Integrand size = 24, antiderivative size = 340 \[ \int \frac {(2+e x)^{5/2}}{\sqrt [4]{12-3 e^2 x^2}} \, dx=-\frac {5\ 3^{3/4} (2-e x)^{3/4} \sqrt [4]{2+e x}}{2 e}-\frac {3^{3/4} (2-e x)^{3/4} (2+e x)^{5/4}}{2 e}-\frac {(2-e x)^{3/4} (2+e x)^{9/4}}{3 \sqrt [4]{3} e}+\frac {5\ 3^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{2+e x}}\right )}{\sqrt {2} e}-\frac {5\ 3^{3/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{2+e x}}\right )}{\sqrt {2} e}-\frac {5\ 3^{3/4} \log \left (\frac {\sqrt {6-3 e x}-\sqrt {6} \sqrt [4]{2-e x} \sqrt [4]{2+e x}+\sqrt {3} \sqrt {2+e x}}{\sqrt {2+e x}}\right )}{2 \sqrt {2} e}+\frac {5\ 3^{3/4} \log \left (\frac {\sqrt {6-3 e x}+\sqrt {6} \sqrt [4]{2-e x} \sqrt [4]{2+e x}+\sqrt {3} \sqrt {2+e x}}{\sqrt {2+e x}}\right )}{2 \sqrt {2} e} \]
-5/2*3^(3/4)*(-e*x+2)^(3/4)*(e*x+2)^(1/4)/e-1/2*3^(3/4)*(-e*x+2)^(3/4)*(e* x+2)^(5/4)/e-1/9*(-e*x+2)^(3/4)*(e*x+2)^(9/4)*3^(3/4)/e-5/2*3^(3/4)*arctan (-1+(-e*x+2)^(1/4)*2^(1/2)/(e*x+2)^(1/4))/e*2^(1/2)-5/2*3^(3/4)*arctan(1+( -e*x+2)^(1/4)*2^(1/2)/(e*x+2)^(1/4))/e*2^(1/2)-5/4*3^(3/4)*ln(3^(1/2)-(-e* x+2)^(1/4)*6^(1/2)/(e*x+2)^(1/4)+3^(1/2)*(-e*x+2)^(1/2)/(e*x+2)^(1/2))/e*2 ^(1/2)+5/4*3^(3/4)*ln(3^(1/2)+(-e*x+2)^(1/4)*6^(1/2)/(e*x+2)^(1/4)+3^(1/2) *(-e*x+2)^(1/2)/(e*x+2)^(1/2))/e*2^(1/2)
Time = 1.02 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.50 \[ \int \frac {(2+e x)^{5/2}}{\sqrt [4]{12-3 e^2 x^2}} \, dx=\frac {-\left (4-e^2 x^2\right )^{3/4} \left (71+17 e x+2 e^2 x^2\right )-45 \sqrt {4+2 e x} \arctan \left (\frac {\sqrt {4+2 e x} \sqrt [4]{4-e^2 x^2}}{2+e x-\sqrt {4-e^2 x^2}}\right )+45 \sqrt {4+2 e x} \text {arctanh}\left (\frac {2+e x+\sqrt {4-e^2 x^2}}{\sqrt {4+2 e x} \sqrt [4]{4-e^2 x^2}}\right )}{6 \sqrt [4]{3} e \sqrt {2+e x}} \]
(-((4 - e^2*x^2)^(3/4)*(71 + 17*e*x + 2*e^2*x^2)) - 45*Sqrt[4 + 2*e*x]*Arc Tan[(Sqrt[4 + 2*e*x]*(4 - e^2*x^2)^(1/4))/(2 + e*x - Sqrt[4 - e^2*x^2])] + 45*Sqrt[4 + 2*e*x]*ArcTanh[(2 + e*x + Sqrt[4 - e^2*x^2])/(Sqrt[4 + 2*e*x] *(4 - e^2*x^2)^(1/4))])/(6*3^(1/4)*e*Sqrt[2 + e*x])
Time = 0.43 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.81, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {456, 60, 27, 60, 60, 73, 854, 826, 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.
