Integrand size = 34, antiderivative size = 592 \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right ) \left (a-c x^4\right )^{5/2}} \, dx=\frac {x \left ((A c+a C) d-a B e+(B c d-(A c+a C) e) x^2\right )}{6 a \left (c d^2-a e^2\right ) \left (a-c x^4\right )^{3/2}}+\frac {x \left (A c d \left (5 c d^2-11 a e^2\right )-a (C d-B e) \left (c d^2+5 a e^2\right )+3 \left (B c d \left (c d^2-3 a e^2\right )-A c e \left (c d^2-3 a e^2\right )+a C e \left (c d^2+a e^2\right )\right ) x^2\right )}{12 a^2 \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}}-\frac {\left (B c d \left (c d^2-3 a e^2\right )-A c e \left (c d^2-3 a e^2\right )+a C e \left (c d^2+a e^2\right )\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{4 a^{5/4} c^{3/4} \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}}+\frac {\left (5 A c^2 d^2-3 a^2 C e^2+\sqrt {a} c^{3/2} d (3 B d+2 A e)+a^{3/2} \sqrt {c} e (2 C d-5 B e)-a c \left (C d^2-e (4 B d-9 A e)\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{12 a^{7/4} c^{3/4} \left (\sqrt {c} d+\sqrt {a} e\right ) \left (c d^2-a e^2\right ) \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} e^2 \left (C d^2-B d e+A e^2\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} d \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}} \] Output:
1/6*x*((A*c+C*a)*d-B*a*e+(B*c*d-(A*c+C*a)*e)*x^2)/a/(-a*e^2+c*d^2)/(-c*x^4 +a)^(3/2)+1/12*x*(A*c*d*(-11*a*e^2+5*c*d^2)-a*(-B*e+C*d)*(5*a*e^2+c*d^2)+3 *(B*c*d*(-3*a*e^2+c*d^2)-A*c*e*(-3*a*e^2+c*d^2)+a*C*e*(a*e^2+c*d^2))*x^2)/ a^2/(-a*e^2+c*d^2)^2/(-c*x^4+a)^(1/2)-1/4*(B*c*d*(-3*a*e^2+c*d^2)-A*c*e*(- 3*a*e^2+c*d^2)+a*C*e*(a*e^2+c*d^2))*(1-c*x^4/a)^(1/2)*EllipticE(c^(1/4)*x/ a^(1/4),I)/a^(5/4)/c^(3/4)/(-a*e^2+c*d^2)^2/(-c*x^4+a)^(1/2)+1/12*(5*A*c^2 *d^2-3*a^2*C*e^2+a^(1/2)*c^(3/2)*d*(2*A*e+3*B*d)+a^(3/2)*c^(1/2)*e*(-5*B*e +2*C*d)-a*c*(C*d^2-e*(-9*A*e+4*B*d)))*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4)* x/a^(1/4),I)/a^(7/4)/c^(3/4)/(c^(1/2)*d+a^(1/2)*e)/(-a*e^2+c*d^2)/(-c*x^4+ a)^(1/2)+a^(1/4)*e^2*(A*e^2-B*d*e+C*d^2)*(1-c*x^4/a)^(1/2)*EllipticPi(c^(1 /4)*x/a^(1/4),-a^(1/2)*e/c^(1/2)/d,I)/c^(1/4)/d/(-a*e^2+c*d^2)^2/(-c*x^4+a )^(1/2)
Result contains complex when optimal does not.
Time = 12.32 (sec) , antiderivative size = 545, normalized size of antiderivative = 0.92 \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right ) \left (a-c x^4\right )^{5/2}} \, dx=-\frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} \sqrt {c} d \left (2 a \left (c d^2-a e^2\right ) x \left (B c d x^2+A c \left (d-e x^2\right )+a \left (C d-B e-C e x^2\right )\right )+x \left (a-c x^4\right ) \left (3 B c^2 d^3 x^2+a^2 e^2 \left (-5 C d+5 B e+3 C e x^2\right )+a c d \left (B e \left (d-9 e x^2\right )-C d \left (d-3 e x^2\right )\right )+A c \left (c d^2 \left (5 d-3 e x^2\right )+a e^2 \left (-11 d+9 e x^2\right )\right )\right )\right )+i \left (a-c x^4\right ) \sqrt {1-\frac {c x^4}{a}} \left (3 \sqrt {a} d \left (B c d \left (c d^2-3 a e^2\right )+a C e \left (c d^2+a e^2\right )+A c e \left (-c d^2+3 a e^2\right )\right ) E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )-d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (5 A c^2 d^2-3 a^2 C e^2+\sqrt {a} c^{3/2} d (3 B d+2 A e)+a^{3/2} \sqrt {c} e (2 C d-5 B e)-a c \left (C d^2+e (-4 B d+9 A e)\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-12 a^2 \sqrt {c} e^2 \left (C d^2+e (-B d+A e)\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )\right )}{12 a^{5/2} \left (-\frac {\sqrt {c}}{\sqrt {a}}\right )^{3/2} d \left (c d^2-a e^2\right )^2 \left (a-c x^4\right )^{3/2}} \] Input:
Integrate[(A + B*x^2 + C*x^4)/((d + e*x^2)*(a - c*x^4)^(5/2)),x]
Output:
-1/12*(Sqrt[-(Sqrt[c]/Sqrt[a])]*Sqrt[c]*d*(2*a*(c*d^2 - a*e^2)*x*(B*c*d*x^ 2 + A*c*(d - e*x^2) + a*(C*d - B*e - C*e*x^2)) + x*(a - c*x^4)*(3*B*c^2*d^ 3*x^2 + a^2*e^2*(-5*C*d + 5*B*e + 3*C*e*x^2) + a*c*d*(B*e*(d - 9*e*x^2) - C*d*(d - 3*e*x^2)) + A*c*(c*d^2*(5*d - 3*e*x^2) + a*e^2*(-11*d + 9*e*x^2)) )) + I*(a - c*x^4)*Sqrt[1 - (c*x^4)/a]*(3*Sqrt[a]*d*(B*c*d*(c*d^2 - 3*a*e^ 2) + a*C*e*(c*d^2 + a*e^2) + A*c*e*(-(c*d^2) + 3*a*e^2))*EllipticE[I*ArcSi nh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] - d*(Sqrt[c]*d - Sqrt[a]*e)*(5*A*c^2*d ^2 - 3*a^2*C*e^2 + Sqrt[a]*c^(3/2)*d*(3*B*d + 2*A*e) + a^(3/2)*Sqrt[c]*e*( 2*C*d - 5*B*e) - a*c*(C*d^2 + e*(-4*B*d + 9*A*e)))*EllipticF[I*ArcSinh[Sqr t[-(Sqrt[c]/Sqrt[a])]*x], -1] - 12*a^2*Sqrt[c]*e^2*(C*d^2 + e*(-(B*d) + A* e))*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a ])]*x], -1]))/(a^(5/2)*(-(Sqrt[c]/Sqrt[a]))^(3/2)*d*(c*d^2 - a*e^2)^2*(a - c*x^4)^(3/2))
Time = 1.17 (sec) , antiderivative size = 689, normalized size of antiderivative = 1.16, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2259, 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.
