\(\int \frac {1}{(d+e x)^2 (a+b x^2+c x^4)^2} \, dx\) [246]

Optimal result

Integrand size = 22, antiderivative size = 1689 \[ \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )^2} \, dx =\text {Too large to display} \] Output:

-e^7/(a*e^4+b*d^2*e^2+c*d^4)^2/(e*x+d)+d*e*(2*b^2*c*d^2*e^2-4*a*c^2*d^2*e^ 
2+b^3*e^4+b*c*(-3*a*e^4+c*d^4)+c*(-2*a*c*e^4+b^2*e^4+2*b*c*d^2*e^2+2*c^2*d 
^4)*x^2)/(-4*a*c+b^2)/(a*e^4+b*d^2*e^2+c*d^4)^2/(c*x^4+b*x^2+a)-1/2*x*(a*b 
*c*e^2*(-a*e^4+b*d^2*e^2+3*c*d^4)-(-2*a*c+b^2)*(-a*b*e^6-3*a*c*d^2*e^4+b^2 
*d^2*e^4+2*b*c*d^4*e^2+c^2*d^6)-c*(b^3*d^2*e^4+b*c*d^2*(-5*a*e^4+c*d^4)-2* 
a*c*e^2*(-a*e^4+3*c*d^4)+b^2*(-a*e^6+2*c*d^4*e^2))*x^2)/a/(-4*a*c+b^2)/(a* 
e^4+b*d^2*e^2+c*d^4)^2/(c*x^4+b*x^2+a)+1/2*c^(1/2)*e^4*(10*c^2*d^6+c*d^2*e 
^2*(9*b*d^2+7*(-4*a*c+b^2)^(1/2)*d^2-6*a*e^2)+(b+(-4*a*c+b^2)^(1/2))*e^4*( 
-a*e^2+3*b*d^2))*arctan(2^(1/2)*c^(1/2)*x/(b-(-4*a*c+b^2)^(1/2))^(1/2))*2^ 
(1/2)/(-4*a*c+b^2)^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)/(a*e^4+b*d^2*e^2+c*d 
^4)^3+1/4*c^(1/2)*(b^4*d^2*e^4+b^3*(2*c*d^4*e^2+(-4*a*c+b^2)^(1/2)*d^2*e^4 
-a*e^6)+b*c*(c*(-4*a*c+b^2)^(1/2)*d^6-12*a*c*d^4*e^2-5*a*(-4*a*c+b^2)^(1/2 
)*d^2*e^4+8*a^2*e^6)-2*a*c*(6*c^2*d^6+3*c*(-4*a*c+b^2)^(1/2)*d^4*e^2-18*a* 
c*d^2*e^4-a*(-4*a*c+b^2)^(1/2)*e^6)+b^2*(c^2*d^6+2*c*(-4*a*c+b^2)^(1/2)*d^ 
4*e^2-11*a*c*d^2*e^4-a*(-4*a*c+b^2)^(1/2)*e^6))*arctan(2^(1/2)*c^(1/2)*x/( 
b-(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)/a/(-4*a*c+b^2)^(3/2)/(b-(-4*a*c+b^2)^ 
(1/2))^(1/2)/(a*e^4+b*d^2*e^2+c*d^4)^2-1/2*c^(1/2)*e^4*(10*c^2*d^6+c*d^2*e 
^2*(9*b*d^2-7*(-4*a*c+b^2)^(1/2)*d^2-6*a*e^2)+(b-(-4*a*c+b^2)^(1/2))*e^4*( 
-a*e^2+3*b*d^2))*arctan(2^(1/2)*c^(1/2)*x/(b+(-4*a*c+b^2)^(1/2))^(1/2))*2^ 
(1/2)/(-4*a*c+b^2)^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)/(a*e^4+b*d^2*e^2+...
 

Mathematica [A] (verified)

Time = 8.08 (sec) , antiderivative size = 2086, normalized size of antiderivative = 1.24 \[ \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )^2} \, dx=\text {Result too large to show} \] Input:

Integrate[1/((d + e*x)^2*(a + b*x^2 + c*x^4)^2),x]
 

Output:

-(e^7/((c*d^4 + b*d^2*e^2 + a*e^4)^2*(d + e*x))) + (-2*a*b*c^2*d^5*e - 4*a 
*b^2*c*d^3*e^3 + 8*a^2*c^2*d^3*e^3 - 2*a*b^3*d*e^5 + 6*a^2*b*c*d*e^5 - b^2 
*c^2*d^6*x + 2*a*c^3*d^6*x - 2*b^3*c*d^4*e^2*x + 7*a*b*c^2*d^4*e^2*x - b^4 
*d^2*e^4*x + 6*a*b^2*c*d^2*e^4*x - 6*a^2*c^2*d^2*e^4*x + a*b^3*e^6*x - 3*a 
^2*b*c*e^6*x - 4*a*c^3*d^5*e*x^2 - 4*a*b*c^2*d^3*e^3*x^2 - 2*a*b^2*c*d*e^5 
*x^2 + 4*a^2*c^2*d*e^5*x^2 - b*c^3*d^6*x^3 - 2*b^2*c^2*d^4*e^2*x^3 + 6*a*c 
^3*d^4*e^2*x^3 - b^3*c*d^2*e^4*x^3 + 5*a*b*c^2*d^2*e^4*x^3 + a*b^2*c*e^6*x 
^3 - 2*a^2*c^2*e^6*x^3)/(2*a*(-b^2 + 4*a*c)*(c*d^4 + b*d^2*e^2 + a*e^4)^2* 
(a + b*x^2 + c*x^4)) + (Sqrt[c]*(b^2*c^3*d^10 - 12*a*c^4*d^10 + b*c^3*Sqrt 
[b^2 - 4*a*c]*d^10 + 3*b^3*c^2*d^8*e^2 - 24*a*b*c^3*d^8*e^2 + 3*b^2*c^2*Sq 
rt[b^2 - 4*a*c]*d^8*e^2 - 6*a*c^3*Sqrt[b^2 - 4*a*c]*d^8*e^2 + 3*b^4*c*d^6* 
e^4 - 2*a*b^2*c^2*d^6*e^4 - 56*a^2*c^3*d^6*e^4 + 3*b^3*c*Sqrt[b^2 - 4*a*c] 
*d^6*e^4 - 10*a*b*c^2*Sqrt[b^2 - 4*a*c]*d^6*e^4 + b^5*d^4*e^6 + 8*a*b^3*c* 
d^4*e^6 - 40*a^2*b*c^2*d^4*e^6 + b^4*Sqrt[b^2 - 4*a*c]*d^4*e^6 + 10*a*b^2* 
c*Sqrt[b^2 - 4*a*c]*d^4*e^6 - 60*a^2*c^2*Sqrt[b^2 - 4*a*c]*d^4*e^6 + 6*a*b 
^4*d^2*e^8 - 39*a^2*b^2*c*d^2*e^8 + 84*a^3*c^2*d^2*e^8 + 6*a*b^3*Sqrt[b^2 
- 4*a*c]*d^2*e^8 - 27*a^2*b*c*Sqrt[b^2 - 4*a*c]*d^2*e^8 - 3*a^2*b^3*e^10 + 
 16*a^3*b*c*e^10 - 3*a^2*b^2*Sqrt[b^2 - 4*a*c]*e^10 + 10*a^3*c*Sqrt[b^2 - 
4*a*c]*e^10)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(2*S 
qrt[2]*a*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]*(c*d^4 + b*d^2...
 

Rubi [F]

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.

\(\displaystyle \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )^2} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^4 \left (-2 d e x \left (-a c e^4+2 b^2 e^4+5 b c d^2 e^2+3 c^2 d^4\right )+c e^2 x^2 \left (-a e^4+3 b d^2 e^2+7 c d^4\right )-a b e^6-3 a c d^2 e^4+3 b^2 d^2 e^4+8 b c d^4 e^2-4 c d e^3 x^3 \left (b e^2+2 c d^2\right )+5 c^2 d^6\right )}{\left (a+b x^2+c x^4\right ) \left (a e^4+b d^2 e^2+c d^4\right )^3}+\frac {-2 d e x \left (-a c e^4+b^2 e^4+2 b c d^2 e^2+c^2 d^4\right )+c e^2 x^2 \left (-a e^4+b d^2 e^2+3 c d^4\right )-a b e^6-3 a c d^2 e^4+b^2 d^2 e^4+2 b c d^4 e^2-2 c d e^3 x^3 \left (b e^2+2 c d^2\right )+c^2 d^6}{\left (a+b x^2+c x^4\right )^2 \left (a e^4+b d^2 e^2+c d^4\right )^2}+\frac {4 d e^8 \left (b e^2+2 c d^2\right )}{(d+e x) \left (a e^4+b d^2 e^2+c d^4\right )^3}+\frac {e^8}{(d+e x)^2 \left (a e^4+b d^2 e^2+c d^4\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^4 \left (-2 d e x \left (-a c e^4+2 b^2 e^4+5 b c d^2 e^2+3 c^2 d^4\right )+c e^2 x^2 \left (-a e^4+3 b d^2 e^2+7 c d^4\right )-a b e^6-3 a c d^2 e^4+3 b^2 d^2 e^4+8 b c d^4 e^2-4 c d e^3 x^3 \left (b e^2+2 c d^2\right )+5 c^2 d^6\right )}{\left (a+b x^2+c x^4\right ) \left (a e^4+b d^2 e^2+c d^4\right )^3}+\frac {-2 d e x \left (-a c e^4+b^2 e^4+2 b c d^2 e^2+c^2 d^4\right )+c e^2 x^2 \left (-a e^4+b d^2 e^2+3 c d^4\right )-a b e^6-3 a c d^2 e^4+b^2 d^2 e^4+2 b c d^4 e^2-2 c d e^3 x^3 \left (b e^2+2 c d^2\right )+c^2 d^6}{\left (a+b x^2+c x^4\right )^2 \left (a e^4+b d^2 e^2+c d^4\right )^2}+\frac {4 d e^8 \left (b e^2+2 c d^2\right )}{(d+e x) \left (a e^4+b d^2 e^2+c d^4\right )^3}+\frac {e^8}{(d+e x)^2 \left (a e^4+b d^2 e^2+c d^4\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^4 \left (-2 d e x \left (-a c e^4+2 b^2 e^4+5 b c d^2 e^2+3 c^2 d^4\right )+c e^2 x^2 \left (-a e^4+3 b d^2 e^2+7 c d^4\right )-a b e^6-3 a c d^2 e^4+3 b^2 d^2 e^4+8 b c d^4 e^2-4 c d e^3 x^3 \left (b e^2+2 c d^2\right )+5 c^2 d^6\right )}{\left (a+b x^2+c x^4\right ) \left (a e^4+b d^2 e^2+c d^4\right )^3}+\frac {-2 d e x \left (-a c e^4+b^2 e^4+2 b c d^2 e^2+c^2 d^4\right )+c e^2 x^2 \left (-a e^4+b d^2 e^2+3 c d^4\right )-a b e^6-3 a c d^2 e^4+b^2 d^2 e^4+2 b c d^4 e^2-2 c d e^3 x^3 \left (b e^2+2 c d^2\right )+c^2 d^6}{\left (a+b x^2+c x^4\right )^2 \left (a e^4+b d^2 e^2+c d^4\right )^2}+\frac {4 d e^8 \left (b e^2+2 c d^2\right )}{(d+e x) \left (a e^4+b d^2 e^2+c d^4\right )^3}+\frac {e^8}{(d+e x)^2 \left (a e^4+b d^2 e^2+c d^4\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^4 \left (-2 d e x \left (-a c e^4+2 b^2 e^4+5 b c d^2 e^2+3 c^2 d^4\right )+c e^2 x^2 \left (-a e^4+3 b d^2 e^2+7 c d^4\right )-a b e^6-3 a c d^2 e^4+3 b^2 d^2 e^4+8 b c d^4 e^2-4 c d e^3 x^3 \left (b e^2+2 c d^2\right )+5 c^2 d^6\right )}{\left (a+b x^2+c x^4\right ) \left (a e^4+b d^2 e^2+c d^4\right )^3}+\frac {-2 d e x \left (-a c e^4+b^2 e^4+2 b c d^2 e^2+c^2 d^4\right )+c e^2 x^2 \left (-a e^4+b d^2 e^2+3 c d^4\right )-a b e^6-3 a c d^2 e^4+b^2 d^2 e^4+2 b c d^4 e^2-2 c d e^3 x^3 \left (b e^2+2 