| \(\displaystyle \int \frac {(e x+2)^{5/2}}{\sqrt [4]{12-3 e^2 x^2}} \, dx\) |
| \(\Big \downarrow \) 456 |
| \(\displaystyle \int \frac {(e x+2)^{9/4}}{\sqrt [4]{6-3 e x}}dx\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle 3 \int \frac {(e x+2)^{5/4}}{\sqrt [4]{3} \sqrt [4]{2-e x}}dx-\frac {(2-e x)^{3/4} (e x+2)^{9/4}}{3 \sqrt [4]{3} e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 3^{3/4} \int \frac {(e x+2)^{5/4}}{\sqrt [4]{2-e x}}dx-\frac {(2-e x)^{3/4} (e x+2)^{9/4}}{3 \sqrt [4]{3} e}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle 3^{3/4} \left (\frac {5}{2} \int \frac {\sqrt [4]{e x+2}}{\sqrt [4]{2-e x}}dx-\frac {(2-e x)^{3/4} (e x+2)^{5/4}}{2 e}\right )-\frac {(2-e x)^{3/4} (e x+2)^{9/4}}{3 \sqrt [4]{3} e}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle 3^{3/4} \left (\frac {5}{2} \left (\int \frac {1}{\sqrt [4]{2-e x} (e x+2)^{3/4}}dx-\frac {(2-e x)^{3/4} \sqrt [4]{e x+2}}{e}\right )-\frac {(2-e x)^{3/4} (e x+2)^{5/4}}{2 e}\right )-\frac {(2-e x)^{3/4} (e x+2)^{9/4}}{3 \sqrt [4]{3} e}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle 3^{3/4} \left (\frac {5}{2} \left (-\frac {4 \int \frac {\sqrt {2-e x}}{(e x+2)^{3/4}}d\sqrt [4]{2-e x}}{e}-\frac {(2-e x)^{3/4} \sqrt [4]{e x+2}}{e}\right )-\frac {(2-e x)^{3/4} (e x+2)^{5/4}}{2 e}\right )-\frac {(2-e x)^{3/4} (e x+2)^{9/4}}{3 \sqrt [4]{3} e}\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle 3^{3/4} \left (\frac {5}{2} \left (-\frac {4 \int \frac {\sqrt {2-e x}}{3-e x}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}}{e}-\frac {(2-e x)^{3/4} \sqrt [4]{e x+2}}{e}\right )-\frac {(2-e x)^{3/4} (e x+2)^{5/4}}{2 e}\right )-\frac {(2-e x)^{3/4} (e x+2)^{9/4}}{3 \sqrt [4]{3} e}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle 3^{3/4} \left (\frac {5}{2} \left (-\frac {4 \left (\frac {1}{2} \int \frac {\sqrt {2-e x}+1}{3-e x}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}-\frac {1}{2} \int \frac {1-\sqrt {2-e x}}{3-e x}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{e}-\frac {(2-e x)^{3/4} \sqrt [4]{e x+2}}{e}\right )-\frac {(2-e x)^{3/4} (e x+2)^{5/4}}{2 e}\right )-\frac {(2-e x)^{3/4} (e x+2)^{9/4}}{3 \sqrt [4]{3} e}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle 3^{3/4} \left (\frac {5}{2} \left (-\frac {4 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\sqrt {2-e x}-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+\frac {1}{2} \int \frac {1}{\sqrt {2-e x}+\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )-\frac {1}{2} \int \frac {1-\sqrt {2-e x}}{3-e x}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{e}-\frac {(2-e x)^{3/4} \sqrt [4]{e x+2}}{e}\right )-\frac {(2-e x)^{3/4} (e x+2)^{5/4}}{2 e}\right )-\frac {(2-e x)^{3/4} (e x+2)^{9/4}}{3 \sqrt [4]{3} e}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle 3^{3/4} \left (\frac {5}{2} \left (-\frac {4 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\sqrt {2-e x}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\sqrt {2-e x}-1}d\left (\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\sqrt {2-e x}}{3-e x}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{e}-\frac {(2-e x)^{3/4} \sqrt [4]{e x+2}}{e}\right )-\frac {(2-e x)^{3/4} (e x+2)^{5/4}}{2 e}\right )-\frac {(2-e x)^{3/4} (e x+2)^{9/4}}{3 \sqrt [4]{3} e}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 