\(\int \frac {1}{(d+e x)^2 (a+b x^2+c x^4)} \, dx\) [240]

Optimal result

Integrand size = 22, antiderivative size = 569 \[ \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )} \, dx=-\frac {e^3}{\left (c d^4+b d^2 e^2+a e^4\right ) (d+e x)}+\frac {\sqrt {c} \left (2 c^2 d^6+c d^2 e^2 \left (b d^2+3 \sqrt {b^2-4 a c} d^2-6 a e^2\right )+\left (b+\sqrt {b^2-4 a c}\right ) e^4 \left (b d^2-a e^2\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}} \left (c d^4+b d^2 e^2+a e^4\right )^2}-\frac {\sqrt {c} \left (2 c^2 d^6+c d^2 e^2 \left (b d^2-3 \sqrt {b^2-4 a c} d^2-6 a e^2\right )+\left (b-\sqrt {b^2-4 a c}\right ) e^4 \left (b d^2-a e^2\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}} \left (c d^4+b d^2 e^2+a e^4\right )^2}+\frac {d e \left (2 c^2 d^4+2 b c d^2 e^2+b^2 e^4-2 a c e^4\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (c d^4+b d^2 e^2+a e^4\right )^2}+\frac {2 d e^3 \left (2 c d^2+b e^2\right ) \log (d+e x)}{\left (c d^4+b d^2 e^2+a e^4\right )^2}-\frac {d e^3 \left (2 c d^2+b e^2\right ) \log \left (a+b x^2+c x^4\right )}{2 \left (c d^4+b d^2 e^2+a e^4\right )^2} \] Output:

-e^3/(a*e^4+b*d^2*e^2+c*d^4)/(e*x+d)+1/2*c^(1/2)*(2*c^2*d^6+c*d^2*e^2*(b*d 
^2+3*(-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+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)^2-1/2*c 
^(1/2)*(2*c^2*d^6+c*d^2*e^2*(b*d^2-3*(-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+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)^2+d*e*(-2*a*c*e^4+b^2*e^4+2*b*c*d^2*e^2+2*c^2*d^4 
)*arctanh((2*c*x^2+b)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2)/(a*e^4+b*d^2* 
e^2+c*d^4)^2+2*d*e^3*(b*e^2+2*c*d^2)*ln(e*x+d)/(a*e^4+b*d^2*e^2+c*d^4)^2-1 
/2*d*e^3*(b*e^2+2*c*d^2)*ln(c*x^4+b*x^2+a)/(a*e^4+b*d^2*e^2+c*d^4)^2
 

Mathematica [A] (verified)

Time = 1.39 (sec) , antiderivative size = 585, normalized size of antiderivative = 1.03 \[ \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )} \, dx=\frac {-\frac {2 e^3 \left (c d^4+b d^2 e^2+a e^4\right )}{d+e x}+\frac {\sqrt {2} \sqrt {c} \left (2 c^2 d^6+c d^2 e^2 \left (b d^2+3 \sqrt {b^2-4 a c} d^2-6 a e^2\right )+\left (b+\sqrt {b^2-4 a c}\right ) e^4 \left (b d^2-a e^2\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \sqrt {c} \left (-2 c^2 d^6+\left (-b+\sqrt {b^2-4 a c}\right ) e^4 \left (b d^2-a e^2\right )+c d^2 e^2 \left (-b d^2+3 \sqrt {b^2-4 a c} d^2+6 a e^2\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}+4 \left (2 c d^3 e^3+b d e^5\right ) \log (d+e x)-\frac {d e \left (2 c^2 d^4+b \left (b+\sqrt {b^2-4 a c}\right ) e^4+2 c e^2 \left (b d^2+\sqrt {b^2-4 a c} d^2-a e^2\right )\right ) \log \left (-b+\sqrt {b^2-4 a c}-2 c x^2\right )}{\sqrt {b^2-4 a c}}+\frac {d e \left (2 c^2 d^4+b \left (b-\sqrt {b^2-4 a c}\right ) e^4-2 c e^2 \left (-b d^2+\sqrt {b^2-4 a c} d^2+a e^2\right )\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{\sqrt {b^2-4 a c}}}{2 \left (c d^4+b d^2 e^2+a e^4\right )^2} \] Input:

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

Output:

((-2*e^3*(c*d^4 + b*d^2*e^2 + a*e^4))/(d + e*x) + (Sqrt[2]*Sqrt[c]*(2*c^2* 
d^6 + c*d^2*e^2*(b*d^2 + 3*Sqrt[b^2 - 4*a*c]*d^2 - 6*a*e^2) + (b + Sqrt[b^ 
2 - 4*a*c])*e^4*(b*d^2 - a*e^2))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[ 
b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2] 
*Sqrt[c]*(-2*c^2*d^6 + (-b + Sqrt[b^2 - 4*a*c])*e^4*(b*d^2 - a*e^2) + c*d^ 
2*e^2*(-(b*d^2) + 3*Sqrt[b^2 - 4*a*c]*d^2 + 6*a*e^2))*ArcTan[(Sqrt[2]*Sqrt 
[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 
- 4*a*c]]) + 4*(2*c*d^3*e^3 + b*d*e^5)*Log[d + e*x] - (d*e*(2*c^2*d^4 + b* 
(b + Sqrt[b^2 - 4*a*c])*e^4 + 2*c*e^2*(b*d^2 + Sqrt[b^2 - 4*a*c]*d^2 - a*e 
^2))*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/Sqrt[b^2 - 4*a*c] + (d*e*(2*c^ 
2*d^4 + b*(b - Sqrt[b^2 - 4*a*c])*e^4 - 2*c*e^2*(-(b*d^2) + Sqrt[b^2 - 4*a 
*c]*d^2 + a*e^2))*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/Sqrt[b^2 - 4*a*c]) 
/(2*(c*d^4 + b*d^2*e^2 + a*e^4)^2)
 

Rubi [A] (verified)

Time = 3.13 (sec) , antiderivative size = 569, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {7279, 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.

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

\(\Big \downarrow \) 7279

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (c d^2 e^2 \left (3 d^2 \sqrt {b^2-4 a c}-6 a e^2+b d^2\right )+e^4 \left (\sqrt {b^2-4 a c}+b\right ) \left (b d^2-a e^2\right )+2 c^2 d^6\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}} \left (a e^4+b d^2 e^2+c d^4\right )^2}-\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (c d^2 e^2 \left (-3 d^2 \sqrt {b^2-4 a c}-6 a e^2+b d^2\right )+e^4 \left (b-\sqrt {b^2-4 a c}\right ) \left (b d^2-a e^2\right )+2 c^2 d^6\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b} \left (a e^4+b d^2 e^2+c d^4\right )^2}+\frac {d e \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (-2 a c e^4+b^2 e^4+2 b c d^2 e^2+2 c^2 d^4\right )}{\sqrt {b^2-4 a c} \left (a e^4+b d^2 e^2+c d^4\right )^2}-\frac {d e^3 \left (b e^2+2 c d^2\right ) \log \left (a+b x^2+c x^4\right )}{2 \left (a e^4+b d^2 e^2+c d^4\right )^2}-\frac {e^3}{(d+e x) \left (a e^4+b d^2 e^2+c d^4\right )}+\frac {2 d e^3 \left (b e^2+2 c d^2\right ) \log (d+e x)}{\left (a e^4+b d^2 e^2+c d^4\right )^2}\)

Input:

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

Output:

-(e^3/((c*d^4 + b*d^2*e^2 + a*e^4)*(d + e*x))) + (Sqrt[c]*(2*c^2*d^6 + c*d 
^2*e^2*(b*d^2 + 3*Sqrt[b^2 - 4*a*c]*d^2 - 6*a*e^2) + (b + Sqrt[b^2 - 4*a*c 
])*e^4*(b*d^2 - a*e^2))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a 
*c]]])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]*(c*d^4 + b*d 
^2*e^2 + a*e^4)^2) - (Sqrt[c]*(2*c^2*d^6 + c*d^2*e^2*(b*d^2 - 3*Sqrt[b^2 - 
 4*a*c]*d^2 - 6*a*e^2) + (b - Sqrt[b^2 - 4*a*c])*e^4*(b*d^2 - a*e^2))*ArcT 
an[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b^2 - 4 
*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]*(c*d^4 + b*d^2*e^2 + a*e^4)^2) + (d*e*(2 
*c^2*d^4 + 2*b*c*d^2*e^2 + b^2*e^4 - 2*a*c*e^4)*ArcTanh[(b + 2*c*x^2)/Sqrt 
[b^2 - 4*a*c]])/(Sqrt[b^2 - 4*a*c]*(c*d^4 + b*d^2*e^2 + a*e^4)^2) + (2*d*e 
^3*(2*c*d^2 + b*e^2)*Log[d + e*x])/(c*d^4 + b*d^2*e^2 + a*e^4)^2 - (d*e^3* 
(2*c*d^2 + b*e^2)*Log[a + b*x^2 + c*x^4])/(2*(c*d^4 + b*d^2*e^2 + a*e^4)^2 
)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7279

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ 
{v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su 
mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
 

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 637, normalized size of antiderivative = 1.12

method	result	size
default	\(\frac {4 c \left (\frac {\sqrt {-4 a c +b^{2}}\, \left (-\frac {\left (-2 \sqrt {-4 a c +b^{2}}\, b d \,e^{5}-4 \sqrt {-4 a c +b^{2}}\, c \,d^{3} e^{3}+4 a c d \,e^{5}-2 b^{2} d \,e^{5}-4 b c \,d^{3} e^{3}-4 c^{2} d^{5} e \right ) \ln \left (-2 c \,x^{2}+\sqrt {-4 a c +b^{2}}-b \right )}{4 c}+\frac {\left (-a \sqrt {-4 a c +b^{2}}\, e^{6}+\sqrt {-4 a c +b^{2}}\, b \,d^{2} e^{4}+3 c \sqrt {-4 a c +b^{2}}\, d^{4} e^{2}-a b \,e^{6}-6 a c \,d^{2} e^{4}+b^{2} d^{2} e^{4}+b c \,d^{4} e^{2}+2 c^{2} d^{6}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{16 a c -4 b^{2}}+\frac {\sqrt {-4 a c +b^{2}}\, \left (\frac {\left (2 \sqrt {-4 a c +b^{2}}\, b d \,e^{5}+4 \sqrt {-4 a c +b^{2}}\, c \,d^{3} e^{3}+4 a c d \,e^{5}-2 b^{2} d \,e^{5}-4 b c \,d^{3} e^{3}-4 c^{2} d^{5} e \right ) \ln \left (2 c \,x^{2}+\sqrt {-4 a c +b^{2}}+b \right )}{4 c}+\frac {\left (a \sqrt {-4 a c +b^{2}}\, e^{6}-\sqrt {-4 a c +b^{2}}\, b \,d^{2} e^{4}-3 c \sqrt {-4 a c +b^{2}}\, d^{4} e^{2}-a b \,e^{6}-6 a c \,d^{2} e^{4}+b^{2} d^{2} e^{4}+b c \,d^{4} e^{2}+2 c^{2} d^{6}\right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{16 a c -4 b^{2}}\right )}{\left (e^{4} a +b \,d^{2} e^{2}+c \,d^{4}\right )^{2}}-\frac {e^{3}}{\left (e^{4} a +b \,d^{2} e^{2}+c \,d^{4}\right ) \left (e x +d \right )}+\frac {2 d \,e^{3} \left (b \,e^{2}+2 c \,d^{2}\right ) \ln \left (e x +d \right )}{\left (e^{4} a +b \,d^{2} e^{2}+c \,d^{4}\right )^{2}}\)	\(637\)
risch	\(\text {Expression too large to display}\)	\(2101\)

