\(\int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{3/2}} \, dx\) [932]

Optimal result

Integrand size = 24, antiderivative size = 270 \[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{3/2}} \, dx=-\frac {4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{2+e x}}-\frac {\sqrt {2} \sqrt [4]{3} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{2+e x}}\right )}{e}+\frac {\sqrt {2} \sqrt [4]{3} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{2+e x}}\right )}{e}-\frac {\sqrt [4]{3} \log \left (\frac {\sqrt {6-3 e x}-\sqrt {6} \sqrt [4]{2-e x} \sqrt [4]{2+e x}+\sqrt {3} \sqrt {2+e x}}{\sqrt {2+e x}}\right )}{\sqrt {2} e}+\frac {\sqrt [4]{3} \log \left (\frac {\sqrt {6-3 e x}+\sqrt {6} \sqrt [4]{2-e x} \sqrt [4]{2+e x}+\sqrt {3} \sqrt {2+e x}}{\sqrt {2+e x}}\right )}{\sqrt {2} e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {689, 49, 65, 246, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{3/2}} \, dx=-\frac {\sqrt {2} \sqrt [4]{3} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{e}+\frac {\sqrt {2} \sqrt [4]{3} \arctan \left (\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{e}-\frac {4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{e x+2}}-\frac {\sqrt [4]{3} \log \left (\frac {\sqrt {6-3 e x}+\sqrt {3} \sqrt {e x+2}-\sqrt {6} \sqrt [4]{2-e x} \sqrt [4]{e x+2}}{\sqrt {e x+2}}\right )}{\sqrt {2} e}+\frac {\sqrt [4]{3} \log \left (\frac {\sqrt {6-3 e x}+\sqrt {3} \sqrt {e x+2}+\sqrt {6} \sqrt [4]{2-e x} \sqrt [4]{e x+2}}{\sqrt {e x+2}}\right )}{\sqrt {2} e} \]

[In]

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

[Out]

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

Rule 49

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] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && 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 65

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 210

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 217

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 246

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 631

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 642

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 689

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c/e)*x)^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 1176

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 1179

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*} \text {integral}& = \int \frac {\sqrt [4]{6-3 e x}}{(2+e x)^{5/4}} \, dx \\ & = -\frac {4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{2+e x}}-3 \int \frac {1}{(6-3 e x)^{3/4} \sqrt [4]{2+e x}} \, dx \\ & = -\frac {4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{2+e x}}+\frac {4 \text {Subst}\left (\int \frac {1}{\sqrt [4]{4-\frac {x^4}{3}}} \, dx,x,\sqrt [4]{6-3 e x}\right )}{e} \\ & = -\frac {4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{2+e x}}+\frac {4 \text {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 {4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{2+e x}}+\frac {2 \text {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 \text {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 {4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{2+e x}}-\frac {\sqrt [4]{3} \text {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} \text {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} \text {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} \text {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 {4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{2+e x}}-\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 ) \text {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 ) \text {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 {4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{2+e x}}-\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*}

Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.56 \[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{3/2}} \, dx=\frac {\sqrt [4]{3} \left (-4 \sqrt [4]{4-e^2 x^2}+\sqrt {4+2 e x} \arctan \left (\frac {\sqrt {4+2 e x} \sqrt [4]{4-e^2 x^2}}{2+e x-\sqrt {4-e^2 x^2}}\right )+\sqrt {4+2 e x} \text {arctanh}\left (\frac {2+e x+\sqrt {4-e^2 x^2}}{\sqrt {4+2 e x} \sqrt [4]{4-e^2 x^2}}\right )\right )}{e \sqrt {2+e x}} \]

[In]

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

[Out]

(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 - Sqr
t[4 - e^2*x^2])] + 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))
]))/(e*Sqrt[2 + e*x])

Maple [F]

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

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.32 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{3/2}} \, dx=\frac {3^{\frac {1}{4}} {\left (e^{2} x + 2 \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} \log \left (\frac {3^{\frac {1}{4}} {\left (e^{2} x + 2 \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} + {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} \sqrt {e x + 2}}{e x + 2}\right ) - 3^{\frac {1}{4}} {\left (e^{2} x + 2 \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {3^{\frac {1}{4}} {\left (e^{2} x + 2 \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} - {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} \sqrt {e x + 2}}{e x + 2}\right ) + 3^{\frac {1}{4}} {\left (i \, e^{2} x + 2 i \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} \log \left (\frac {3^{\frac {1}{4}} {\left (i \, e^{2} x + 2 i \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} + {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} \sqrt {e x + 2}}{e x + 2}\right ) + 3^{\frac {1}{4}} {\left (-i \, e^{2} x - 2 i \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} \log \left (\frac {3^{\frac {1}{4}} {\left (-i \, e^{2} x - 2 i \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} + {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} \sqrt {e x + 2}}{e x + 2}\right ) - 4 \, {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} \sqrt {e x + 2}}{e^{2} x + 2 \, e} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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