\(\int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{3/2}} \, dx\) [306]

Optimal result

Integrand size = 24, antiderivative size = 184 \[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{3/2}} \, dx=-\frac {4 \sqrt [4]{3} \sqrt [4]{4-e^2 x^2}}{e \sqrt {2+e x}}+\frac {\sqrt {2} \sqrt [4]{3} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{2+e x}}{\sqrt [4]{2-e x}}\right )}{e}-\frac {\sqrt {2} \sqrt [4]{3} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{2+e x}}{\sqrt [4]{2-e x}}\right )}{e}+\frac {\sqrt {2} \sqrt [4]{3} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{2+e x}}{\sqrt [4]{2-e x} \left (1+\frac {\sqrt {2+e x}}{\sqrt {2-e x}}\right )}\right )}{e} \] Output:

-4*3^(1/4)*(-e^2*x^2+4)^(1/4)/e/(e*x+2)^(1/2)+3^(1/4)*arctan(1-2^(1/2)*(e* 
x+2)^(1/4)/(-e*x+2)^(1/4))*2^(1/2)/e-3^(1/4)*arctan(1+2^(1/2)*(e*x+2)^(1/4 
)/(-e*x+2)^(1/4))*2^(1/2)/e+3^(1/4)*arctanh(2^(1/2)*(e*x+2)^(1/4)/(-e*x+2) 
^(1/4)/(1+(e*x+2)^(1/2)/(-e*x+2)^(1/2)))*2^(1/2)/e
 

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{3/2}} \, dx=\frac {\sqrt [4]{3} \left (-4 \sqrt [4]{4-e^2 x^2}+\sqrt {4+2 e x} \arctan \left (\frac {\sqrt {4+2 e x} \sqrt [4]{4-e^2 x^2}}{2+e x-\sqrt {4-e^2 x^2}}\right )+\sqrt {4+2 e x} \text {arctanh}\left (\frac {2+e x+\sqrt {4-e^2 x^2}}{\sqrt {4+2 e x} \sqrt [4]{4-e^2 x^2}}\right )\right )}{e \sqrt {2+e x}} \] Input:

Integrate[(12 - 3*e^2*x^2)^(1/4)/(2 + e*x)^(3/2),x]
 

Output:

(3^(1/4)*(-4*(4 - e^2*x^2)^(1/4) + Sqrt[4 + 2*e*x]*ArcTan[(Sqrt[4 + 2*e*x] 
*(4 - e^2*x^2)^(1/4))/(2 + e*x - Sqrt[4 - e^2*x^2])] + Sqrt[4 + 2*e*x]*Arc 
Tanh[(2 + e*x + Sqrt[4 - e^2*x^2])/(Sqrt[4 + 2*e*x]*(4 - e^2*x^2)^(1/4))]) 
)/(e*Sqrt[2 + e*x])
 

Rubi [A] (warning: unable to verify)

Time = 0.62 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.17, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {456, 57, 27, 73, 770, 755, 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.

\(\displaystyle \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(e x+2)^{3/2}} \, dx\)

\(\Big \downarrow \) 456

\(\displaystyle \int \frac {\sqrt [4]{6-3 e x}}{(e x+2)^{5/4}}dx\)

\(\Big \downarrow \) 57

\(\displaystyle -3 \int \frac {1}{3^{3/4} (2-e x)^{3/4} \sqrt [4]{e x+2}}dx-\frac {4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{e x+2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\sqrt [4]{3} \int \frac {1}{(2-e x)^{3/4} \sqrt [4]{e x+2}}dx-\frac {4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{e x+2}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {4 \sqrt [4]{3} \int \frac {1}{\sqrt [4]{e x+2}}d\sqrt [4]{2-e x}}{e}-\frac {4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{e x+2}}\)

\(\Big \downarrow \) 770

\(\displaystyle \frac {4 \sqrt [4]{3} \int \frac {1}{3-e x}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}}{e}-\frac {4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{e x+2}}\)

