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

Optimal. Leaf size=269 \[ \frac {\sqrt [4]{3} \sqrt [4]{2-e x} (e x+2)^{3/4}}{e}+\frac {\sqrt [4]{3} \log \left (\frac {\sqrt {6-3 e x}+\sqrt {3} \sqrt {e x+2}-\sqrt {6} \sqrt [4]{2-e x} \sqrt [4]{e x+2}}{\sqrt {e x+2}}\right )}{\sqrt {2} e}-\frac {\sqrt [4]{3} \log \left (\frac {\sqrt {6-3 e x}+\sqrt {3} \sqrt {e x+2}+\sqrt {6} \sqrt [4]{2-e x} \sqrt [4]{e x+2}}{\sqrt {e x+2}}\right )}{\sqrt {2} e}+\frac {\sqrt {2} \sqrt [4]{3} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{e}-\frac {\sqrt {2} \sqrt [4]{3} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{e} \]

[Out]

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

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Rubi [A]  time = 0.26, antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {675, 50, 63, 240, 211, 1165, 628, 1162, 617, 204} \[ \frac {\sqrt [4]{3} \sqrt [4]{2-e x} (e x+2)^{3/4}}{e}+\frac {\sqrt [4]{3} \log \left (\frac {\sqrt {6-3 e x}+\sqrt {3} \sqrt {e x+2}-\sqrt {6} \sqrt [4]{2-e x} \sqrt [4]{e x+2}}{\sqrt {e x+2}}\right )}{\sqrt {2} e}-\frac {\sqrt [4]{3} \log \left (\frac {\sqrt {6-3 e x}+\sqrt {3} \sqrt {e x+2}+\sqrt {6} \sqrt [4]{2-e x} \sqrt [4]{e x+2}}{\sqrt {e x+2}}\right )}{\sqrt {2} e}+\frac {\sqrt {2} \sqrt [4]{3} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{e}-\frac {\sqrt {2} \sqrt [4]{3} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3^(1/4)*(2 - e*x)^(1/4)*(2 + e*x)^(3/4))/e + (Sqrt[2]*3^(1/4)*ArcTan[1 - (Sqrt[2]*(2 - e*x)^(1/4))/(2 + e*x)^
(1/4)])/e - (Sqrt[2]*3^(1/4)*ArcTan[1 + (Sqrt[2]*(2 - e*x)^(1/4))/(2 + e*x)^(1/4)])/e + (3^(1/4)*Log[(Sqrt[6 -
3*e*x] - Sqrt[6]*(2 - e*x)^(1/4)*(2 + e*x)^(1/4) + Sqrt[3]*Sqrt[2 + e*x])/Sqrt[2 + e*x]])/(Sqrt[2]*e) - (3^(1
/4)*Log[(Sqrt[6 - 3*e*x] + Sqrt[6]*(2 - e*x)^(1/4)*(2 + e*x)^(1/4) + Sqrt[3]*Sqrt[2 + e*x])/Sqrt[2 + e*x]])/(S
qrt[2]*e)

Rule 50

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] + Dist[(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] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[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 240

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[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 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[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])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(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 675

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^p,
x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && GtQ[a, 0] && GtQ[d, 0] &&  !I
GtQ[m, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[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 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps

\begin {align*} \int \frac {\sqrt [4]{12-3 e^2 x^2}}{\sqrt {2+e x}} \, dx &=\int \frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{2+e x}} \, dx\\ &=\frac {\sqrt [4]{3} \sqrt [4]{2-e x} (2+e x)^{3/4}}{e}+3 \int \frac {1}{(6-3 e x)^{3/4} \sqrt [4]{2+e x}} \, dx\\ &=\frac {\sqrt [4]{3} \sqrt [4]{2-e x} (2+e x)^{3/4}}{e}-\frac {4 \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{4-\frac {x^4}{3}}} \, dx,x,\sqrt [4]{6-3 e x}\right )}{e}\\ &=\frac {\sqrt [4]{3} \sqrt [4]{2-e x} (2+e x)^{3/4}}{e}-\frac {4 \operatorname {Subst}\left (\int \frac {1}{1+\frac {x^4}{3}} \, dx,x,\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{2+e x}}\right )}{e}\\ &=\frac {\sqrt [4]{3} \sqrt [4]{2-e x} (2+e x)^{3/4}}{e}-\frac {2 \operatorname {Subst}\left (\int \frac {\sqrt {3}-x^2}{1+\frac {x^4}{3}} \, dx,x,\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{2+e x}}\right )}{\sqrt {3} e}-\frac {2 \operatorname {Subst}\left (\int \frac {\sqrt {3}+x^2}{1+\frac {x^4}{3}} \, dx,x,\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{2+e x}}\right )}{\sqrt {3} e}\\ &=\frac {\sqrt [4]{3} \sqrt [4]{2-e x} (2+e x)^{3/4}}{e}+\frac {\sqrt [4]{3} \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{3}+2 x}{-\sqrt {3}-\sqrt {2} \sqrt [4]{3} x-x^2} \, dx,x,\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{2+e x}}\right )}{\sqrt {2} e}+\frac {\sqrt [4]{3} \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{3}-2 x}{-\sqrt {3}+\sqrt {2} \sqrt [4]{3} x-x^2} \, dx,x,\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{2+e x}}\right )}{\sqrt {2} e}-\frac {\sqrt {3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {3}-\sqrt {2} \sqrt [4]{3} x+x^2} \, dx,x,\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{2+e x}}\right )}{e}-\frac {\sqrt {3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {3}+\sqrt {2} \sqrt [4]{3} x+x^2} \, dx,x,\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{2+e x}}\right )}{e}\\ &=\frac {\sqrt [4]{3} \sqrt [4]{2-e x} (2+e x)^{3/4}}{e}+\frac {\sqrt [4]{3} \log \left (\frac {\sqrt {2-e x}-\sqrt {2} \sqrt [4]{2-e x} \sqrt [4]{2+e x}+\sqrt {2+e x}}{\sqrt {2+e x}}\right )}{\sqrt {2} e}-\frac {\sqrt [4]{3} \log \left (\frac {\sqrt {2-e x}+\sqrt {2} \sqrt [4]{2-e x} \sqrt [4]{2+e x}+\sqrt {2+e x}}{\sqrt {2+e x}}\right )}{\sqrt {2} e}-\frac {\left (\sqrt {2} \sqrt [4]{3}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{2+e x}}\right )}{e}+\frac {\left (\sqrt {2} \sqrt [4]{3}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{2+e x}}\right )}{e}\\ &=\frac {\sqrt [4]{3} \sqrt [4]{2-e x} (2+e x)^{3/4}}{e}+\frac {\sqrt {2} \sqrt [4]{3} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{2+e x}}\right )}{e}-\frac {\sqrt {2} \sqrt [4]{3} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{2+e x}}\right )}{e}+\frac {\sqrt [4]{3} \log \left (\frac {\sqrt {2-e x}-\sqrt {2} \sqrt [4]{2-e x} \sqrt [4]{2+e x}+\sqrt {2+e x}}{\sqrt {2+e x}}\right )}{\sqrt {2} e}-\frac {\sqrt [4]{3} \log \left (\frac {\sqrt {2-e x}+\sqrt {2} \sqrt [4]{2-e x} \sqrt [4]{2+e x}+\sqrt {2+e x}}{\sqrt {2+e x}}\right )}{\sqrt {2} e}\\ \end {align*}

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Mathematica [C]  time = 0.05, size = 60, normalized size = 0.22 \[ \frac {2 \sqrt {2} (e x-2) \sqrt [4]{12-3 e^2 x^2} \, _2F_1\left (\frac {1}{4},\frac {5}{4};\frac {9}{4};\frac {1}{2}-\frac {e x}{4}\right )}{5 e \sqrt [4]{e x+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[2]*(-2 + e*x)*(12 - 3*e^2*x^2)^(1/4)*Hypergeometric2F1[1/4, 5/4, 9/4, 1/2 - (e*x)/4])/(5*e*(2 + e*x)^(
1/4))