| \(\displaystyle \int \frac {A+B x^2+C x^4}{\left (a-c x^4\right )^{5/2} \left (d+e x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2259 |
| \(\displaystyle \int \left (\frac {e^2 \left (A e^2-B d e+C d^2\right )}{\sqrt {a-c x^4} \left (d+e x^2\right ) \left (c d^2-a e^2\right )^2}+\frac {c \left (d-e x^2\right ) \left (A e^2-B d e+C d^2\right )}{\sqrt {a-c x^4} \left (c x^4-a\right ) \left (c d^2-a e^2\right )^2}+\frac {x^2 (B c d-e (a C+A c))-a B e+a C d+A c d}{\left (a-c x^4\right )^{5/2} \left (c d^2-a e^2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right ) \left (\frac {5 \sqrt {c} (-a B e+a C d+A c d)}{\sqrt {a}}-3 e (a C+A c)+3 B c d\right )}{12 a^{5/4} c^{3/4} \sqrt {a-c x^4} \left (c d^2-a e^2\right )}-\frac {\sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right ) (-a C e-A c e+B c d)}{4 a^{5/4} c^{3/4} \sqrt {a-c x^4} \left (c d^2-a e^2\right )}-\frac {\sqrt [4]{c} \sqrt {1-\frac {c x^4}{a}} \left (\sqrt {c} d-\sqrt {a} e\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right ) \left (A e^2-B d e+C d^2\right )}{2 a^{3/4} \sqrt {a-c x^4} \left (c d^2-a e^2\right )^2}+\frac {x \left (3 x^2 (B c d-e (a C+A c))+5 (-a B e+a C d+A c d)\right )}{12 a^2 \sqrt {a-c x^4} \left (c d^2-a e^2\right )}-\frac {\sqrt [4]{c} e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right ) \left (A e^2-B d e+C d^2\right )}{2 \sqrt [4]{a} \sqrt {a-c x^4} \left (c d^2-a e^2\right )^2}+\frac {\sqrt [4]{a} e^2 \sqrt {1-\frac {c x^4}{a}} \left (A e^2-B d e+C d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} d \sqrt {a-c x^4} \left (c d^2-a e^2\right )^2}-\frac {c x \left (d-e x^2\right ) \left (A e^2-B d e+C d^2\right )}{2 a \sqrt {a-c x^4} \left (c d^2-a e^2\right )^2}+\frac {x \left (x^2 (B c d-e (a C+A c))-a B e+a C d+A c d\right )}{6 a \left (a-c x^4\right )^{3/2} \left (c d^2-a e^2\right )}\) |
Input:
Int[(A + B*x^2 + C*x^4)/((d + e*x^2)*(a - c*x^4)^(5/2)),x]
Output:
(x*(A*c*d + a*C*d - a*B*e + (B*c*d - (A*c + a*C)*e)*x^2))/(6*a*(c*d^2 - a* e^2)*(a - c*x^4)^(3/2)) - (c*(C*d^2 - B*d*e + A*e^2)*x*(d - e*x^2))/(2*a*( c*d^2 - a*e^2)^2*Sqrt[a - c*x^4]) + (x*(5*(A*c*d + a*C*d - a*B*e) + 3*(B*c *d - (A*c + a*C)*e)*x^2))/(12*a^2*(c*d^2 - a*e^2)*Sqrt[a - c*x^4]) - ((B*c *d - A*c*e - a*C*e)*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/ 4)], -1])/(4*a^(5/4)*c^(3/4)*(c*d^2 - a*e^2)*Sqrt[a - c*x^4]) - (c^(1/4)*e *(C*d^2 - B*d*e + A*e^2)*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/ a^(1/4)], -1])/(2*a^(1/4)*(c*d^2 - a*e^2)^2*Sqrt[a - c*x^4]) - (c^(1/4)*(S qrt[c]*d - Sqrt[a]*e)*(C*d^2 - B*d*e + A*e^2)*Sqrt[1 - (c*x^4)/a]*Elliptic F[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(2*a^(3/4)*(c*d^2 - a*e^2)^2*Sqrt[a - c*x^4]) + ((3*B*c*d - 3*(A*c + a*C)*e + (5*Sqrt[c]*(A*c*d + a*C*d - a*B*e) )/Sqrt[a])*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1]) /(12*a^(5/4)*c^(3/4)*(c*d^2 - a*e^2)*Sqrt[a - c*x^4]) + (a^(1/4)*e^2*(C*d^ 2 - B*d*e + A*e^2)*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d )), ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(1/4)*d*(c*d^2 - a*e^2)^2*Sqrt[a - c*x^4])
Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[1/Sqrt[a + c*x^4], Px*(d + e*x^2)^q*(a + c*x^4)^(p + 1/2), x], x] /; FreeQ[{a, c, d, e}, x] && PolyQ[Px, x] && IntegerQ[p + 1/ 2] && IntegerQ[q]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1220 vs. \(2 (528 ) = 1056\).
