3.1904 \(\int \frac{(d+e x)^2}{(a d e+(c d^2+a e^2) x+c d e x^2)^4} \, dx\)

Optimal. Leaf size=173 \[ -\frac{e^3}{(d+e x) \left (c d^2-a e^2\right )^4}-\frac{3 c d e^2}{\left (c d^2-a e^2\right )^4 (a e+c d x)}+\frac{c d e}{\left (c d^2-a e^2\right )^3 (a e+c d x)^2}-\frac{c d}{3 \left (c d^2-a e^2\right )^2 (a e+c d x)^3}-\frac{4 c d e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^5}+\frac{4 c d e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^5} \]

[Out]

-(c*d)/(3*(c*d^2 - a*e^2)^2*(a*e + c*d*x)^3) + (c*d*e)/((c*d^2 - a*e^2)^3*(a*e + c*d*x)^2) - (3*c*d*e^2)/((c*d
^2 - a*e^2)^4*(a*e + c*d*x)) - e^3/((c*d^2 - a*e^2)^4*(d + e*x)) - (4*c*d*e^3*Log[a*e + c*d*x])/(c*d^2 - a*e^2
)^5 + (4*c*d*e^3*Log[d + e*x])/(c*d^2 - a*e^2)^5

________________________________________________________________________________________

Rubi [A]  time = 0.152527, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {626, 44} \[ -\frac{e^3}{(d+e x) \left (c d^2-a e^2\right )^4}-\frac{3 c d e^2}{\left (c d^2-a e^2\right )^4 (a e+c d x)}+\frac{c d e}{\left (c d^2-a e^2\right )^3 (a e+c d x)^2}-\frac{c d}{3 \left (c d^2-a e^2\right )^2 (a e+c d x)^3}-\frac{4 c d e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^5}+\frac{4 c d e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^5} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

-(c*d)/(3*(c*d^2 - a*e^2)^2*(a*e + c*d*x)^3) + (c*d*e)/((c*d^2 - a*e^2)^3*(a*e + c*d*x)^2) - (3*c*d*e^2)/((c*d
^2 - a*e^2)^4*(a*e + c*d*x)) - e^3/((c*d^2 - a*e^2)^4*(d + e*x)) - (4*c*d*e^3*Log[a*e + c*d*x])/(c*d^2 - a*e^2
)^5 + (4*c*d*e^3*Log[d + e*x])/(c*d^2 - a*e^2)^5

Rule 626

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c*x)/e)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&
IntegerQ[p]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx &=\int \frac{1}{(a e+c d x)^4 (d+e x)^2} \, dx\\ &=\int \left (\frac{c^2 d^2}{\left (c d^2-a e^2\right )^2 (a e+c d x)^4}-\frac{2 c^2 d^2 e}{\left (c d^2-a e^2\right )^3 (a e+c d x)^3}+\frac{3 c^2 d^2 e^2}{\left (c d^2-a e^2\right )^4 (a e+c d x)^2}-\frac{4 c^2 d^2 e^3}{\left (c d^2-a e^2\right )^5 (a e+c d x)}+\frac{e^4}{\left (c d^2-a e^2\right )^4 (d+e x)^2}+\frac{4 c d e^4}{\left (c d^2-a e^2\right )^5 (d+e x)}\right ) \, dx\\ &=-\frac{c d}{3 \left (c d^2-a e^2\right )^2 (a e+c d x)^3}+\frac{c d e}{\left (c d^2-a e^2\right )^3 (a e+c d x)^2}-\frac{3 c d e^2}{\left (c d^2-a e^2\right )^4 (a e+c d x)}-\frac{e^3}{\left (c d^2-a e^2\right )^4 (d+e x)}-\frac{4 c d e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^5}+\frac{4 c d e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^5}\\ \end{align*}

Mathematica [A]  time = 0.149459, size = 157, normalized size = 0.91 \[ \frac{\frac{9 c d e^2 \left (c d^2-a e^2\right )}{a e+c d x}-\frac{3 c d e \left (c d^2-a e^2\right )^2}{(a e+c d x)^2}+\frac{3 c d^2 e^3-3 a e^5}{d+e x}+\frac{c d \left (c d^2-a e^2\right )^3}{(a e+c d x)^3}+12 c d e^3 \log (a e+c d x)-12 c d e^3 \log (d+e x)}{3 \left (a e^2-c d^2\right )^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^2/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