c d^2\right )+c^2 d^6}{\left (a+b x^2+c x^4\right )^2 \left (a e^4+b d^2 e^2+c d^4\right )^2}+\frac {4 d e^8 \left (b e^2+2 c d^2\right )}{(d+e x) \left (a e^4+b d^2 e^2+c d^4\right )^3}+\frac {e^8}{(d+e x)^2 \left (a e^4+b d^2 e^2+c d^4\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^4 \left (-2 d e x \left (-a c e^4+2 b^2 e^4+5 b c d^2 e^2+3 c^2 d^4\right )+c e^2 x^2 \left (-a e^4+3 b d^2 e^2+7 c d^4\right )-a b e^6-3 a c d^2 e^4+3 b^2 d^2 e^4+8 b c d^4 e^2-4 c d e^3 x^3 \left (b e^2+2 c d^2\right )+5 c^2 d^6\right )}{\left (a+b x^2+c x^4\right ) \left (a e^4+b d^2 e^2+c d^4\right )^3}+\frac {-2 d e x \left (-a c e^4+b^2 e^4+2 b c d^2 e^2+c^2 d^4\right )+c e^2 x^2 \left (-a e^4+b d^2 e^2+3 c d^4\right )-a b e^6-3 a c d^2 e^4+b^2 d^2 e^4+2 b c d^4 e^2-2 c d e^3 x^3 \left (b e^2+2 c d^2\right )+c^2 d^6}{\left (a+b x^2+c x^4\right )^2 \left (a e^4+b d^2 e^2+c d^4\right )^2}+\frac {4 d e^8 \left (b e^2+2 c d^2\right )}{(d+e x) \left (a e^4+b d^2 e^2+c d^4\right )^3}+\frac {e^8}{(d+e x)^2 \left (a e^4+b d^2 e^2+c d^4\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^4 \left (-2 d e x \left (-a c e^4+2 b^2 e^4+5 b c d^2 e^2+3 c^2 d^4\right )+c e^2 x^2 \left (-a e^4+3 b d^2 e^2+7 c d^4\right )-a b e^6-3 a c d^2 e^4+3 b^2 d^2 e^4+8 b c d^4 e^2-4 c d e^3 x^3 \left (b e^2+2 c d^2\right )+5 c^2 d^6\right )}{\left (a+b x^2+c x^4\right ) \left (a e^4+b d^2 e^2+c d^4\right )^3}+\frac {-2 d e x \left (-a c e^4+b^2 e^4+2 b c d^2 e^2+c^2 d^4\right )+c e^2 x^2 \left (-a e^4+b d^2 e^2+3 c d^4\right )-a b e^6-3 a c d^2 e^4+b^2 d^2 e^4+2 b c d^4 e^2-2 c d e^3 x^3 \left (b e^2+2 c d^2\right )+c^2 d^6}{\left (a+b x^2+c x^4\right )^2 \left (a e^4+b d^2 e^2+c d^4\right )^2}+\frac {4 d e^8 \left (b e^2+2 c d^2\right )}{(d+e x) \left (a e^4+b d^2 e^2+c d^4\right )^3}+\frac {e^8}{(d+e x)^2 \left (a e^4+b d^2 e^2+c d^4\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^4 \left (-2 d e x \left (-a c e^4+2 b^2 e^4+5 b c d^2 e^2+3 c^2 d^4\right )+c e^2 x^2 \left (-a e^4+3 b d^2 e^2+7 c d^4\right )-a b e^6-3 a c d^2 e^4+3 b^2 d^2 e^4+8 b c d^4 e^2-4 c d e^3 x^3 \left (b e^2+2 c d^2\right )+5 c^2 d^6\right )}{\left (a+b x^2+c x^4\right ) \left (a e^4+b d^2 e^2+c d^4\right )^3}+\frac {-2 d e x \left (-a c e^4+b^2 e^4+2 b c d^2 e^2+c^2 d^4\right )+c e^2 x^2 \left (-a e^4+b d^2 e^2+3 c d^4\right )-a b e^6-3 a c d^2 e^4+b^2 d^2 e^4+2 b c d^4 e^2-2 c d e^3 x^3 \left (b e^2+2 c d^2\right )+c^2 d^6}{\left (a+b x^2+c x^4\right )^2 \left (a e^4+b d^2 e^2+c d^4\right )^2}+\frac {4 d e^8 \left (b e^2+2 c d^2\right )}{(d+e x) \left (a e^4+b d^2 e^2+c d^4\right )^3}+\frac {e^8}{(d+e x)^2 \left (a e^4+b d^2 e^2+c d^4\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^4 \left (-2 d e x \left (-a c e^4+2 b^2 e^4+5 b c d^2 e^2+3 c^2 d^4\right )+c e^2 x^2 \left (-a e^4+3 b d^2 e^2+7 c d^4\right )-a b e^6-3 a c d^2 e^4+3 b^2 d^2 e^4+8 b c d^4 e^2-4 c d e^3 x^3 \left (b e^2+2 c d^2\right )+5 c^2 d^6\right )}{\left (a+b x^2+c x^4\right ) \left (a e^4+b d^2 e^2+c d^4\right )^3}+\frac {-2 d e x \left (-a c e^4+b^2 e^4+2 b c d^2 e^2+c^2 d^4\right )+c e^2 x^2 \left (-a e^4+b d^2 e^2+3 c d^4\right )-a b e^6-3 a c d^2 e^4+b^2 d^2 e^4+2 b c d^4 e^2-2 c d e^3 x^3 \left (b e^2+2 c d^2\right )+c^2 d^6}{\left (a+b x^2+c x^4\right )^2 \left (a e^4+b d^2 e^2+c d^4\right )^2}+\frac {4 d e^8 \left (b e^2+2 c d^2\right )}{(d+e x) \left (a e^4+b d^2 e^2+c d^4\right )^3}+\frac {e^8}{(d+e x)^2 \left (a e^4+b d^2 e^2+c d^4\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^4 \left (-2 d e x \left (-a c e^4+2 b^2 e^4+5 b c d^2 e^2+3 c^2 d^4\right )+c e^2 x^2 \left (-a e^4+3 b d^2 e^2+7 c d^4\right )-a b e^6-3 a c d^2 e^4+3 b^2 d^2 e^4+8 b c d^4 e^2-4 c d e^3 x^3 \left (b e^2+2 c d^2\right )+5 c^2 d^6\right )}{\left (a+b x^2+c x^4\right ) \left (a e^4+b d^2 e^2+c d^4\right )^3}+\frac {-2 d e x \left (-a c e^4+b^2 e^4+2 b c d^2 e^2+c^2 d^4\right )+c e^2 x^2 \left (-a e^4+b d^2 e^2+3 c d^4\right )-a b e^6-3 a c d^2 e^4+b^2 d^2 e^4+2 b c d^4 e^2-2 c d e^3 x^3 \left (b e^2+2 c d^2\right )+c^2 d^6}{\left (a+b x^2+c x^4\right )^2 \left (a e^4+b d^2 e^2+c d^4\right )^2}+\frac {4 d e^8 \left (b e^2+2 c d^2\right )}{(d+e x) \left (a e^4+b d^2 e^2+c d^4\right )^3}+\frac {e^8}{(d+e x)^2 \left (a e^4+b d^2 e^2+c d^4\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^4 \left (-2 d e x \left (-a c e^4+2 b^2 e^4+5 b c d^2 e^2+3 c^2 d^4\right )+c e^2 x^2 \left (-a e^4+3 b d^2 e^2+7 c d^4\right )-a b e^6-3 a c d^2 e^4+3 b^2 d^2 e^4+8 b c d^4 e^2-4 c d e^3 x^3 \left (b e^2+2 c d^2\right )+5 c^2 d^6\right )}{\left (a+b x^2+c x^4\right ) \left (a e^4+b d^2 e^2+c d^4\right )^3}+\frac {-2 d e x \left (-a c e^4+b^2 e^4+2 b c d^2 e^2+c^2 d^4\right )+c e^2 x^2 \left (-a e^4+b d^2 e^2+3 c d^4\right )-a b e^6-3 a c d^2 e^4+b^2 d^2 e^4+2 b c d^4 e^2-2 c d e^3 x^3 \left (b e^2+2 c d^2\right )+c^2 d^6}{\left (a+b x^2+c x^4\right )^2 \left (a e^4+b d^2 e^2+c d^4\right )^2}+\frac {4 d e^8 \left (b e^2+2 c d^2\right )}{(d+e x) \left (a e^4+b d^2 e^2+c d^4\right )^3}+\frac {e^8}{(d+e x)^2 \left (a e^4+b d^2 e^2+c d^4\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^4 \left (-2 d e x \left (-a c e^4+2 b^2 e^4+5 b c d^2 e^2+3 c^2 d^4\right )+c e^2 x^2 \left (-a e^4+3 b d^2 e^2+7 c d^4\right )-a b e^6-3 a c d^2 e^4+3 b^2 d^2 e^4+8 b c d^4 e^2-4 c d e^3 x^3 \left (b e^2+2 c d^2\right )+5 c^2 d^6\right )}{\left (a+b x^2+c x^4\right ) \left (a e^4+b d^2 e^2+c d^4\right )^3}+\frac {-2 d e x \left (-a c e^4+b^2 e^4+2 b c d^2 e^2+c^2 d^4\right )+c e^2 x^2 \left (-a e^4+b d^2 e^2+3 c d^4\right )-a b e^6-3 a c d^2 e^4+b^2 d^2 e^4+2 b c d^4 e^2-2 c d e^3 x^3 \left (b e^2+2 c d^2\right )+c^2 d^6}{\left (a+b x^2+c x^4\right )^2 \left (a e^4+b d^2 e^2+c d^4\right )^2}+\frac {4 d e^8 \left (b e^2+2 c d^2\right )}{(d+e x) \left (a e^4+b d^2 e^2+c d^4\right )^3}+\frac {e^8}{(d+e x)^2 \left (a e^4+b d^2 e^2+c d^4\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^4 \left (-2 d e x \left (-a c e^4+2 b^2 e^4+5 b c d^2 e^2+3 c^2 d^4\right )+c e^2 x^2 \left (-a e^4+3 b d^2 e^2+7 c d^4\right )-a b e^6-3 a c d^2 e^4+3 b^2 d^2 e^4+8 b c d^4 e^2-4 c d e^3 x^3 \left (b e^2+2 c d^2\right )+5 c^2 d^6\right )}{\left (a+b x^2+c x^4\right ) \left (a e^4+b d^2 e^2+c d^4\right )^3}+\frac {-2 d e x \left (-a c e^4+b^2 e^4+2 b c d^2 e^2+c^2 d^4\right )+c e^2 x^2 \left (-a e^4+b d^2 e^2+3 c d^4\right )-a b e^6-3 a c d^2 e^4+b^2 d^2 e^4+2 b c d^4 e^2-2 c d e^3 x^3 \left (b e^2+2 c d^2\right )+c^2 d^6}{\left (a+b x^2+c x^4\right )^2 \left (a e^4+b d^2 e^2+c d^4\right )^2}+\frac {4 d e^8 \left (b e^2+2 c d^2\right )}{(d+e x) \left (a e^4+b d^2 e^2+c d^4\right )^3}+\frac {e^8}{(d+e x)^2 \left (a e^4+b d^2 e^2+c d^4\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^4 \left (-2 d e x \left (-a c e^4+2 b^2 e^4+5 b c d^2 e^2+3 c^2 d^4\right )+c e^2 x^2 \left (-a e^4+3 b d^2 e^2+7 c d^4\right )-a b e^6-3 a c d^2 e^4+3 b^2 d^2 e^4+8 b c d^4 e^2-4 c d e^3 x^3 \left (b e^2+2 c d^2\right )+5 c^2 d^6\right )}{\left (a+b x^2+c x^4\right ) \left (a e^4+b d^2 e^2+c d^4\right )^3}+\frac {-2 d e x \left (-a c e^4+b^2 e^4+2 b c d^2 e^2+c^2 d^4\right )+c e^2 x^2 \left (-a e^4+b d^2 e^2+3 c d^4\right )-a b e^6-3 a c d^2 e^4+b^2 d^2 e^4+2 b c d^4 e^2-2 c d e^3 x^3 \left (b e^2+2 c d^2\right )+c^2 d^6}{\left (a+b x^2+c x^4\right )^2 \left (a e^4+b d^2 e^2+c d^4\right )^2}+\frac {4 d e^8 \left (b e^2+2 c d^2\right )}{(d+e x) \left (a e^4+b d^2 e^2+c d^4\right )^3}+\frac {e^8}{(d+e x)^2 \left (a e^4+b d^2 e^2+c d^4\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^4 \left (-2 d e x \left (-a c e^4+2 b^2 e^4+5 b c d^2 e^2+3 c^2 d^4\right )+c e^2 x^2 \left (-a e^4+3 b d^2 e^2+7 c d^4\right )-a b e^6-3 a c d^2 e^4+3 b^2 d^2 e^4+8 b c d^4 e^2-4 c d e^3 x^3 \left (b e^2+2 c d^2\right )+5 c^2 d^6\right )}{\left (a+b x^2+c x^4\right ) \left (a e^4+b d^2 e^2+c d^4\right )^3}+\frac {-2 d e x \left (-a c e^4+b^2 e^4+2 b c d^2 e^2+c^2 d^4\right )+c e^2 x^2 \left (-a e^4+b d^2 e^2+3 c d^4\right )-a b e^6-3 a c d^2 e^4+b^2 d^2 e^4+2 b c d^4 e^2-2 c d e^3 x^3 \left (b e^2+2 c d^2\right )+c^2 d^6}{\left (a+b x^2+c x^4\right )^2 \left (a e^4+b d^2 e^2+c d^4\right )^2}+\frac {4 d e^8 \left (b e^2+2 c d^2\right )}{(d+e x) \left (a e^4+b d^2 e^2+c d^4\right )^3}+\frac {e^8}{(d+e x)^2 \left (a e^4+b d^2 e^2+c d^4\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^4 \left (-2 d e x \left (-a c e^4+2 b^2 e^4+5 b c d^2 e^2+3 c^2 d^4\right )+c e^2 x^2 \left (-a e^4+3 b d^2 e^2+7 c d^4\right )-a b e^6-3 a c d^2 e^4+3 b^2 d^2 e^4+8 b c d^4 e^2-4 c d e^3 x^3 \left (b e^2+2 c d^2\right )+5 c^2 d^6\right )}{\left (a+b x^2+c x^4\right ) \left (a e^4+b d^2 e^2+c d^4\right )^3}+\frac {-2 d e x \left (-a c e^4+b^2 e^4+2 b c d^2 e^2+c^2 d^4\right )+c e^2 x^2 \left (-a e^4+b d^2 e^2+3 c d^4\right )-a b e^6-3 a c d^2 e^4+b^2 d^2 e^4+2 b c d^4 e^2-2 c d e^3 x^3 \left (b e^2+2 c d^2\right )+c^2 d^6}{\left (a+b x^2+c x^4\right )^2 \left (a e^4+b d^2 e^2+c d^4\right )^2}+\frac {4 d e^8 \left (b e^2+2 c d^2\right )}{(d+e x) \left (a e^4+b d^2 e^2+c d^4\right )^3}+\frac {e^8}{(d+e x)^2 \left (a e^4+b d^2 e^2+c d^4\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )^2}dx\)

