Integrand size = 27, antiderivative size = 305 \[ \int \frac {A+B x+C x^2}{(d+e x)^3 \left (a+c x^2\right )} \, dx=-\frac {C d^2-B d e+A e^2}{2 e \left (c d^2+a e^2\right ) (d+e x)^2}+\frac {B c d^2-2 A c d e+2 a C d e-a B e^2}{\left (c d^2+a e^2\right )^2 (d+e x)}+\frac {\sqrt {c} \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \left (c d^2+a e^2\right )^3}-\frac {\left (B c d \left (c d^2-3 a e^2\right )-(A c-a C) e \left (3 c d^2-a e^2\right )\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^3}+\frac {\left (B c d \left (c d^2-3 a e^2\right )-(A c-a C) e \left (3 c d^2-a e^2\right )\right ) \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^3} \]
1/2*(-A*e^2+B*d*e-C*d^2)/e/(a*e^2+c*d^2)/(e*x+d)^2+(-2*A*c*d*e-B*a*e^2+B*c *d^2+2*C*a*d*e)/(a*e^2+c*d^2)^2/(e*x+d)-(B*c*d*(-3*a*e^2+c*d^2)-(A*c-C*a)* e*(-a*e^2+3*c*d^2))*ln(e*x+d)/(a*e^2+c*d^2)^3+1/2*(B*c*d*(-3*a*e^2+c*d^2)- (A*c-C*a)*e*(-a*e^2+3*c*d^2))*ln(c*x^2+a)/(a*e^2+c*d^2)^3+(A*c*d*(-3*a*e^2 +c*d^2)-a*(c*d^2*(-3*B*e+C*d)-a*e^2*(-B*e+3*C*d)))*arctan(x*c^(1/2)/a^(1/2 ))*c^(1/2)/(a*e^2+c*d^2)^3/a^(1/2)
Time = 0.18 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.91 \[ \int \frac {A+B x+C x^2}{(d+e x)^3 \left (a+c x^2\right )} \, dx=\frac {-\frac {\left (c d^2+a e^2\right )^2 \left (C d^2+e (-B d+A e)\right )}{e (d+e x)^2}+\frac {2 \left (c d^2+a e^2\right ) \left (B c d^2-2 A c d e+2 a C d e-a B e^2\right )}{d+e x}+\frac {2 \sqrt {c} \left (A c d \left (c d^2-3 a e^2\right )+a \left (a e^2 (3 C d-B e)+c d^2 (-C d+3 B e)\right )\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a}}-2 \left (B c d \left (c d^2-3 a e^2\right )-(A c-a C) e \left (3 c d^2-a e^2\right )\right ) \log (d+e x)+\left (B c d \left (c d^2-3 a e^2\right )-(A c-a C) e \left (3 c d^2-a e^2\right )\right ) \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^3} \]
(-(((c*d^2 + a*e^2)^2*(C*d^2 + e*(-(B*d) + A*e)))/(e*(d + e*x)^2)) + (2*(c *d^2 + a*e^2)*(B*c*d^2 - 2*A*c*d*e + 2*a*C*d*e - a*B*e^2))/(d + e*x) + (2* Sqrt[c]*(A*c*d*(c*d^2 - 3*a*e^2) + a*(a*e^2*(3*C*d - B*e) + c*d^2*(-(C*d) + 3*B*e)))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/Sqrt[a] - 2*(B*c*d*(c*d^2 - 3*a*e^ 2) - (A*c - a*C)*e*(3*c*d^2 - a*e^2))*Log[d + e*x] + (B*c*d*(c*d^2 - 3*a*e ^2) - (A*c - a*C)*e*(3*c*d^2 - a*e^2))*Log[a + c*x^2])/(2*(c*d^2 + a*e^2)^ 3)