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fricas [B]  time = 0.99, size = 536, normalized size = 1.99 \[ \frac {4 \cdot 3^{\frac {1}{4}} \sqrt {2} e \frac {1}{e^{4}}^{\frac {1}{4}} \arctan \left (-\frac {3^{\frac {3}{4}} \sqrt {2} {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} \sqrt {e x + 2} e^{3} \frac {1}{e^{4}}^{\frac {3}{4}} - 3^{\frac {3}{4}} \sqrt {2} {\left (e^{4} x + 2 \, e^{3}\right )} \sqrt {\frac {3^{\frac {1}{4}} \sqrt {2} {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} \sqrt {e x + 2} e \frac {1}{e^{4}}^{\frac {1}{4}} + \sqrt {3} {\left (e^{3} x + 2 \, e^{2}\right )} \sqrt {\frac {1}{e^{4}}} + \sqrt {-3 \, e^{2} x^{2} + 12}}{e x + 2}} \frac {1}{e^{4}}^{\frac {3}{4}} + 3 \, e x + 6}{3 \, {\left (e x + 2\right )}}\right ) + 4 \cdot 3^{\frac {1}{4}} \sqrt {2} e \frac {1}{e^{4}}^{\frac {1}{4}} \arctan \left (-\frac {3^{\frac {3}{4}} \sqrt {2} {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} \sqrt {e x + 2} e^{3} \frac {1}{e^{4}}^{\frac {3}{4}} - 3^{\frac {3}{4}} \sqrt {2} {\left (e^{4} x + 2 \, e^{3}\right )} \sqrt {-\frac {3^{\frac {1}{4}} \sqrt {2} {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} \sqrt {e x + 2} e \frac {1}{e^{4}}^{\frac {1}{4}} - \sqrt {3} {\left (e^{3} x + 2 \, e^{2}\right )} \sqrt {\frac {1}{e^{4}}} - \sqrt {-3 \, e^{2} x^{2} + 12}}{e x + 2}} \frac {1}{e^{4}}^{\frac {3}{4}} - 3 \, e x - 6}{3 \, {\left (e x + 2\right )}}\right ) - 3^{\frac {1}{4}} \sqrt {2} e \frac {1}{e^{4}}^{\frac {1}{4}} \log \left (\frac {3^{\frac {1}{4}} \sqrt {2} {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} \sqrt {e x + 2} e \frac {1}{e^{4}}^{\frac {1}{4}} + \sqrt {3} {\left (e^{3} x + 2 \, e^{2}\right )} \sqrt {\frac {1}{e^{4}}} + \sqrt {-3 \, e^{2} x^{2} + 12}}{e x + 2}\right ) + 3^{\frac {1}{4}} \sqrt {2} e \frac {1}{e^{4}}^{\frac {1}{4}} \log \left (-\frac {3^{\frac {1}{4}} \sqrt {2} {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} \sqrt {e x + 2} e \frac {1}{e^{4}}^{\frac {1}{4}} - \sqrt {3} {\left (e^{3} x + 2 \, e^{2}\right )} \sqrt {\frac {1}{e^{4}}} - \sqrt {-3 \, e^{2} x^{2} + 12}}{e x + 2}\right ) + 2 \, {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} \sqrt {e x + 2}}{2 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.39, size = 172, normalized size = 0.64 \[ -\frac {1}{2} \cdot 3^{\frac {1}{4}} {\left (2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (\frac {4}{x e + 2} - 1\right )}^{\frac {1}{4}}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (\frac {4}{x e + 2} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \sqrt {2} \log \left (\sqrt {2} {\left (\frac {4}{x e + 2} - 1\right )}^{\frac {1}{4}} + \sqrt {\frac {4}{x e + 2} - 1} + 1\right ) - \sqrt {2} \log \left (-\sqrt {2} {\left (\frac {4}{x e + 2} - 1\right )}^{\frac {1}{4}} + \sqrt {\frac {4}{x e + 2} - 1} + 1\right ) - 2 \, {\left (x e + 2\right )} {\left (\frac {4}{x e + 2} - 1\right )}^{\frac {1}{4}}\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {\left (-3 e^{2} x^{2}+12\right )^{\frac {1}{4}}}{\sqrt {e x +2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}}}{\sqrt {e x + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (12-3\,e^2\,x^2\right )}^{1/4}}{\sqrt {e\,x+2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \sqrt [4]{3} \int \frac {\sqrt [4]{- e^{2} x^{2} + 4}}{\sqrt {e x + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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