3^{3/4} \left (\frac {5}{2} \left (-\frac {4 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\sqrt {2-e x}}{3-e x}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{e}-\frac {(2-e x)^{3/4} \sqrt [4]{e x+2}}{e}\right )-\frac {(2-e x)^{3/4} (e x+2)^{5/4}}{2 e}\right )-\frac {(2-e x)^{3/4} (e x+2)^{9/4}}{3 \sqrt [4]{3} e}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle 3^{3/4} \left (\frac {5}{2} \left (-\frac {4 \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}}{\sqrt {2-e x}-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt {2-e x}+\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{\sqrt {2}}\right )\right )}{e}-\frac {(2-e x)^{3/4} \sqrt [4]{e x+2}}{e}\right )-\frac {(2-e x)^{3/4} (e x+2)^{5/4}}{2 e}\right )-\frac {(2-e x)^{3/4} (e x+2)^{9/4}}{3 \sqrt [4]{3} e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 3^{3/4} \left (\frac {5}{2} \left (-\frac {4 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}}{\sqrt {2-e x}-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt {2-e x}+\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{\sqrt {2}}\right )\right )}{e}-\frac {(2-e x)^{3/4} \sqrt [4]{e x+2}}{e}\right )-\frac {(2-e x)^{3/4} (e x+2)^{5/4}}{2 e}\right )-\frac {(2-e x)^{3/4} (e x+2)^{9/4}}{3 \sqrt [4]{3} e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 3^{3/4} \left (\frac {5}{2} \left (-\frac {4 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}}{\sqrt {2-e x}-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1}{\sqrt {2-e x}+\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{\sqrt {2}}\right )\right )}{e}-\frac {(2-e x)^{3/4} \sqrt [4]{e x+2}}{e}\right )-\frac {(2-e x)^{3/4} (e x+2)^{5/4}}{2 e}\right )-\frac {(2-e x)^{3/4} (e x+2)^{9/4}}{3 \sqrt [4]{3} e}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 3^{3/4} \left (\frac {5}{2} \left (-\frac {4 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\sqrt {2-e x}-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\sqrt {2-e x}+\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{2 \sqrt {2}}\right )\right )}{e}-\frac {(2-e x)^{3/4} \sqrt [4]{e x+2}}{e}\right )-\frac {(2-e x)^{3/4} (e x+2)^{5/4}}{2 e}\right )-\frac {(2-e x)^{3/4} (e x+2)^{9/4}}{3 \sqrt [4]{3} e}\) |
-1/3*((2 - e*x)^(3/4)*(2 + e*x)^(9/4))/(3^(1/4)*e) + 3^(3/4)*(-1/2*((2 - e *x)^(3/4)*(2 + e*x)^(5/4))/e + (5*(-(((2 - e*x)^(3/4)*(2 + e*x)^(1/4))/e) - (4*((-(ArcTan[1 - (Sqrt[2]*(2 - e*x)^(1/4))/(2 + e*x)^(1/4)]/Sqrt[2]) + ArcTan[1 + (Sqrt[2]*(2 - e*x)^(1/4))/(2 + e*x)^(1/4)]/Sqrt[2])/2 + (Log[1 + Sqrt[2 - e*x] - (Sqrt[2]*(2 - e*x)^(1/4))/(2 + e*x)^(1/4)]/(2*Sqrt[2]) - Log[1 + Sqrt[2 - e*x] + (Sqrt[2]*(2 - e*x)^(1/4))/(2 + e*x)^(1/4)]/(2*Sqr t[2]))/2))/e))/2)
3.10.37.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ (c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] && !Integ erQ[n]))
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) 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[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 1)/n) Subst[Int[x^m/(1 - b*x^n)^(p + (m + 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)] && IntegersQ[m, p + (m + 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]
\[\int \frac {\left (e x +2\right )^{\frac {5}{2}}}{\left (-3 x^{2} e^{2}+12\right )^{\frac {1}{4}}}d x\]
Result contains complex when optimal does not.