Input:

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

Output:

4/(a*e^4+b*d^2*e^2+c*d^4)^2*c*((-4*a*c+b^2)^(1/2)/(16*a*c-4*b^2)*(-1/4*(-2 
*(-4*a*c+b^2)^(1/2)*b*d*e^5-4*(-4*a*c+b^2)^(1/2)*c*d^3*e^3+4*a*c*d*e^5-2*b 
^2*d*e^5-4*b*c*d^3*e^3-4*c^2*d^5*e)/c*ln(-2*c*x^2+(-4*a*c+b^2)^(1/2)-b)+1/ 
2*(-a*(-4*a*c+b^2)^(1/2)*e^6+(-4*a*c+b^2)^(1/2)*b*d^2*e^4+3*c*(-4*a*c+b^2) 
^(1/2)*d^4*e^2-a*b*e^6-6*a*c*d^2*e^4+b^2*d^2*e^4+b*c*d^4*e^2+2*c^2*d^6)*2^ 
(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x*2^(1/2)/((-b+(-4*a*c+b 
^2)^(1/2))*c)^(1/2)))+(-4*a*c+b^2)^(1/2)/(16*a*c-4*b^2)*(1/4*(2*(-4*a*c+b^ 
2)^(1/2)*b*d*e^5+4*(-4*a*c+b^2)^(1/2)*c*d^3*e^3+4*a*c*d*e^5-2*b^2*d*e^5-4* 
b*c*d^3*e^3-4*c^2*d^5*e)/c*ln(2*c*x^2+(-4*a*c+b^2)^(1/2)+b)+1/2*(a*(-4*a*c 
+b^2)^(1/2)*e^6-(-4*a*c+b^2)^(1/2)*b*d^2*e^4-3*c*(-4*a*c+b^2)^(1/2)*d^4*e^ 
2-a*b*e^6-6*a*c*d^2*e^4+b^2*d^2*e^4+b*c*d^4*e^2+2*c^2*d^6)*2^(1/2)/((b+(-4 
*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1 
/2))))-e^3/(a*e^4+b*d^2*e^2+c*d^4)/(e*x+d)+2*d*e^3*(b*e^2+2*c*d^2)*ln(e*x+ 
d)/(a*e^4+b*d^2*e^2+c*d^4)^2
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-e^3/(c*d^5 + b*d^3*e^2 + a*d*e^4 + (c*d^4*e + b*d^2*e^3 + a*e^5)*x) + 2*( 
2*c*d^3*e^3 + b*d*e^5)*log(e*x + d)/(c^2*d^8 + 2*b*c*d^6*e^2 + 2*a*b*d^2*e 
^6 + a^2*e^8 + (b^2 + 2*a*c)*d^4*e^4) - integrate(-(c^2*d^6 + 2*b*c*d^4*e^ 
2 - a*b*e^6 + (b^2 - 3*a*c)*d^2*e^4 - 2*(2*c^2*d^3*e^3 + b*c*d*e^5)*x^3 + 
(3*c^2*d^4*e^2 + b*c*d^2*e^4 - a*c*e^6)*x^2 - 2*(c^2*d^5*e + 2*b*c*d^3*e^3 
 + (b^2 - a*c)*d*e^5)*x)/(c*x^4 + b*x^2 + a), x)/(c^2*d^8 + 2*b*c*d^6*e^2 
+ 2*a*b*d^2*e^6 + a^2*e^8 + (b^2 + 2*a*c)*d^4*e^4)
 

Giac [F]

