### 3.1921 $$\int (d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2} \, dx$$

Optimal. Leaf size=283 $-\frac{3 \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 c^3 d^3 e^2}+\frac{\left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{16 c^2 d^2 e}+\frac{3 \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{256 c^{7/2} d^{7/2} e^{5/2}}+\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 c d}$

[Out]

(-3*(c*d^2 - a*e^2)^3*(c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(128*c^3*d^3*e^
2) + ((c*d^2 - a*e^2)*(c*d^2 + a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(16*c^2*d^2*e
) + (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(5*c*d) + (3*(c*d^2 - a*e^2)^5*ArcTanh[(c*d^2 + a*e^2 + 2*c*
d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(256*c^(7/2)*d^(7/2)*e^(5/2))

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Rubi [A]  time = 0.161692, antiderivative size = 283, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 35, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.114, Rules used = {640, 612, 621, 206} $-\frac{3 \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 c^3 d^3 e^2}+\frac{\left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{16 c^2 d^2 e}+\frac{3 \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{256 c^{7/2} d^{7/2} e^{5/2}}+\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 c d}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]

[Out]

(-3*(c*d^2 - a*e^2)^3*(c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(128*c^3*d^3*e^
2) + ((c*d^2 - a*e^2)*(c*d^2 + a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(16*c^2*d^2*e
) + (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(5*c*d) + (3*(c*d^2 - a*e^2)^5*ArcTanh[(c*d^2 + a*e^2 + 2*c*
d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(256*c^(7/2)*d^(7/2)*e^(5/2))

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 621

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

Rule 206

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

Rubi steps

\begin{align*} \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx &=\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 c d}+\frac{\left (d^2-\frac{a e^2}{c}\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{2 d}\\ &=\frac{\left (c d^2-a e^2\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{16 c^2 d^2 e}+\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 c d}-\frac{\left (3 \left (c d^2-a e^2\right )^3\right ) \int \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{32 c^2 d^2 e}\\ &=-\frac{3 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^3 d^3 e^2}+\frac{\left (c d^2-a e^2\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{16 c^2 d^2 e}+\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 c d}+\frac{\left (3 \left (c d^2-a e^2\right )^5\right ) \int \frac{1}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{256 c^3 d^3 e^2}\\ &=-\frac{3 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^3 d^3 e^2}+\frac{\left (c d^2-a e^2\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{16 c^2 d^2 e}+\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 c d}+\frac{\left (3 \left (c d^2-a e^2\right )^5\right ) \operatorname{Subst}\left (\int \frac{1}{4 c d e-x^2} \, dx,x,\frac{c d^2+a e^2+2 c d e x}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{128 c^3 d^3 e^2}\\ &=-\frac{3 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^3 d^3 e^2}+\frac{\left (c d^2-a e^2\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{16 c^2 d^2 e}+\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 c d}+\frac{3 \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac{c d^2+a e^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{256 c^{7/2} d^{7/2} e^{5/2}}\\ \end{align*}

Mathematica [A]  time = 2.23107, size = 299, normalized size = 1.06 $\frac{(a e+c d x)^2 \sqrt{(d+e x) (a e+c d x)} \left (-\frac{15 c^2 d^2 \left (c d^2-a e^2\right )^4}{e^2 (a e+c d x)^2}+\frac{10 c^2 d^2 \left (c d^2-a e^2\right )^3}{e (a e+c d x)}+80 c^3 d^3 (d+e x) \left (c d^2-a e^2\right )+\frac{15 c^{3/2} d^{3/2} \sqrt{c d} \left (c d^2-a e^2\right )^{9/2} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right )}{e^{5/2} (a e+c d x)^{5/2} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}+40 \left (c^2 d^3-a c d e^2\right )^2+128 c^4 d^4 (d+e x)^2\right )}{640 c^5 d^5}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]

[Out]

((a*e + c*d*x)^2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(40*(c^2*d^3 - a*c*d*e^2)^2 - (15*c^2*d^2*(c*d^2 - a*e^2)^4)/(e
^2*(a*e + c*d*x)^2) + (10*c^2*d^2*(c*d^2 - a*e^2)^3)/(e*(a*e + c*d*x)) + 80*c^3*d^3*(c*d^2 - a*e^2)*(d + e*x)
+ 128*c^4*d^4*(d + e*x)^2 + (15*c^(3/2)*d^(3/2)*Sqrt[c*d]*(c*d^2 - a*e^2)^(9/2)*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[
e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/(e^(5/2)*(a*e + c*d*x)^(5/2)*Sqrt[(c*d*(d + e*x))/(c*d
^2 - a*e^2)])))/(640*c^5*d^5)

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Maple [B]  time = 0.051, size = 917, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)

[Out]

-3/64/d*e^4/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^3+9/64*e*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)
*x*a-3/128*d^5/e^2*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+15/128*d^3*e^2*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(
d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)*a^2+3/256*d^7/e^2*c^2*ln((1/2*a*e^2+1/2*c*
d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)-1/16/d^2*e^3/c^2*(a*d*e+(a*e
^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^2-15/256*d^5*c*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)*a-3/64*d^4/e*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+3/128/d^3*e^6/c
^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^4-9/64*e^3/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^2+3/64*d
^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a+1/5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d+1/8*d*(a*d*e+(a*e
^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x+1/16*d^2/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-1/8/d*e^2/c*(a*d*e+(a*e^2+c*
d^2)*x+c*d*e*x^2)^(3/2)*x*a+3/64/d^2*e^5/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^3-3/256/d^3*e^8/c^3*l
n((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)*a^5+15/25
6/d*e^6/c^2*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1
/2)*a^4-15/128*d*e^4/c*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))
/(d*e*c)^(1/2)*a^3