Time = 0.95 (sec) , antiderivative size = 1221, normalized size of antiderivative = 2.06
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1221\) |
| elliptic | \(\text {Expression too large to display}\) | \(2188\) |
Input:
int((C*x^4+B*x^2+A)/(e*x^2+d)/(-c*x^4+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/e^2*(B*e*(1/6/a*x/c^2*(-c*x^4+a)^(1/2)/(x^4-a/c)^2+5/12/a^2*x/(-(x^4-a/c )*c)^(1/2)+5/12/a^2/(c^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2)*x^2/a^(1/2))^(1/2)* (1+c^(1/2)*x^2/a^(1/2))^(1/2)/(-c*x^4+a)^(1/2)*EllipticF(x*(c^(1/2)/a^(1/2 ))^(1/2),I))+C*e*(1/6/a*x^3/c^2*(-c*x^4+a)^(1/2)/(x^4-a/c)^2+1/4/a^2*x^3/( -(x^4-a/c)*c)^(1/2)+1/4/a^(3/2)/(c^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2)*x^2/a^( 1/2))^(1/2)*(1+c^(1/2)*x^2/a^(1/2))^(1/2)/(-c*x^4+a)^(1/2)/c^(1/2)*(Ellipt icF(x*(c^(1/2)/a^(1/2))^(1/2),I)-EllipticE(x*(c^(1/2)/a^(1/2))^(1/2),I)))- C*d*(1/6/a*x/c^2*(-c*x^4+a)^(1/2)/(x^4-a/c)^2+5/12/a^2*x/(-(x^4-a/c)*c)^(1 /2)+5/12/a^2/(c^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2)*x^2/a^(1/2))^(1/2)*(1+c^(1 /2)*x^2/a^(1/2))^(1/2)/(-c*x^4+a)^(1/2)*EllipticF(x*(c^(1/2)/a^(1/2))^(1/2 ),I)))+(A*e^2-B*d*e+C*d^2)/e^2*((1/6/c/a*e/(a*e^2-c*d^2)*x^3-1/6/c/a*d/(a* e^2-c*d^2)*x)*(-c*x^4+a)^(1/2)/(x^4-a/c)^2+2*c*(1/8*e*(3*a*e^2-c*d^2)/a^2/ (a*e^2-c*d^2)^2*x^3-1/24*d*(11*a*e^2-5*c*d^2)/a^2/(a*e^2-c*d^2)^2*x)/(-(x^ 4-a/c)*c)^(1/2)-11/12*c*d/a/(a*e^2-c*d^2)^2/(c^(1/2)/a^(1/2))^(1/2)*(1-c^( 1/2)*x^2/a^(1/2))^(1/2)*(1+c^(1/2)*x^2/a^(1/2))^(1/2)/(-c*x^4+a)^(1/2)*Ell ipticF(x*(c^(1/2)/a^(1/2))^(1/2),I)*e^2+5/12*c^2*d^3/a^2/(a*e^2-c*d^2)^2/( c^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2)*x^2/a^(1/2))^(1/2)*(1+c^(1/2)*x^2/a^(1/2 ))^(1/2)/(-c*x^4+a)^(1/2)*EllipticF(x*(c^(1/2)/a^(1/2))^(1/2),I)+3/4*c^(1/ 2)*e^3/a^(1/2)/(a*e^2-c*d^2)^2/(c^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2)*x^2/a^(1 /2))^(1/2)*(1+c^(1/2)*x^2/a^(1/2))^(1/2)/(-c*x^4+a)^(1/2)*EllipticF(x*(...
