### 3.932 $$\int \frac{\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{3/2}} \, dx$$

Optimal. Leaf size=270 $-\frac{4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{e x+2}}-\frac{\sqrt [4]{3} \log \left (\frac{\sqrt{6-3 e x}+\sqrt{3} \sqrt{e x+2}-\sqrt{6} \sqrt [4]{2-e x} \sqrt [4]{e x+2}}{\sqrt{e x+2}}\right )}{\sqrt{2} e}+\frac{\sqrt [4]{3} \log \left (\frac{\sqrt{6-3 e x}+\sqrt{3} \sqrt{e x+2}+\sqrt{6} \sqrt [4]{2-e x} \sqrt [4]{e x+2}}{\sqrt{e x+2}}\right )}{\sqrt{2} e}-\frac{\sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{e}+\frac{\sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{e}$

[Out]

(-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)

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Rubi [A]  time = 0.262523, antiderivative size = 270, normalized size of antiderivative = 1., 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, 47, 63, 240, 211, 1165, 628, 1162, 617, 204} $-\frac{4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{e x+2}}-\frac{\sqrt [4]{3} \log \left (\frac{\sqrt{6-3 e x}+\sqrt{3} \sqrt{e x+2}-\sqrt{6} \sqrt [4]{2-e x} \sqrt [4]{e x+2}}{\sqrt{e x+2}}\right )}{\sqrt{2} e}+\frac{\sqrt [4]{3} \log \left (\frac{\sqrt{6-3 e x}+\sqrt{3} \sqrt{e x+2}+\sqrt{6} \sqrt [4]{2-e x} \sqrt [4]{e x+2}}{\sqrt{e x+2}}\right )}{\sqrt{2} e}-\frac{\sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{e}+\frac{\sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{e}$

Antiderivative was successfully veriﬁed.

[In]

Int[(12 - 3*e^2*x^2)^(1/4)/(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 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 47

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 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 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 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 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]

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 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 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 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])

Rubi steps

\begin{align*} \int \frac{\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{3/2}} \, dx &=\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 \operatorname{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 \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{4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{2+e x}}+\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{4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{2+e x}}-\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{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 ) \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{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 [C]  time = 0.056021, size = 60, normalized size = 0.22 $\frac{(e x-2) \sqrt [4]{12-3 e^2 x^2} \, _2F_1\left (\frac{5}{4},\frac{5}{4};\frac{9}{4};\frac{1}{2}-\frac{e x}{4}\right )}{5 \sqrt{2} e \sqrt [4]{e x+2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [F]  time = 0.485, size = 0, normalized size = 0. \begin{align*} \int{\sqrt [4]{-3\,{e}^{2}{x}^{2}+12} \left ( ex+2 \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}}}{{\left (e x + 2\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B]  time = 2.05574, size = 1593, normalized size = 5.9 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(4*3^(1/4)*sqrt(2)*(e^2*x + 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)*s
qrt(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^2*x + 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)*sq
rt(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^2*x + 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^2*x + 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)) + 8*(-3*e^2*x^2 + 12)^(1/4)*sqrt(e*x + 2))/(e^2*x + 2*e)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \sqrt [4]{3} \int \frac{\sqrt [4]{- e^{2} x^{2} + 4}}{e x \sqrt{e x + 2} + 2 \sqrt{e x + 2}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError