {e x} \sqrt {e}}{\sqrt [4]{1-x^2}}}d\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}}{2 \sqrt {e}}\right )+\frac {1}{2} e \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )-\frac {1}{2} e \left (1-x^2\right )^{3/4} \sqrt {e x}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{2} e \left (\frac {1}{2} e \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}\right )}{\sqrt {2} \sqrt {e}}\right )+\frac {1}{2} e \left (\frac {\log \left (\frac {\sqrt {2} \sqrt {e} \sqrt {e x}}{\sqrt [4]{1-x^2}}+e x+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {e} \sqrt {e x}}{\sqrt [4]{1-x^2}}+e x+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )-\frac {1}{2} e \left (1-x^2\right )^{3/4} \sqrt {e x}\) |
-1/2*(e*Sqrt[e*x]*(1 - x^2)^(3/4)) + (e*((e*(-(ArcTan[1 - (Sqrt[2]*Sqrt[e* x])/(Sqrt[e]*(1 - x^2)^(1/4))]/(Sqrt[2]*Sqrt[e])) + ArcTan[1 + (Sqrt[2]*Sq rt[e*x])/(Sqrt[e]*(1 - x^2)^(1/4))]/(Sqrt[2]*Sqrt[e])))/2 + (e*(-1/2*Log[e + e*x - (Sqrt[2]*Sqrt[e]*Sqrt[e*x])/(1 - x^2)^(1/4)]/(Sqrt[2]*Sqrt[e]) + Log[e + e*x + (Sqrt[2]*Sqrt[e]*Sqrt[e*x])/(1 - x^2)^(1/4)]/(2*Sqrt[2]*Sqrt [e])))/2))/2
3.10.8.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n) Subst[In t[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p + 1 /n]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
\[\int \frac {\left (e x \right )^{\frac {3}{2}}}{\left (1-x \right )^{\frac {1}{4}} \left (1+x \right )^{\frac {1}{4}}}d x\]
Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.94 \[ \int \frac {(e x)^{3/2}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx=-\frac {1}{2} \, \sqrt {e x} e {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - \frac {1}{8} \, \left (-e^{6}\right )^{\frac {1}{4}} \log \left (\frac {\sqrt {e x} e {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} + \left (-e^{6}\right )^{\frac {1}{4}} {\left (x^{2} - 1\right )}}{x^{2} - 1}\right ) + \frac {1}{8} \, \left (-e^{6}\right )^{\frac {1}{4}} \log \left (\frac {\sqrt {e x} e {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - \left (-e^{6}\right )^{\frac {1}{4}} {\left (x^{2} - 1\right )}}{x^{2} - 1}\right ) - \frac {1}{8} i \, \left (-e^{6}\right )^{\frac {1}{4}} \log \left (\frac {\sqrt {e x} e {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} + \left (-e^{6}\right )^{\frac {1}{4}} {\left (i \, x^{2} - i\right )}}{x^{2} - 1}\right ) + \frac {1}{8} i \, \left (-e^{6}\right )^{\frac {1}{4}} \log \left (\frac {\sqrt {e x} e {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} + \left (-e^{6}\right )^{\frac {1}{4}} {\left (-i \, x^{2} + i\right )}}{x^{2} - 1}\right ) \]
-1/2*sqrt(e*x)*e*(x + 1)^(3/4)*(-x + 1)^(3/4) - 1/8*(-e^6)^(1/4)*log((sqrt (e*x)*e*(x + 1)^(3/4)*(-x + 1)^(3/4) + (-e^6)^(1/4)*(x^2 - 1))/(x^2 - 1)) + 1/8*(-e^6)^(1/4)*log((sqrt(e*x)*e*(x + 1)^(3/4)*(-x + 1)^(3/4) - (-e^6)^ (1/4)*(x^2 - 1))/(x^2 - 1)) - 1/8*I*(-e^6)^(1/4)*log((sqrt(e*x)*e*(x + 1)^ (3/4)*(-x + 1)^(3/4) + (-e^6)^(1/4)*(I*x^2 - I))/(x^2 - 1)) + 1/8*I*(-e^6) ^(1/4)*log((sqrt(e*x)*e*(x + 1)^(3/4)*(-x + 1)^(3/4) + (-e^6)^(1/4)*(-I*x^ 2 + I))/(x^2 - 1))
Timed out. \[ \int \frac {(e x)^{3/2}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx=\text {Timed out} \]
\[ \int \frac {(e x)^{3/2}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx=\int { \frac {\left (e x\right )^{\frac {3}{2}}}{{\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}} \,d x } \]
\[ \int \frac {(e x)^{3/2}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx=\int { \frac {\left (e x\right )^{\frac {3}{2}}}{{\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}} \,d x } \]
Timed out. \[ \int \frac {(e x)^{3/2}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx=\int \frac {{\left (e\,x\right )}^{3/2}}{{\left (1-x\right )}^{1/4}\,{\left (x+1\right )}^{1/4}} \,d x \]