Time = 0.70 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2160, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {A+B x+C x^2}{\left (a+c x^2\right ) (d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 2160 |
| \(\displaystyle \int \left (\frac {c \left (x \left (B c d \left (c d^2-3 a e^2\right )-e (A c-a C) \left (3 c d^2-a e^2\right )\right )+A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right )}{\left (a+c x^2\right ) \left (a e^2+c d^2\right )^3}+\frac {A e^2-B d e+C d^2}{(d+e x)^3 \left (a e^2+c d^2\right )}+\frac {e \left (e (A c-a C) \left (3 c d^2-a e^2\right )-B c d \left (c d^2-3 a e^2\right )\right )}{(d+e x) \left (a e^2+c d^2\right )^3}+\frac {e \left (a B e^2-2 a C d e+2 A c d e-B c d^2\right )}{(d+e x)^2 \left (a e^2+c d^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {c} \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right )}{\sqrt {a} \left (a e^2+c d^2\right )^3}+\frac {\log \left (a+c x^2\right ) \left (B c d \left (c d^2-3 a e^2\right )-e (A c-a C) \left (3 c d^2-a e^2\right )\right )}{2 \left (a e^2+c d^2\right )^3}-\frac {A e^2-B d e+C d^2}{2 e (d+e x)^2 \left (a e^2+c d^2\right )}+\frac {-a B e^2+2 a C d e-2 A c d e+B c d^2}{(d+e x) \left (a e^2+c d^2\right )^2}-\frac {\log (d+e x) \left (B c d \left (c d^2-3 a e^2\right )-e (A c-a C) \left (3 c d^2-a e^2\right )\right )}{\left (a e^2+c d^2\right )^3}\) |
-1/2*(C*d^2 - B*d*e + A*e^2)/(e*(c*d^2 + a*e^2)*(d + e*x)^2) + (B*c*d^2 - 2*A*c*d*e + 2*a*C*d*e - a*B*e^2)/((c*d^2 + a*e^2)^2*(d + e*x)) + (Sqrt[c]* (A*c*d*(c*d^2 - 3*a*e^2) - a*(c*d^2*(C*d - 3*B*e) - a*e^2*(3*C*d - B*e)))* ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[a]*(c*d^2 + a*e^2)^3) - ((B*c*d*(c*d^2 - 3*a*e^2) - (A*c - a*C)*e*(3*c*d^2 - a*e^2))*Log[d + e*x])/(c*d^2 + a*e^2 )^3 + ((B*c*d*(c*d^2 - 3*a*e^2) - (A*c - a*C)*e*(3*c*d^2 - a*e^2))*Log[a + c*x^2])/(2*(c*d^2 + a*e^2)^3)
3.1.49.3.1 Defintions of rubi rules used
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Time = 0.60 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.04
| method | result | size |