\(\Big \downarrow \) 755

\(\displaystyle \frac {4 \sqrt [4]{3} \left (\frac {1}{2} \int \frac {1-\sqrt {2-e x}}{3-e x}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+\frac {1}{2} \int \frac {\sqrt {2-e x}+1}{3-e x}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{e}-\frac {4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{e x+2}}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {4 \sqrt [4]{3} \left (\frac {1}{2} \int \frac {1-\sqrt {2-e x}}{3-e x}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\sqrt {2-e x}-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+\frac {1}{2} \int \frac {1}{\sqrt {2-e x}+\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )\right )}{e}-\frac {4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{e x+2}}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {4 \sqrt [4]{3} \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\sqrt {2-e x}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\sqrt {2-e x}-1}d\left (\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt {2}}\right )+\frac {1}{2} \int \frac {1-\sqrt {2-e x}}{3-e x}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{e}-\frac {4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{e x+2}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {4 \sqrt [4]{3} \left (\frac {1}{2} \int \frac {1-\sqrt {2-e x}}{3-e x}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{\sqrt {2}}\right )\right )}{e}-\frac {4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{e x+2}}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {4 \sqrt [4]{3} \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}}{\sqrt {2-e x}-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt {2-e x}+\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{\sqrt {2}}\right )\right )}{e}-\frac {4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{e x+2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {4 \sqrt [4]{3} \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}}{\sqrt {2-e x}-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt {2-e x}+\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{\sqrt {2}}\right )\right )}{e}-\frac {4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{e x+2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4 \sqrt [4]{3} \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}}{\sqrt {2-e x}-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1}{\sqrt {2-e x}+\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1}d\frac {\sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{\sqrt {2}}\right )\right )}{e}-\frac {4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{e x+2}}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {4 \sqrt [4]{3} \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\sqrt {2-e x}+\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\sqrt {2-e x}-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{2 \sqrt {2}}\right )\right )}{e}-\frac {4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{e x+2}}\)

Input:

Int[(12 - 3*e^2*x^2)^(1/4)/(2 + e*x)^(3/2),x]
 

Output:

(-4*3^(1/4)*(2 - e*x)^(1/4))/(e*(2 + e*x)^(1/4)) + (4*3^(1/4)*((-(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 + (-1/2*Log[1 + Sqrt[2 - e 
*x] - (Sqrt[2]*(2 - e*x)^(1/4))/(2 + e*x)^(1/4)]/Sqrt[2] + Log[1 + Sqrt[2 
- e*x] + (Sqrt[2]*(2 - e*x)^(1/4))/(2 + e*x)^(1/4)]/(2*Sqrt[2]))/2))/e
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 57

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & 
& GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m 
+ n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c 
, d, m, n, x]
 

rule 73

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]
 

rule 217

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])
 

rule 456

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]))
 

rule 755

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]]))
 

rule 770

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]
 

rule 1082

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]
 

rule 1103

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]
 

rule 1476

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]
 

rule 1479

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]
 

Maple [F]

\[\int \frac {\left (-3 e^{2} x^{2}+12\right )^{\frac {1}{4}}}{\left (e x +2\right )^{\frac {3}{2}}}d x\]

Input:

int((-3*e^2*x^2+12)^(1/4)/(e*x+2)^(3/2),x)
 

Output:

int((-3*e^2*x^2+12)^(1/4)/(e*x+2)^(3/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.43 \[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{3/2}} \, dx=\frac {2 \, \left (\frac {3}{4}\right )^{\frac {1}{4}} {\left (e x + 2\right )} \arctan \left (\frac {3 \, e x + 4 \, \left (\frac {3}{4}\right )^{\frac {3}{4}} {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} \sqrt {e x + 2} + 6}{3 \, {\left (e x + 2\right )}}\right ) + 2 \, \left (\frac {3}{4}\right )^{\frac {1}{4}} {\left (e x + 2\right )} \arctan \left (-\frac {3 \, e x - 4 \, \left (\frac {3}{4}\right )^{\frac {3}{4}} {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} \sqrt {e x + 2} + 6}{3 \, {\left (e x + 2\right )}}\right ) + \left (\frac {3}{4}\right )^{\frac {1}{4}} {\left (e x + 2\right )} \log \left (\frac {\sqrt {3} {\left (e x + 2\right )} + 2 \, \left (\frac {3}{4}\right )^{\frac {1}{4}} {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} \sqrt {e x + 2} + \sqrt {-3 \, e^{2} x^{2} + 12}}{e x + 2}\right ) - \left (\frac {3}{4}\right )^{\frac {1}{4}} {\left (e x + 2\right )} \log \left (\frac {\sqrt {3} {\left (e x + 2\right )} - 2 \, \left (\frac {3}{4}\right )^{\frac {1}{4}} {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} \sqrt {e x + 2} + \sqrt {-3 \, e^{2} x^{2} + 12}}{e x + 2}\right ) - 4 \, {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} \sqrt {e x + 2}}{e^{2} x + 2 \, e} \] Input:

integrate((-3*e^2*x^2+12)^(1/4)/(e*x+2)^(3/2),x, algorithm="fricas")
 

Output:

(2*(3/4)^(1/4)*(e*x + 2)*arctan(1/3*(3*e*x + 4*(3/4)^(3/4)*(-3*e^2*x^2 + 1 
2)^(1/4)*sqrt(e*x + 2) + 6)/(e*x + 2)) + 2*(3/4)^(1/4)*(e*x + 2)*arctan(-1 
/3*(3*e*x - 4*(3/4)^(3/4)*(-3*e^2*x^2 + 12)^(1/4)*sqrt(e*x + 2) + 6)/(e*x 
+ 2)) + (3/4)^(1/4)*(e*x + 2)*log((sqrt(3)*(e*x + 2) + 2*(3/4)^(1/4)*(-3*e 
^2*x^2 + 12)^(1/4)*sqrt(e*x + 2) + sqrt(-3*e^2*x^2 + 12))/(e*x + 2)) - (3/ 
4)^(1/4)*(e*x + 2)*log((sqrt(3)*(e*x + 2) - 2*(3/4)^(1/4)*(-3*e^2*x^2 + 12 
)^(1/4)*sqrt(e*x + 2) + sqrt(-3*e^2*x^2 + 12))/(e*x + 2)) - 4*(-3*e^2*x^2 
+ 12)^(1/4)*sqrt(e*x + 2))/(e^2*x + 2*e)
 

Sympy [F]

\[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{3/2}} \, dx=\sqrt [4]{3} \int \frac {\sqrt [4]{- e^{2} x^{2} + 4}}{e x \sqrt {e x + 2} + 2 \sqrt {e x + 2}}\, dx \] Input:

integrate((-3*e**2*x**2+12)**(1/4)/(e*x+2)**(3/2),x)
 

Output:

3**(1/4)*Integral((-e**2*x**2 + 4)**(1/4)/(e*x*sqrt(e*x + 2) + 2*sqrt(e*x 
+ 2)), x)
 

Maxima [F]

\[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{3/2}} \, dx=\int { \frac {{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}}}{{\left (e x + 2\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((-3*e^2*x^2+12)^(1/4)/(e*x+2)^(3/2),x, algorithm="maxima")
 

Output:

integrate((-3*e^2*x^2 + 12)^(1/4)/(e*x + 2)^(3/2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-3*e^2*x^2+12)^(1/4)/(e*x+2)^(3/2),x, algorithm="giac")
 

Output:

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
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{3/2}} \, dx=\int \frac {{\left (12-3\,e^2\,x^2\right )}^{1/4}}{{\left (e\,x+2\right )}^{3/2}} \,d x \] Input:

int((12 - 3*e^2*x^2)^(1/4)/(e*x + 2)^(3/2),x)
 

Output:

int((12 - 3*e^2*x^2)^(1/4)/(e*x + 2)^(3/2), x)
 

Reduce [F]

\[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{3/2}} \, dx=\int \frac {\left (-3 e^{2} x^{2}+12\right )^{\frac {1}{4}}}{\left (e x +2\right )^{\frac {3}{2}}}d x \] Input:

int((-3*e^2*x^2+12)^(1/4)/(e*x+2)^(3/2),x)
 

Output:

int((-3*e^2*x^2+12)^(1/4)/(e*x+2)^(3/2),x)