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

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

Output:

sage0*x
 

Mupad [B] (verification not implemented)

Time = 23.99 (sec) , antiderivative size = 4722, normalized size of antiderivative = 8.30 \[ \int \frac {1}{(d+e x)^2 \left (a+b x^2+c x^4\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

symsum(log(root(512*a^4*b*c^2*d^2*e^6*z^4 + 512*a^3*b*c^3*d^6*e^2*z^4 - 25 
6*a^3*b^3*c*d^2*e^6*z^4 - 96*a^2*b^4*c*d^4*e^4*z^4 + 32*a*b^5*c*d^6*e^2*z^ 
4 - 256*a^2*b^3*c^2*d^6*e^2*z^4 + 16*a*b^6*d^4*e^4*z^4 - 128*a^4*b^2*c*e^8 
*z^4 + 16*a*b^4*c^2*d^8*z^4 + 512*a^4*c^3*d^4*e^4*z^4 + 32*a^2*b^5*d^2*e^6 
*z^4 - 128*a^2*b^2*c^3*d^8*z^4 + 256*a^5*c^2*e^8*z^4 + 256*a^3*c^4*d^8*z^4 
 + 16*a^3*b^4*e^8*z^4 + 512*a^3*b*c^2*d*e^5*z^3 - 256*a^2*b^3*c*d*e^5*z^3 
+ 64*a*b^4*c*d^3*e^3*z^3 - 512*a^2*b^2*c^2*d^3*e^3*z^3 + 32*a*b^5*d*e^5*z^ 
3 + 1024*a^3*c^3*d^3*e^3*z^3 - 112*a*b^2*c^2*d^2*e^2*z^2 + 8*b^4*c*d^2*e^2 
*z^2 + 48*a^2*b*c^2*e^4*z^2 - 28*a*b^3*c*e^4*z^2 - 16*a*b*c^3*d^4*z^2 + 32 
0*a^2*c^3*d^2*e^2*z^2 + 4*b^3*c^2*d^4*z^2 + 4*b^5*e^4*z^2 + 32*a*c^3*d*e*z 
 - 8*b^2*c^2*d*e*z + c^3, z, k)*((32*a*c^5*d^2*e^7 + 16*b*c^5*d^4*e^5)/(a^ 
2*e^8 + c^2*d^8 + b^2*d^4*e^4 + 2*a*b*d^2*e^6 + 2*a*c*d^4*e^4 + 2*b*c*d^6* 
e^2) + root(512*a^4*b*c^2*d^2*e^6*z^4 + 512*a^3*b*c^3*d^6*e^2*z^4 - 256*a^ 
3*b^3*c*d^2*e^6*z^4 - 96*a^2*b^4*c*d^4*e^4*z^4 + 32*a*b^5*c*d^6*e^2*z^4 - 
256*a^2*b^3*c^2*d^6*e^2*z^4 + 16*a*b^6*d^4*e^4*z^4 - 128*a^4*b^2*c*e^8*z^4 
 + 16*a*b^4*c^2*d^8*z^4 + 512*a^4*c^3*d^4*e^4*z^4 + 32*a^2*b^5*d^2*e^6*z^4 
 - 128*a^2*b^2*c^3*d^8*z^4 + 256*a^5*c^2*e^8*z^4 + 256*a^3*c^4*d^8*z^4 + 1 
6*a^3*b^4*e^8*z^4 + 512*a^3*b*c^2*d*e^5*z^3 - 256*a^2*b^3*c*d*e^5*z^3 + 64 
*a*b^4*c*d^3*e^3*z^3 - 512*a^2*b^2*c^2*d^3*e^3*z^3 + 32*a*b^5*d*e^5*z^3 + 
1024*a^3*c^3*d^3*e^3*z^3 - 112*a*b^2*c^2*d^2*e^2*z^2 + 8*b^4*c*d^2*e^2*...
 

Reduce [F]

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

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

Output:

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