| default | \(-\frac {c \left (\frac {\left (-A a c \,e^{3}+3 A \,c^{2} d^{2} e +3 B a c d \,e^{2}-B \,c^{2} d^{3}+C \,a^{2} e^{3}-3 C a c \,d^{2} e \right ) \ln \left (c \,x^{2}+a \right )}{2 c}+\frac {\left (3 A a c d \,e^{2}-A \,d^{3} c^{2}+a^{2} B \,e^{3}-3 B a c \,d^{2} e -3 C \,a^{2} d \,e^{2}+C a c \,d^{3}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}\right )}{\left (e^{2} a +c \,d^{2}\right )^{3}}-\frac {A \,e^{2}-B d e +C \,d^{2}}{2 \left (e^{2} a +c \,d^{2}\right ) e \left (e x +d \right )^{2}}-\frac {2 A c d e +B a \,e^{2}-B c \,d^{2}-2 a d e C}{\left (e^{2} a +c \,d^{2}\right )^{2} \left (e x +d \right )}-\frac {\left (A a c \,e^{3}-3 A \,c^{2} d^{2} e -3 B a c d \,e^{2}+B \,c^{2} d^{3}-C \,a^{2} e^{3}+3 C a c \,d^{2} e \right ) \ln \left (e x +d \right )}{\left (e^{2} a +c \,d^{2}\right )^{3}}\) | \(317\) |
| risch | \(\text {Expression too large to display}\) | \(1358\) |
-c/(a*e^2+c*d^2)^3*(1/2*(-A*a*c*e^3+3*A*c^2*d^2*e+3*B*a*c*d*e^2-B*c^2*d^3+ C*a^2*e^3-3*C*a*c*d^2*e)/c*ln(c*x^2+a)+(3*A*a*c*d*e^2-A*c^2*d^3+B*a^2*e^3- 3*B*a*c*d^2*e-3*C*a^2*d*e^2+C*a*c*d^3)/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2)) )-1/2*(A*e^2-B*d*e+C*d^2)/(a*e^2+c*d^2)/e/(e*x+d)^2-(2*A*c*d*e+B*a*e^2-B*c *d^2-2*C*a*d*e)/(a*e^2+c*d^2)^2/(e*x+d)-(A*a*c*e^3-3*A*c^2*d^2*e-3*B*a*c*d *e^2+B*c^2*d^3-C*a^2*e^3+3*C*a*c*d^2*e)/(a*e^2+c*d^2)^3*ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 868 vs. \(2 (291) = 582\).
Time = 76.07 (sec) , antiderivative size = 1759, normalized size of antiderivative = 5.77 \[ \int \frac {A+B x+C x^2}{(d+e x)^3 \left (a+c x^2\right )} \, dx=\text {Too large to display} \]
[-1/2*(C*c^2*d^6 - 3*B*c^2*d^5*e - 2*B*a*c*d^3*e^3 + B*a^2*d*e^5 + A*a^2*e ^6 - (2*C*a*c - 5*A*c^2)*d^4*e^2 - 3*(C*a^2 - 2*A*a*c)*d^2*e^4 + (3*B*a*c* d^4*e^2 - B*a^2*d^2*e^4 - (C*a*c - A*c^2)*d^5*e + 3*(C*a^2 - A*a*c)*d^3*e^ 3 + (3*B*a*c*d^2*e^4 - B*a^2*e^6 - (C*a*c - A*c^2)*d^3*e^3 + 3*(C*a^2 - A* a*c)*d*e^5)*x^2 + 2*(3*B*a*c*d^3*e^3 - B*a^2*d*e^5 - (C*a*c - A*c^2)*d^4*e ^2 + 3*(C*a^2 - A*a*c)*d^2*e^4)*x)*sqrt(-c/a)*log((c*x^2 - 2*a*x*sqrt(-c/a ) - a)/(c*x^2 + a)) - 2*(B*c^2*d^4*e^2 - B*a^2*e^6 + 2*(C*a*c - A*c^2)*d^3 *e^3 + 2*(C*a^2 - A*a*c)*d*e^5)*x - (B*c^2*d^5*e - 3*B*a*c*d^3*e^3 + 3*(C* a*c - A*c^2)*d^4*e^2 - (C*a^2 - A*a*c)*d^2*e^4 + (B*c^2*d^3*e^3 - 3*B*a*c* d*e^5 + 3*(C*a*c - A*c^2)*d^2*e^4 - (C*a^2 - A*a*c)*e^6)*x^2 + 2*(B*c^2*d^ 4*e^2 - 3*B*a*c*d^2*e^4 + 3*(C*a*c - A*c^2)*d^3*e^3 - (C*a^2 - A*a*c)*d*e^ 5)*x)*log(c*x^2 + a) + 2*(B*c^2*d^5*e - 3*B*a*c*d^3*e^3 + 3*(C*a*c - A*c^2 )*d^4*e^2 - (C*a^2 - A*a*c)*d^2*e^4 + (B*c^2*d^3*e^3 - 3*B*a*c*d*e^5 + 3*( C*a*c - A*c^2)*d^2*e^4 - (C*a^2 - A*a*c)*e^6)*x^2 + 2*(B*c^2*d^4*e^2 - 3*B *a*c*d^2*e^4 + 3*(C*a*c - A*c^2)*d^3*e^3 - (C*a^2 - A*a*c)*d*e^5)*x)*log(e *x + d))/(c^3*d^8*e + 3*a*c^2*d^6*e^3 + 3*a^2*c*d^4*e^5 + a^3*d^2*e^7 + (c ^3*d^6*e^3 + 3*a*c^2*d^4*e^5 + 3*a^2*c*d^2*e^7 + a^3*e^9)*x^2 + 2*(c^3*d^7 *e^2 + 3*a*c^2*d^5*e^4 + 3*a^2*c*d^3*e^6 + a^3*d*e^8)*x), -1/2*(C*c^2*d^6 - 3*B*c^2*d^5*e - 2*B*a*c*d^3*e^3 + B*a^2*d*e^5 + A*a^2*e^6 - (2*C*a*c - 5 *A*c^2)*d^4*e^2 - 3*(C*a^2 - 2*A*a*c)*d^2*e^4 - 2*(3*B*a*c*d^4*e^2 - B*...
Timed out. \[ \int \frac {A+B x+C x^2}{(d+e x)^3 \left (a+c x^2\right )} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.62 \[ \int \frac {A+B x+C x^2}{(d+e x)^3 \left (a+c x^2\right )} \, dx=\frac {{\left (B c^{2} d^{3} - 3 \, B a c d e^{2} + 3 \, {\left (C a c - A c^{2}\right )} d^{2} e - {\left (C a^{2} - A a c\right )} e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \, {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )}} - \frac {{\left (B c^{2} d^{3} - 3 \, B a c d e^{2} + 3 \, {\left (C a c - A c^{2}\right )} d^{2} e - {\left (C a^{2} - A a c\right )} e^{3}\right )} \log \left (e x + d\right )}{c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}} + \frac {{\left (3 \, B a c^{2} d^{2} e - B a^{2} c e^{3} - {\left (C a c^{2} - A c^{3}\right )} d^{3} + 3 \, {\left (C a^{2} c - A a c^{2}\right )} d e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} \sqrt {a c}} - \frac {C c d^{4} - 3 \, B c d^{3} e + B a d e^{3} + A a e^{4} - {\left (3 \, C a - 5 \, A c\right )} d^{2} e^{2} - 2 \, {\left (B c d^{2} e^{2} - B a e^{4} + 2 \, {\left (C a - A c\right )} d e^{3}\right )} x}{2 \, {\left (c^{2} d^{6} e + 2 \, a c d^{4} e^{3} + a^{2} d^{2} e^{5} + {\left (c^{2} d^{4} e^{3} + 2 \, a c d^{2} e^{5} + a^{2} e^{7}\right )} x^{2} + 2 \, {\left (c^{2} d^{5} e^{2} + 2 \, a c d^{3} e^{4} + a^{2} d e^{6}\right )} x\right )}} \]
1/2*(B*c^2*d^3 - 3*B*a*c*d*e^2 + 3*(C*a*c - A*c^2)*d^2*e - (C*a^2 - A*a*c) *e^3)*log(c*x^2 + a)/(c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^ 6) - (B*c^2*d^3 - 3*B*a*c*d*e^2 + 3*(C*a*c - A*c^2)*d^2*e - (C*a^2 - A*a*c )*e^3)*log(e*x + d)/(c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6 ) + (3*B*a*c^2*d^2*e - B*a^2*c*e^3 - (C*a*c^2 - A*c^3)*d^3 + 3*(C*a^2*c - A*a*c^2)*d*e^2)*arctan(c*x/sqrt(a*c))/((c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2* c*d^2*e^4 + a^3*e^6)*sqrt(a*c)) - 1/2*(C*c*d^4 - 3*B*c*d^3*e + B*a*d*e^3 + A*a*e^4 - (3*C*a - 5*A*c)*d^2*e^2 - 2*(B*c*d^2*e^2 - B*a*e^4 + 2*(C*a - A *c)*d*e^3)*x)/(c^2*d^6*e + 2*a*c*d^4*e^3 + a^2*d^2*e^5 + (c^2*d^4*e^3 + 2* a*c*d^2*e^5 + a^2*e^7)*x^2 + 2*(c^2*d^5*e^2 + 2*a*c*d^3*e^4 + a^2*d*e^6)*x )
Time = 0.27 (sec) , antiderivative size = 516, normalized size of antiderivative = 1.69 \[ \int \frac {A+B x+C x^2}{(d+e x)^3 \left (a+c x^2\right )} \, dx=\frac {{\left (B c^{2} d^{3} + 3 \, C a c d^{2} e - 3 \, A c^{2} d^{2} e - 3 \, B a c d e^{2} - C a^{2} e^{3} + A a c e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \, {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )}} - \frac {{\left (B c^{2} d^{3} e + 3 \, C a c d^{2} e^{2} - 3 \, A c^{2} d^{2} e^{2} - 3 \, B a c d e^{3} - C a^{2} e^{4} + A a c e^{4}\right )} \log \left ({\left | e x + d \right |}\right )}{c^{3} d^{6} e + 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} + a^{3} e^{7}} - \frac {{\left (C a c^{2} d^{3} - A c^{3} d^{3} - 3 \, B a c^{2} d^{2} e - 3 \, C a^{2} c d e^{2} + 3 \, A a c^{2} d e^{2} + B a^{2} c e^{3}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} \sqrt {a c}} - \frac {C c^{2} d^{6} - 3 \, B c^{2} d^{5} e - 2 \, C a c d^{4} e^{2} + 5 \, A c^{2} d^{4} e^{2} - 2 \, B a c d^{3} e^{3} - 3 \, C a^{2} d^{2} e^{4} + 6 \, A a c d^{2} e^{4} + B a^{2} d e^{5} + A a^{2} e^{6} - 2 \, {\left (B c^{2} d^{4} e^{2} + 2 \, C a c d^{3} e^{3} - 2 \, A c^{2} d^{3} e^{3} + 2 \, C a^{2} d e^{5} - 2 \, A a c d e^{5} - B a^{2} e^{6}\right )} x}{2 \, {\left (c d^{2} + a e^{2}\right )}^{3} {\left (e x + d\right )}^{2} e} \]