((c*d*(c*d^2 - a*e^2)^3)/(a*e + c*d*x)^3 - (3*c*d*e*(c*d^2 - a*e^2)^2)/(a*e + c*d*x)^2 + (9*c*d*e^2*(c*d^2 - a
*e^2))/(a*e + c*d*x) + (3*c*d^2*e^3 - 3*a*e^5)/(d + e*x) + 12*c*d*e^3*Log[a*e + c*d*x] - 12*c*d*e^3*Log[d + e*
x])/(3*(-(c*d^2) + a*e^2)^5)

________________________________________________________________________________________

Maple [A]  time = 0.056, size = 173, normalized size = 1. \begin{align*} -{\frac{{e}^{3}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{4} \left ( ex+d \right ) }}-4\,{\frac{{e}^{3}cd\ln \left ( ex+d \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{5}}}-{\frac{cd}{3\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{2} \left ( cdx+ae \right ) ^{3}}}+4\,{\frac{{e}^{3}cd\ln \left ( cdx+ae \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{5}}}-3\,{\frac{{e}^{2}cd}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{4} \left ( cdx+ae \right ) }}-{\frac{dec}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{3} \left ( cdx+ae \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x)

[Out]

-e^3/(a*e^2-c*d^2)^4/(e*x+d)-4*e^3/(a*e^2-c*d^2)^5*c*d*ln(e*x+d)-1/3*c*d/(a*e^2-c*d^2)^2/(c*d*x+a*e)^3+4*e^3/(
a*e^2-c*d^2)^5*c*d*ln(c*d*x+a*e)-3*c*d/(a*e^2-c*d^2)^4*e^2/(c*d*x+a*e)-c*d/(a*e^2-c*d^2)^3*e/(c*d*x+a*e)^2

________________________________________________________________________________________

Maxima [B]  time = 1.23983, size = 886, normalized size = 5.12 \begin{align*} -\frac{4 \, c d e^{3} \log \left (c d x + a e\right )}{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}} + \frac{4 \, c d e^{3} \log \left (e x + d\right )}{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}} - \frac{12 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} - 5 \, a c^{2} d^{4} e^{2} + 13 \, a^{2} c d^{2} e^{4} + 3 \, a^{3} e^{6} + 6 \,{\left (c^{3} d^{4} e^{2} + 5 \, a c^{2} d^{2} e^{4}\right )} x^{2} - 2 \,{\left (c^{3} d^{5} e - 8 \, a c^{2} d^{3} e^{3} - 11 \, a^{2} c d e^{5}\right )} x}{3 \,{\left (a^{3} c^{4} d^{9} e^{3} - 4 \, a^{4} c^{3} d^{7} e^{5} + 6 \, a^{5} c^{2} d^{5} e^{7} - 4 \, a^{6} c d^{3} e^{9} + a^{7} d e^{11} +{\left (c^{7} d^{11} e - 4 \, a c^{6} d^{9} e^{3} + 6 \, a^{2} c^{5} d^{7} e^{5} - 4 \, a^{3} c^{4} d^{5} e^{7} + a^{4} c^{3} d^{3} e^{9}\right )} x^{4} +{\left (c^{7} d^{12} - a c^{6} d^{10} e^{2} - 6 \, a^{2} c^{5} d^{8} e^{4} + 14 \, a^{3} c^{4} d^{6} e^{6} - 11 \, a^{4} c^{3} d^{4} e^{8} + 3 \, a^{5} c^{2} d^{2} e^{10}\right )} x^{3} + 3 \,{\left (a c^{6} d^{11} e - 3 \, a^{2} c^{5} d^{9} e^{3} + 2 \, a^{3} c^{4} d^{7} e^{5} + 2 \, a^{4} c^{3} d^{5} e^{7} - 3 \, a^{5} c^{2} d^{3} e^{9} + a^{6} c d e^{11}\right )} x^{2} +{\left (3 \, a^{2} c^{5} d^{10} e^{2} - 11 \, a^{3} c^{4} d^{8} e^{4} + 14 \, a^{4} c^{3} d^{6} e^{6} - 6 \, a^{5} c^{2} d^{4} e^{8} - a^{6} c d^{2} e^{10} + a^{7} e^{12}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="maxima")