Timed out. \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right ) \left (a-c x^4\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((C*x^4+B*x^2+A)/(e*x^2+d)/(-c*x^4+a)^(5/2),x, algorithm="fricas" )
Output:
Timed out
Timed out. \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right ) \left (a-c x^4\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((C*x**4+B*x**2+A)/(e*x**2+d)/(-c*x**4+a)**(5/2),x)
Output:
Timed out
\[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right ) \left (a-c x^4\right )^{5/2}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{{\left (-c x^{4} + a\right )}^{\frac {5}{2}} {\left (e x^{2} + d\right )}} \,d x } \] Input:
integrate((C*x^4+B*x^2+A)/(e*x^2+d)/(-c*x^4+a)^(5/2),x, algorithm="maxima" )
Output:
integrate((C*x^4 + B*x^2 + A)/((-c*x^4 + a)^(5/2)*(e*x^2 + d)), x)
\[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right ) \left (a-c x^4\right )^{5/2}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{{\left (-c x^{4} + a\right )}^{\frac {5}{2}} {\left (e x^{2} + d\right )}} \,d x } \] Input:
integrate((C*x^4+B*x^2+A)/(e*x^2+d)/(-c*x^4+a)^(5/2),x, algorithm="giac")
Output:
integrate((C*x^4 + B*x^2 + A)/((-c*x^4 + a)^(5/2)*(e*x^2 + d)), x)
Timed out. \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right ) \left (a-c x^4\right )^{5/2}} \, dx=\int \frac {C\,x^4+B\,x^2+A}{{\left (a-c\,x^4\right )}^{5/2}\,\left (e\,x^2+d\right )} \,d x \] Input:
int((A + B*x^2 + C*x^4)/((a - c*x^4)^(5/2)*(d + e*x^2)),x)
Output:
int((A + B*x^2 + C*x^4)/((a - c*x^4)^(5/2)*(d + e*x^2)), x)
\[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right ) \left (a-c x^4\right )^{5/2}} \, dx=\left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c^{3} e \,x^{14}-c^{3} d \,x^{12}+3 a \,c^{2} e \,x^{10}+3 a \,c^{2} d \,x^{8}-3 a^{2} c e \,x^{6}-3 a^{2} c d \,x^{4}+a^{3} e \,x^{2}+a^{3} d}d x \right ) a +\left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{4}}{-c^{3} e \,x^{14}-c^{3} d \,x^{12}+3 a \,c^{2} e \,x^{10}+3 a \,c^{2} d \,x^{8}-3 a^{2} c e \,x^{6}-3 a^{2} c d \,x^{4}+a^{3} e \,x^{2}+a^{3} d}d x \right ) c +\left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c^{3} e \,x^{14}-c^{3} d \,x^{12}+3 a \,c^{2} e \,x^{10}+3 a \,c^{2} d \,x^{8}-3 a^{2} c e \,x^{6}-3 a^{2} c d \,x^{4}+a^{3} e \,x^{2}+a^{3} d}d x \right ) b \] Input:
int((C*x^4+B*x^2+A)/(e*x^2+d)/(-c*x^4+a)^(5/2),x)
Output:
int(sqrt(a - c*x**4)/(a**3*d + a**3*e*x**2 - 3*a**2*c*d*x**4 - 3*a**2*c*e* x**6 + 3*a*c**2*d*x**8 + 3*a*c**2*e*x**10 - c**3*d*x**12 - c**3*e*x**14),x )*a + int((sqrt(a - c*x**4)*x**4)/(a**3*d + a**3*e*x**2 - 3*a**2*c*d*x**4 - 3*a**2*c*e*x**6 + 3*a*c**2*d*x**8 + 3*a*c**2*e*x**10 - c**3*d*x**12 - c* *3*e*x**14),x)*c + int((sqrt(a - c*x**4)*x**2)/(a**3*d + a**3*e*x**2 - 3*a **2*c*d*x**4 - 3*a**2*c*e*x**6 + 3*a*c**2*d*x**8 + 3*a*c**2*e*x**10 - c**3 *d*x**12 - c**3*e*x**14),x)*b