1/2*(B*c^2*d^3 + 3*C*a*c*d^2*e - 3*A*c^2*d^2*e - 3*B*a*c*d*e^2 - C*a^2*e^3 + A*a*c*e^3)*log(c*x^2 + a)/(c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6) - (B*c^2*d^3*e + 3*C*a*c*d^2*e^2 - 3*A*c^2*d^2*e^2 - 3*B*a*c*d* e^3 - C*a^2*e^4 + A*a*c*e^4)*log(abs(e*x + d))/(c^3*d^6*e + 3*a*c^2*d^4*e^ 3 + 3*a^2*c*d^2*e^5 + a^3*e^7) - (C*a*c^2*d^3 - A*c^3*d^3 - 3*B*a*c^2*d^2* e - 3*C*a^2*c*d*e^2 + 3*A*a*c^2*d*e^2 + B*a^2*c*e^3)*arctan(c*x/sqrt(a*c)) /((c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)*sqrt(a*c)) - 1/2 *(C*c^2*d^6 - 3*B*c^2*d^5*e - 2*C*a*c*d^4*e^2 + 5*A*c^2*d^4*e^2 - 2*B*a*c* d^3*e^3 - 3*C*a^2*d^2*e^4 + 6*A*a*c*d^2*e^4 + B*a^2*d*e^5 + A*a^2*e^6 - 2* (B*c^2*d^4*e^2 + 2*C*a*c*d^3*e^3 - 2*A*c^2*d^3*e^3 + 2*C*a^2*d*e^5 - 2*A*a *c*d*e^5 - B*a^2*e^6)*x)/((c*d^2 + a*e^2)^3*(e*x + d)^2*e)
Time = 19.08 (sec) , antiderivative size = 2980, normalized size of antiderivative = 9.77 \[ \int \frac {A+B x+C x^2}{(d+e x)^3 \left (a+c x^2\right )} \, dx=\text {Too large to display} \]
(log(d + e*x)*(e^3*(C*a^2 - A*a*c) - B*c^2*d^3 + d^2*e*(3*A*c^2 - 3*C*a*c) + 3*B*a*c*d*e^2))/(a^3*e^6 + c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4) - (log(9*A^2*a^5*e^10*(-a*c)^(5/2) + A^2*c^5*d^10*(-a*c)^(5/2) - B^2*a^7* e^10*(-a*c)^(3/2) - 9*B^2*c^3*d^10*(-a*c)^(7/2) + 9*C^2*a^9*e^10*(-a*c)^(1 /2) + C^2*c*d^10*(-a*c)^(9/2) + 9*C^2*a^9*c*e^10*x - 6*A^2*a*d^4*e^6*(-a*c )^(9/2) - 6*B^2*a*d^6*e^4*(-a*c)^(9/2) + 106*A^2*c*d^6*e^4*(-a*c)^(9/2) + 77*C^2*a*d^8*e^2*(-a*c)^(9/2) - 27*B^2*c*d^8*e^2*(-a*c)^(9/2) + A^2*a^2*c^ 8*d^10*x + 9*A^2*a^7*c^3*e^10*x + 9*B^2*a^3*c^7*d^10*x + B^2*a^8*c^2*e^10* x + C^2*a^4*c^6*d^10*x + 27*A^2*a^3*d^2*e^8*(-a*c)^(7/2) - 106*B^2*a^3*d^4 *e^6*(-a*c)^(7/2) + 77*B^2*a^5*d^2*e^8*(-a*c)^(5/2) - 77*A^2*c^3*d^8*e^2*( -a*c)^(7/2) - 106*C^2*a^3*d^6*e^4*(-a*c)^(7/2) - 6*C^2*a^5*d^4*e^6*(-a*c)^ (5/2) + 27*C^2*a^7*d^2*e^8*(-a*c)^(3/2) + 18*A*C*a^7*e^10*(-a*c)^(3/2) + 2 *A*C*c^3*d^10*(-a*c)^(7/2) + 224*A*B*a*d^5*e^5*(-a*c)^(9/2) - 48*A*B*a^5*d *e^9*(-a*c)^(5/2) - 212*A*C*a*d^6*e^4*(-a*c)^(9/2) + 64*A*B*c*d^7*e^3*(-a* c)^(9/2) + 48*A*B*c^3*d^9*e*(-a*c)^(7/2) - 64*B*C*a*d^7*e^3*(-a*c)^(9/2) - 48*B*C*a^7*d*e^9*(-a*c)^(3/2) - 154*A*C*c*d^8*e^2*(-a*c)^(9/2) + 77*A^2*a ^3*c^7*d^8*e^2*x + 106*A^2*a^4*c^6*d^6*e^4*x - 6*A^2*a^5*c^5*d^4*e^6*x - 2 7*A^2*a^6*c^4*d^2*e^8*x - 27*B^2*a^4*c^6*d^8*e^2*x - 6*B^2*a^5*c^5*d^6*e^4 *x + 106*B^2*a^6*c^4*d^4*e^6*x + 77*B^2*a^7*c^3*d^2*e^8*x + 77*C^2*a^5*c^5 *d^8*e^2*x + 106*C^2*a^6*c^4*d^6*e^4*x - 6*C^2*a^7*c^3*d^4*e^6*x - 27*C...