[Out]

-4*c*d*e^3*log(c*d*x + a*e)/(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) + 4*c*d*e^3*log(e*x + d)/(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) - 1/3*(12*c^3*d^3*e^3*x^3 + c^3*d^6 - 5*a*c^2*d^4*e^2 + 13*a^2*c*d^2*e^4 + 3*
a^3*e^6 + 6*(c^3*d^4*e^2 + 5*a*c^2*d^2*e^4)*x^2 - 2*(c^3*d^5*e - 8*a*c^2*d^3*e^3 - 11*a^2*c*d*e^5)*x)/(a^3*c^4
*d^9*e^3 - 4*a^4*c^3*d^7*e^5 + 6*a^5*c^2*d^5*e^7 - 4*a^6*c*d^3*e^9 + a^7*d*e^11 + (c^7*d^11*e - 4*a*c^6*d^9*e^
3 + 6*a^2*c^5*d^7*e^5 - 4*a^3*c^4*d^5*e^7 + a^4*c^3*d^3*e^9)*x^4 + (c^7*d^12 - a*c^6*d^10*e^2 - 6*a^2*c^5*d^8*
e^4 + 14*a^3*c^4*d^6*e^6 - 11*a^4*c^3*d^4*e^8 + 3*a^5*c^2*d^2*e^10)*x^3 + 3*(a*c^6*d^11*e - 3*a^2*c^5*d^9*e^3
+ 2*a^3*c^4*d^7*e^5 + 2*a^4*c^3*d^5*e^7 - 3*a^5*c^2*d^3*e^9 + a^6*c*d*e^11)*x^2 + (3*a^2*c^5*d^10*e^2 - 11*a^3
*c^4*d^8*e^4 + 14*a^4*c^3*d^6*e^6 - 6*a^5*c^2*d^4*e^8 - a^6*c*d^2*e^10 + a^7*e^12)*x)

________________________________________________________________________________________

Fricas [B]  time = 2.2465, size = 1669, normalized size = 9.65 \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)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="fricas")

[Out]

-1/3*(c^4*d^8 - 6*a*c^3*d^6*e^2 + 18*a^2*c^2*d^4*e^4 - 10*a^3*c*d^2*e^6 - 3*a^4*e^8 + 12*(c^4*d^5*e^3 - a*c^3*
d^3*e^5)*x^3 + 6*(c^4*d^6*e^2 + 4*a*c^3*d^4*e^4 - 5*a^2*c^2*d^2*e^6)*x^2 - 2*(c^4*d^7*e - 9*a*c^3*d^5*e^3 - 3*
a^2*c^2*d^3*e^5 + 11*a^3*c*d*e^7)*x + 12*(c^4*d^4*e^4*x^4 + a^3*c*d^2*e^6 + (c^4*d^5*e^3 + 3*a*c^3*d^3*e^5)*x^
3 + 3*(a*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*x^2 + (3*a^2*c^2*d^3*e^5 + a^3*c*d*e^7)*x)*log(c*d*x + a*e) - 12*(c^4*
d^4*e^4*x^4 + a^3*c*d^2*e^6 + (c^4*d^5*e^3 + 3*a*c^3*d^3*e^5)*x^3 + 3*(a*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*x^2 +
(3*a^2*c^2*d^3*e^5 + a^3*c*d*e^7)*x)*log(e*x + d))/(a^3*c^5*d^11*e^3 - 5*a^4*c^4*d^9*e^5 + 10*a^5*c^3*d^7*e^7
- 10*a^6*c^2*d^5*e^9 + 5*a^7*c*d^3*e^11 - a^8*d*e^13 + (c^8*d^13*e - 5*a*c^7*d^11*e^3 + 10*a^2*c^6*d^9*e^5 - 1
0*a^3*c^5*d^7*e^7 + 5*a^4*c^4*d^5*e^9 - a^5*c^3*d^3*e^11)*x^4 + (c^8*d^14 - 2*a*c^7*d^12*e^2 - 5*a^2*c^6*d^10*
e^4 + 20*a^3*c^5*d^8*e^6 - 25*a^4*c^4*d^6*e^8 + 14*a^5*c^3*d^4*e^10 - 3*a^6*c^2*d^2*e^12)*x^3 + 3*(a*c^7*d^13*
e - 4*a^2*c^6*d^11*e^3 + 5*a^3*c^5*d^9*e^5 - 5*a^5*c^3*d^5*e^9 + 4*a^6*c^2*d^3*e^11 - a^7*c*d*e^13)*x^2 + (3*a
^2*c^6*d^12*e^2 - 14*a^3*c^5*d^10*e^4 + 25*a^4*c^4*d^8*e^6 - 20*a^5*c^3*d^6*e^8 + 5*a^6*c^2*d^4*e^10 + 2*a^7*c
*d^2*e^12 - a^8*e^14)*x)