Input:

Int[1/((d + e*x)^2*(a + b*x^2 + c*x^4)^2),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [A] (verified)

Time = 0.92 (sec) , antiderivative size = 2539, normalized size of antiderivative = 1.50

method	result	size
default	\(\text {Expression too large to display}\)	\(2539\)
risch	\(\text {Expression too large to display}\)	\(11714\)

Input:

int(1/(e*x+d)^2/(c*x^4+b*x^2+a)^2,x,method=_RETURNVERBOSE)
 

Output:

-1/(a*e^4+b*d^2*e^2+c*d^4)^3*((1/2*c*(2*a^3*c*e^10-a^2*b^2*e^10-3*a^2*b*c* 
d^2*e^8-4*a^2*c^2*d^4*e^6-4*a*b^2*c*d^4*e^6-10*a*b*c^2*d^6*e^4-6*a*c^3*d^8 
*e^2+b^4*d^4*e^6+3*b^3*c*d^6*e^4+3*b^2*c^2*d^8*e^2+b*c^3*d^10)/a/(4*a*c-b^ 
2)*x^3-c*d*e*(2*a^2*c*e^8-a*b^2*e^8-b^3*d^2*e^6-3*b^2*c*d^4*e^4-4*b*c^2*d^ 
6*e^2-2*c^3*d^8)/(4*a*c-b^2)*x^2+1/2*(3*a^3*b*c*e^10+6*a^3*c^2*d^2*e^8-a^2 
*b^3*e^10-3*a^2*b^2*c*d^2*e^8+2*a^2*b*c^2*d^4*e^6+4*a^2*c^3*d^6*e^4-5*a*b^ 
3*c*d^4*e^6-12*a*b^2*c^2*d^6*e^4-9*a*b*c^3*d^8*e^2-2*a*c^4*d^10+b^5*d^4*e^ 
6+3*b^4*c*d^6*e^4+3*b^3*c^2*d^8*e^2+b^2*c^3*d^10)/a/(4*a*c-b^2)*x-d*e*(3*a 
^2*b*c*e^8+4*a^2*c^2*d^2*e^6-a*b^3*e^8+a*b^2*c*d^2*e^6+6*a*b*c^2*d^4*e^4+4 
*a*c^3*d^6*e^2-b^4*d^2*e^6-3*b^3*c*d^4*e^4-3*b^2*c^2*d^6*e^2-b*c^3*d^8)/(4 
*a*c-b^2))/(c*x^4+b*x^2+a)+2/a/(4*a*c-b^2)*c*(1/(16*a*c-4*b^2)*(-1/4*(-48* 
(-4*a*c+b^2)^(1/2)*a^3*c^2*d*e^9+48*(-4*a*c+b^2)^(1/2)*a^2*b^2*c*d*e^9+96* 
(-4*a*c+b^2)^(1/2)*a^2*b*c^2*d^3*e^7+96*(-4*a*c+b^2)^(1/2)*a^2*c^3*d^5*e^5 
-8*(-4*a*c+b^2)^(1/2)*a*b^4*d*e^9-16*(-4*a*c+b^2)^(1/2)*a*b^3*c*d^3*e^7+32 
*(-4*a*c+b^2)^(1/2)*a*b*c^3*d^7*e^3+16*(-4*a*c+b^2)^(1/2)*a*c^4*d^9*e-128* 
a^3*d*e^9*b*c^2-256*a^3*d^3*e^7*c^3+64*a^2*d*e^9*b^3*c+128*a^2*d^3*e^7*c^2 
*b^2-8*a*d*e^9*b^5-16*a*d^3*e^7*b^4*c)/c*ln(-2*c*x^2+(-4*a*c+b^2)^(1/2)-b) 
+1/2*(-4*a*b*c^4*d^10+22*a^3*b^2*c*e^10+240*a^3*c^3*d^4*e^6+6*a*b^5*d^2*e^ 
8+24*a^2*c^4*d^8*e^2+3*b^5*c*d^6*e^4+3*b^4*c^2*d^8*e^2-12*(-4*a*c+b^2)^(1/ 
2)*a*c^4*d^10+(-4*a*c+b^2)^(1/2)*b^5*d^4*e^6+(-4*a*c+b^2)^(1/2)*b^2*c^3...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )^2} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x+d)^2/(c*x^4+b*x^2+a)^2,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )^2} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x+d)**2/(c*x**4+b*x**2+a)**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )^2} \, dx=\int { \frac {1}{{\left (c x^{4} + b x^{2} + a\right )}^{2} {\left (e x + d\right )}^{2}} \,d x } \] Input:

integrate(1/(e*x+d)^2/(c*x^4+b*x^2+a)^2,x, algorithm="maxima")
 

Output:

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

Giac [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )^2} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x+d)^2/(c*x^4+b*x^2+a)^2,x, algorithm="giac")
 

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 39.81 (sec) , antiderivative size = 45031, normalized size of antiderivative = 26.66 \[ \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \] Input:

int(1/((d + e*x)^2*(a + b*x^2 + c*x^4)^2),x)
 

Output:

symsum(log(root(62914560*a^12*b*c^8*d^10*e^14*z^5 + 62914560*a^11*b*c^9*d^ 
14*e^10*z^5 + 31457280*a^13*b*c^7*d^6*e^18*z^5 + 31457280*a^10*b*c^10*d^18 
*e^6*z^5 + 6291456*a^14*b*c^6*d^2*e^22*z^5 + 6291456*a^9*b*c^11*d^22*e^2*z 
^5 - 115200*a^7*b^13*c*d^6*e^18*z^5 - 90624*a^8*b^12*c*d^4*e^20*z^5 - 7680 
0*a^6*b^14*c*d^8*e^16*z^5 - 36864*a^9*b^11*c*d^2*e^22*z^5 - 21504*a^5*b^15 
*c*d^10*e^14*z^5 + 1536*a^4*b^16*c*d^12*e^12*z^5 + 1536*a^3*b^17*c*d^14*e^ 
10*z^5 - 90439680*a^10*b^4*c^7*d^12*e^12*z^5 - 63897600*a^11*b^4*c^6*d^8*e 
^16*z^5 - 63897600*a^9*b^4*c^8*d^16*e^8*z^5 + 62914560*a^11*b^2*c^8*d^12*e 
^12*z^5 + 39321600*a^12*b^2*c^7*d^8*e^16*z^5 + 39321600*a^10*b^2*c^9*d^16* 
e^8*z^5 + 35782656*a^9*b^6*c^6*d^12*e^12*z^5 - 31457280*a^11*b^3*c^7*d^10* 
e^14*z^5 - 31457280*a^10*b^3*c^8*d^14*e^10*z^5 + 30474240*a^10*b^6*c^5*d^8 
*e^16*z^5 + 30474240*a^8*b^6*c^7*d^16*e^8*z^5 + 29884416*a^9*b^7*c^5*d^10* 
e^14*z^5 + 29884416*a^8*b^7*c^6*d^14*e^10*z^5 - 29097984*a^10*b^5*c^6*d^10 
*e^14*z^5 - 29097984*a^9*b^5*c^7*d^14*e^10*z^5 - 26214400*a^12*b^3*c^6*d^6 
*e^18*z^5 - 26214400*a^9*b^3*c^9*d^18*e^6*z^5 - 17694720*a^12*b^4*c^5*d^4* 
e^20*z^5 - 17694720*a^8*b^4*c^9*d^20*e^4*z^5 + 12779520*a^11*b^6*c^4*d^4*e 
^20*z^5 + 12779520*a^7*b^6*c^8*d^20*e^4*z^5 - 10076160*a^8*b^9*c^4*d^10*e^ 
14*z^5 - 10076160*a^7*b^9*c^5*d^14*e^10*z^5 + 9830400*a^10*b^7*c^4*d^6*e^1 
8*z^5 + 9830400*a^7*b^7*c^7*d^18*e^6*z^5 - 9437184*a^13*b^3*c^5*d^2*e^22*z 
^5 - 9437184*a^8*b^3*c^10*d^22*e^2*z^5 + 6291456*a^13*b^2*c^6*d^4*e^20*...
 

Reduce [F]

\[ \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )^2} \, dx=\int \frac {1}{\left (e x +d \right )^{2} \left (c \,x^{4}+b \,x^{2}+a \right )^{2}}d x \] Input:

int(1/(e*x+d)^2/(c*x^4+b*x^2+a)^2,x)
 

Output:

int(1/(e*x+d)^2/(c*x^4+b*x^2+a)^2,x)