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.02347, size = 1790, normalized size = 6.33 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="fricas")

[Out]

[1/2560*(15*(c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 - a^5*e^10
)*sqrt(c*d*e)*log(8*c^2*d^2*e^2*x^2 + c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4 + 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 +
a*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(c*d*e) + 8*(c^2*d^3*e + a*c*d*e^3)*x) + 4*(128*c^5*d^5*e^5*x^4 - 15
*c^5*d^9*e + 70*a*c^4*d^7*e^3 + 128*a^2*c^3*d^5*e^5 - 70*a^3*c^2*d^3*e^7 + 15*a^4*c*d*e^9 + 16*(21*c^5*d^6*e^4
+ 11*a*c^4*d^4*e^6)*x^3 + 8*(31*c^5*d^7*e^3 + 64*a*c^4*d^5*e^5 + a^2*c^3*d^3*e^7)*x^2 + 2*(5*c^5*d^8*e^2 + 23
3*a*c^4*d^6*e^4 + 23*a^2*c^3*d^4*e^6 - 5*a^3*c^2*d^2*e^8)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^4
*d^4*e^3), -1/1280*(15*(c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8
- a^5*e^10)*sqrt(-c*d*e)*arctan(1/2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*s
qrt(-c*d*e)/(c^2*d^2*e^2*x^2 + a*c*d^2*e^2 + (c^2*d^3*e + a*c*d*e^3)*x)) - 2*(128*c^5*d^5*e^5*x^4 - 15*c^5*d^9
*e + 70*a*c^4*d^7*e^3 + 128*a^2*c^3*d^5*e^5 - 70*a^3*c^2*d^3*e^7 + 15*a^4*c*d*e^9 + 16*(21*c^5*d^6*e^4 + 11*a*
c^4*d^4*e^6)*x^3 + 8*(31*c^5*d^7*e^3 + 64*a*c^4*d^5*e^5 + a^2*c^3*d^3*e^7)*x^2 + 2*(5*c^5*d^8*e^2 + 233*a*c^4*
d^6*e^4 + 23*a^2*c^3*d^4*e^6 - 5*a^3*c^2*d^2*e^8)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^4*d^4*e^3
)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 1.31569, size = 536, normalized size = 1.89 \begin{align*} \frac{1}{640} \, \sqrt{c d x^{2} e + c d^{2} x + a x e^{2} + a d e}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \, c d x e^{2} + \frac{{\left (21 \, c^{5} d^{6} e^{5} + 11 \, a c^{4} d^{4} e^{7}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} x + \frac{{\left (31 \, c^{5} d^{7} e^{4} + 64 \, a c^{4} d^{5} e^{6} + a^{2} c^{3} d^{3} e^{8}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} x + \frac{{\left (5 \, c^{5} d^{8} e^{3} + 233 \, a c^{4} d^{6} e^{5} + 23 \, a^{2} c^{3} d^{4} e^{7} - 5 \, a^{3} c^{2} d^{2} e^{9}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} x - \frac{{\left (15 \, c^{5} d^{9} e^{2} - 70 \, a c^{4} d^{7} e^{4} - 128 \, a^{2} c^{3} d^{5} e^{6} + 70 \, a^{3} c^{2} d^{3} e^{8} - 15 \, a^{4} c d e^{10}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} - \frac{3 \,{\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\right )} \sqrt{c d} e^{\left (-\frac{5}{2}\right )} \log \left ({\left | -\sqrt{c d} c d^{2} e^{\frac{1}{2}} - 2 \,{\left (\sqrt{c d} x e^{\frac{1}{2}} - \sqrt{c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} c d e - \sqrt{c d} a e^{\frac{5}{2}} \right |}\right )}{256 \, c^{4} d^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="giac")

[Out]

1/640*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*(2*(4*(2*(8*c*d*x*e^2 + (21*c^5*d^6*e^5 + 11*a*c^4*d^4*e^7)*
e^(-4)/(c^4*d^4))*x + (31*c^5*d^7*e^4 + 64*a*c^4*d^5*e^6 + a^2*c^3*d^3*e^8)*e^(-4)/(c^4*d^4))*x + (5*c^5*d^8*e
^3 + 233*a*c^4*d^6*e^5 + 23*a^2*c^3*d^4*e^7 - 5*a^3*c^2*d^2*e^9)*e^(-4)/(c^4*d^4))*x - (15*c^5*d^9*e^2 - 70*a*
c^4*d^7*e^4 - 128*a^2*c^3*d^5*e^6 + 70*a^3*c^2*d^3*e^8 - 15*a^4*c*d*e^10)*e^(-4)/(c^4*d^4)) - 3/256*(c^5*d^10
- 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 - a^5*e^10)*sqrt(c*d)*e^(-5/2)*l
og(abs(-sqrt(c*d)*c*d^2*e^(1/2) - 2*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))*c*d*e
- sqrt(c*d)*a*e^(5/2)))/(c^4*d^4)