________________________________________________________________________________________

Sympy [B]  time = 4.49505, size = 1005, normalized size = 5.81 \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)**2/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**4,x)

[Out]

-4*c*d*e**3*log(x + (-4*a**6*c*d*e**15/(a*e**2 - c*d**2)**5 + 24*a**5*c**2*d**3*e**13/(a*e**2 - c*d**2)**5 - 6
0*a**4*c**3*d**5*e**11/(a*e**2 - c*d**2)**5 + 80*a**3*c**4*d**7*e**9/(a*e**2 - c*d**2)**5 - 60*a**2*c**5*d**9*
e**7/(a*e**2 - c*d**2)**5 + 24*a*c**6*d**11*e**5/(a*e**2 - c*d**2)**5 + 4*a*c*d*e**5 - 4*c**7*d**13*e**3/(a*e*
*2 - c*d**2)**5 + 4*c**2*d**3*e**3)/(8*c**2*d**2*e**4))/(a*e**2 - c*d**2)**5 + 4*c*d*e**3*log(x + (4*a**6*c*d*
e**15/(a*e**2 - c*d**2)**5 - 24*a**5*c**2*d**3*e**13/(a*e**2 - c*d**2)**5 + 60*a**4*c**3*d**5*e**11/(a*e**2 -
c*d**2)**5 - 80*a**3*c**4*d**7*e**9/(a*e**2 - c*d**2)**5 + 60*a**2*c**5*d**9*e**7/(a*e**2 - c*d**2)**5 - 24*a*
c**6*d**11*e**5/(a*e**2 - c*d**2)**5 + 4*a*c*d*e**5 + 4*c**7*d**13*e**3/(a*e**2 - c*d**2)**5 + 4*c**2*d**3*e**
3)/(8*c**2*d**2*e**4))/(a*e**2 - c*d**2)**5 - (3*a**3*e**6 + 13*a**2*c*d**2*e**4 - 5*a*c**2*d**4*e**2 + c**3*d
**6 + 12*c**3*d**3*e**3*x**3 + x**2*(30*a*c**2*d**2*e**4 + 6*c**3*d**4*e**2) + x*(22*a**2*c*d*e**5 + 16*a*c**2
*d**3*e**3 - 2*c**3*d**5*e))/(3*a**7*d*e**11 - 12*a**6*c*d**3*e**9 + 18*a**5*c**2*d**5*e**7 - 12*a**4*c**3*d**
7*e**5 + 3*a**3*c**4*d**9*e**3 + x**4*(3*a**4*c**3*d**3*e**9 - 12*a**3*c**4*d**5*e**7 + 18*a**2*c**5*d**7*e**5
- 12*a*c**6*d**9*e**3 + 3*c**7*d**11*e) + x**3*(9*a**5*c**2*d**2*e**10 - 33*a**4*c**3*d**4*e**8 + 42*a**3*c**
4*d**6*e**6 - 18*a**2*c**5*d**8*e**4 - 3*a*c**6*d**10*e**2 + 3*c**7*d**12) + x**2*(9*a**6*c*d*e**11 - 27*a**5*
c**2*d**3*e**9 + 18*a**4*c**3*d**5*e**7 + 18*a**3*c**4*d**7*e**5 - 27*a**2*c**5*d**9*e**3 + 9*a*c**6*d**11*e)
+ x*(3*a**7*e**12 - 3*a**6*c*d**2*e**10 - 18*a**5*c**2*d**4*e**8 + 42*a**4*c**3*d**6*e**6 - 33*a**3*c**4*d**8*
e**4 + 9*a**2*c**5*d**10*e**2))