Time = 0.40 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.08 \[ \int \frac {(2+e x)^{5/2}}{\sqrt [4]{12-3 e^2 x^2}} \, dx=-\frac {45 \cdot 27^{\frac {1}{4}} {\left (e^{2} x + 2 \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} \log \left (\frac {27^{\frac {1}{4}} {\left (e^{3} x^{2} - 4 \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} + {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{4}} \sqrt {e x + 2}}{e^{2} x^{2} - 4}\right ) - 45 \cdot 27^{\frac {1}{4}} {\left (e^{2} x + 2 \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {27^{\frac {1}{4}} {\left (e^{3} x^{2} - 4 \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} - {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{4}} \sqrt {e x + 2}}{e^{2} x^{2} - 4}\right ) + 45 \cdot 27^{\frac {1}{4}} {\left (-i \, e^{2} x - 2 i \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {27^{\frac {1}{4}} {\left (i \, e^{3} x^{2} - 4 i \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} - {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{4}} \sqrt {e x + 2}}{e^{2} x^{2} - 4}\right ) + 45 \cdot 27^{\frac {1}{4}} {\left (i \, e^{2} x + 2 i \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {27^{\frac {1}{4}} {\left (-i \, e^{3} x^{2} + 4 i \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} - {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{4}} \sqrt {e x + 2}}{e^{2} x^{2} - 4}\right ) + {\left (2 \, e^{2} x^{2} + 17 \, e x + 71\right )} {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{4}} \sqrt {e x + 2}}{18 \, {\left (e^{2} x + 2 \, e\right )}} \]
-1/18*(45*27^(1/4)*(e^2*x + 2*e)*(-1/e^4)^(1/4)*log((27^(1/4)*(e^3*x^2 - 4 *e)*(-1/e^4)^(1/4) + (-3*e^2*x^2 + 12)^(3/4)*sqrt(e*x + 2))/(e^2*x^2 - 4)) - 45*27^(1/4)*(e^2*x + 2*e)*(-1/e^4)^(1/4)*log(-(27^(1/4)*(e^3*x^2 - 4*e) *(-1/e^4)^(1/4) - (-3*e^2*x^2 + 12)^(3/4)*sqrt(e*x + 2))/(e^2*x^2 - 4)) + 45*27^(1/4)*(-I*e^2*x - 2*I*e)*(-1/e^4)^(1/4)*log(-(27^(1/4)*(I*e^3*x^2 - 4*I*e)*(-1/e^4)^(1/4) - (-3*e^2*x^2 + 12)^(3/4)*sqrt(e*x + 2))/(e^2*x^2 - 4)) + 45*27^(1/4)*(I*e^2*x + 2*I*e)*(-1/e^4)^(1/4)*log(-(27^(1/4)*(-I*e^3* x^2 + 4*I*e)*(-1/e^4)^(1/4) - (-3*e^2*x^2 + 12)^(3/4)*sqrt(e*x + 2))/(e^2* x^2 - 4)) + (2*e^2*x^2 + 17*e*x + 71)*(-3*e^2*x^2 + 12)^(3/4)*sqrt(e*x + 2 ))/(e^2*x + 2*e)
\[ \int \frac {(2+e x)^{5/2}}{\sqrt [4]{12-3 e^2 x^2}} \, dx=\frac {3^{\frac {3}{4}} \left (\int \frac {4 \sqrt {e x + 2}}{\sqrt [4]{- e^{2} x^{2} + 4}}\, dx + \int \frac {4 e x \sqrt {e x + 2}}{\sqrt [4]{- e^{2} x^{2} + 4}}\, dx + \int \frac {e^{2} x^{2} \sqrt {e x + 2}}{\sqrt [4]{- e^{2} x^{2} + 4}}\, dx\right )}{3} \]
3**(3/4)*(Integral(4*sqrt(e*x + 2)/(-e**2*x**2 + 4)**(1/4), x) + Integral( 4*e*x*sqrt(e*x + 2)/(-e**2*x**2 + 4)**(1/4), x) + Integral(e**2*x**2*sqrt( e*x + 2)/(-e**2*x**2 + 4)**(1/4), x))/3
\[ \int \frac {(2+e x)^{5/2}}{\sqrt [4]{12-3 e^2 x^2}} \, dx=\int { \frac {{\left (e x + 2\right )}^{\frac {5}{2}}}{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}}} \,d x } \]
Exception generated. \[ \int \frac {(2+e x)^{5/2}}{\sqrt [4]{12-3 e^2 x^2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {(2+e x)^{5/2}}{\sqrt [4]{12-3 e^2 x^2}} \, dx=\int \frac {{\left (e\,x+2\right )}^{5/2}}{{\left (12-3\,e^2\,x^2\right )}^{1/4}} \,d x \]