________________________________________________________________________________________

Giac [B]  time = 1.28201, size = 907, normalized size = 5.24 \begin{align*} \frac{8 \,{\left (c^{3} d^{5} e^{3} - 2 \, a c^{2} d^{3} e^{5} + a^{2} c d e^{7}\right )} \arctan \left (-\frac{2 \, c d x e + c d^{2} + a e^{2}}{\sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}}\right )}{{\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )} \sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} - \frac{12 \, c^{5} d^{7} x^{5} e^{5} + 30 \, c^{5} d^{8} x^{4} e^{4} + 22 \, c^{5} d^{9} x^{3} e^{3} + 3 \, c^{5} d^{10} x^{2} e^{2} + c^{5} d^{12} - 24 \, a c^{4} d^{5} x^{5} e^{7} - 30 \, a c^{4} d^{6} x^{4} e^{6} + 32 \, a c^{4} d^{7} x^{3} e^{5} + 51 \, a c^{4} d^{8} x^{2} e^{4} + 6 \, a c^{4} d^{9} x e^{3} - 7 \, a c^{4} d^{10} e^{2} + 12 \, a^{2} c^{3} d^{3} x^{5} e^{9} - 30 \, a^{2} c^{3} d^{4} x^{4} e^{8} - 108 \, a^{2} c^{3} d^{5} x^{3} e^{7} - 54 \, a^{2} c^{3} d^{6} x^{2} e^{6} + 36 \, a^{2} c^{3} d^{7} x e^{5} + 24 \, a^{2} c^{3} d^{8} e^{4} + 30 \, a^{3} c^{2} d^{2} x^{4} e^{10} + 32 \, a^{3} c^{2} d^{3} x^{3} e^{9} - 54 \, a^{3} c^{2} d^{4} x^{2} e^{8} - 84 \, a^{3} c^{2} d^{5} x e^{7} - 28 \, a^{3} c^{2} d^{6} e^{6} + 22 \, a^{4} c d x^{3} e^{11} + 51 \, a^{4} c d^{2} x^{2} e^{10} + 36 \, a^{4} c d^{3} x e^{9} + 7 \, a^{4} c d^{4} e^{8} + 3 \, a^{5} x^{2} e^{12} + 6 \, a^{5} d x e^{11} + 3 \, a^{5} d^{2} e^{10}}{3 \,{\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )}{\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="giac")

[Out]

8*(c^3*d^5*e^3 - 2*a*c^2*d^3*e^5 + a^2*c*d*e^7)*arctan(-(2*c*d*x*e + c*d^2 + a*e^2)/sqrt(-c^2*d^4 + 2*a*c*d^2*
e^2 - a^2*e^4))/((c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 -
6*a^5*c*d^2*e^10 + a^6*e^12)*sqrt(-c^2*d^4 + 2*a*c*d^2*e^2 - a^2*e^4)) - 1/3*(12*c^5*d^7*x^5*e^5 + 30*c^5*d^8
*x^4*e^4 + 22*c^5*d^9*x^3*e^3 + 3*c^5*d^10*x^2*e^2 + c^5*d^12 - 24*a*c^4*d^5*x^5*e^7 - 30*a*c^4*d^6*x^4*e^6 +
32*a*c^4*d^7*x^3*e^5 + 51*a*c^4*d^8*x^2*e^4 + 6*a*c^4*d^9*x*e^3 - 7*a*c^4*d^10*e^2 + 12*a^2*c^3*d^3*x^5*e^9 -
30*a^2*c^3*d^4*x^4*e^8 - 108*a^2*c^3*d^5*x^3*e^7 - 54*a^2*c^3*d^6*x^2*e^6 + 36*a^2*c^3*d^7*x*e^5 + 24*a^2*c^3*
d^8*e^4 + 30*a^3*c^2*d^2*x^4*e^10 + 32*a^3*c^2*d^3*x^3*e^9 - 54*a^3*c^2*d^4*x^2*e^8 - 84*a^3*c^2*d^5*x*e^7 - 2
8*a^3*c^2*d^6*e^6 + 22*a^4*c*d*x^3*e^11 + 51*a^4*c*d^2*x^2*e^10 + 36*a^4*c*d^3*x*e^9 + 7*a^4*c*d^4*e^8 + 3*a^5
*x^2*e^12 + 6*a^5*d*x*e^11 + 3*a^5*d^2*e^10)/((c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d
^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12)*(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)^3)