\(\int \frac {(d+e x)^2}{(a+b x^2+c x^4)^2} \, dx\) [242]

Optimal result

Integrand size = 22, antiderivative size = 410 \[ \int \frac {(d+e x)^2}{\left (a+b x^2+c x^4\right )^2} \, dx=-\frac {d e \left (b+2 c x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {x \left (\left (b^2-2 a c\right ) d^2-a b e^2+c \left (b d^2-2 a e^2\right ) x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\sqrt {c} \left (b^2 d^2+b \left (\sqrt {b^2-4 a c} d^2+4 a e^2\right )-2 a \left (6 c d^2+\sqrt {b^2-4 a c} e^2\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (b d^2-2 a e^2-\frac {b^2 d^2-12 a c d^2+4 a b e^2}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}}}+\frac {4 c d e \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \] Output:

-d*e*(2*c*x^2+b)/(-4*a*c+b^2)/(c*x^4+b*x^2+a)+1/2*x*((-2*a*c+b^2)*d^2-a*b* 
e^2+c*(-2*a*e^2+b*d^2)*x^2)/a/(-4*a*c+b^2)/(c*x^4+b*x^2+a)+1/4*c^(1/2)*(b^ 
2*d^2+b*((-4*a*c+b^2)^(1/2)*d^2+4*a*e^2)-2*a*(6*c*d^2+(-4*a*c+b^2)^(1/2)*e 
^2))*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)+1/4*c^(1/2)*(b*d^2-2*a*e^2-(4* 
a*b*e^2-12*a*c*d^2+b^2*d^2)/(-4*a*c+b^2)^(1/2))*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)/(b+(-4*a*c+b^2)^(1/2)) 
^(1/2)+4*c*d*e*arctanh((2*c*x^2+b)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(3/2)
 

Mathematica [A] (verified)

Time = 1.42 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.03 \[ \int \frac {(d+e x)^2}{\left (a+b x^2+c x^4\right )^2} \, dx=\frac {1}{4} \left (\frac {4 a c x (d+e x)^2+2 a b e (2 d+e x)-2 b d^2 x \left (b+c x^2\right )}{a \left (-b^2+4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\sqrt {2} \sqrt {c} \left (b^2 d^2+b \left (\sqrt {b^2-4 a c} d^2+4 a e^2\right )-2 a \left (6 c d^2+\sqrt {b^2-4 a c} e^2\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{a \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \sqrt {c} \left (-b^2 d^2+12 a c d^2+b \sqrt {b^2-4 a c} d^2-4 a b e^2-2 a \sqrt {b^2-4 a c} e^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{a \left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}}-\frac {8 c d e \log \left (-b+\sqrt {b^2-4 a c}-2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac {8 c d e \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}\right ) \] Input:

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

Output:

((4*a*c*x*(d + e*x)^2 + 2*a*b*e*(2*d + e*x) - 2*b*d^2*x*(b + c*x^2))/(a*(- 
b^2 + 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*Sqrt[c]*(b^2*d^2 + b*(Sqrt[b^ 
2 - 4*a*c]*d^2 + 4*a*e^2) - 2*a*(6*c*d^2 + Sqrt[b^2 - 4*a*c]*e^2))*ArcTan[ 
(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(3/2)*S 
qrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(-(b^2*d^2) + 12*a*c*d^2 + 
b*Sqrt[b^2 - 4*a*c]*d^2 - 4*a*b*e^2 - 2*a*Sqrt[b^2 - 4*a*c]*e^2)*ArcTan[(S 
qrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(3/2)*Sqr 
t[b + Sqrt[b^2 - 4*a*c]]) - (8*c*d*e*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2] 
)/(b^2 - 4*a*c)^(3/2) + (8*c*d*e*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^ 
2 - 4*a*c)^(3/2))/4
 

Rubi [A] (verified)

Time = 1.13 (sec) , antiderivative size = 382, normalized size of antiderivative = 0.93, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {2202, 27, 1432, 1086, 1083, 219, 1492, 25, 1480, 218}

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.

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

\(\Big \downarrow \) 2202

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1432

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

\(\Big \downarrow \) 1086

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

\(\Big \downarrow \) 1083

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 1492

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1480

\(\displaystyle \frac {\frac {1}{2} c \left (\frac {4 a b e^2-12 a c d^2+b^2 d^2}{\sqrt {b^2-4 a c}}-2 a e^2+b d^2\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx+\frac {1}{2} c \left (-\frac {4 a b e^2-12 a c d^2+b^2 d^2}{\sqrt {b^2-4 a c}}-2 a e^2+b d^2\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx}{2 a \left (b^2-4 a c\right )}+d e \left (\frac {4 c \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {b+2 c x^2}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )+\frac {x \left (d^2 \left (b^2-2 a c\right )+c x^2 \left (b d^2-2 a e^2\right )-a b e^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (\frac {4 a b e^2-12 a c d^2+b^2 d^2}{\sqrt {b^2-4 a c}}-2 a e^2+b d^2\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (-\frac {4 a b e^2-12 a c d^2+b^2 d^2}{\sqrt {b^2-4 a c}}-2 a e^2+b d^2\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b}}}{2 a \left (b^2-4 a c\right )}+d e \left (\frac {4 c \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {b+2 c x^2}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )+\frac {x \left (d^2 \left (b^2-2 a c\right )+c x^2 \left (b d^2-2 a e^2\right )-a b e^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\)

Input:

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

Output:

(x*((b^2 - 2*a*c)*d^2 - a*b*e^2 + c*(b*d^2 - 2*a*e^2)*x^2))/(2*a*(b^2 - 4* 
a*c)*(a + b*x^2 + c*x^4)) + ((Sqrt[c]*(b*d^2 - 2*a*e^2 + (b^2*d^2 - 12*a*c 
*d^2 + 4*a*b*e^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - S 
qrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[c]*(b*d^ 
2 - 2*a*e^2 - (b^2*d^2 - 12*a*c*d^2 + 4*a*b*e^2)/Sqrt[b^2 - 4*a*c])*ArcTan 
[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b + Sqrt[ 
b^2 - 4*a*c]]))/(2*a*(b^2 - 4*a*c)) + d*e*(-((b + 2*c*x^2)/((b^2 - 4*a*c)* 
(a + b*x^2 + c*x^4))) + (4*c*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(b^ 
2 - 4*a*c)^(3/2))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 1086

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*c*((2*p + 
 3)/((p + 1)*(b^2 - 4*a*c)))   Int[(a + b*x + c*x^2)^(p + 1), x], x] /; Fre 
eQ[{a, b, c}, x] && ILtQ[p, -1]
 

rule 1432

Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[1/2 
 Subst[Int[(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x]
 

rule 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   Int[1/(b/2 
+ q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] 
 && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
 

rule 1492

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb 
ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + 
 c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 
 - 4*a*c))   Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 
7)*(d*b - 2*a*e)*c*x^2, x]*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, 
 b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && 
 LtQ[p, -1] && IntegerQ[2*p]
 

rule 2202

Int[(Pn_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{n 
 = Expon[Pn, x], k}, Int[Sum[Coeff[Pn, x, 2*k]*x^(2*k), {k, 0, n/2}]*(a + b 
*x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pn, x, 2*k + 1]*x^(2*k), {k, 0, (n - 
1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pn, x] 
 &&  !PolyQ[Pn, x^2]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.22 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.62

method	result	size
risch	\(\frac {\frac {c \left (2 a \,e^{2}-b \,d^{2}\right ) x^{3}}{2 \left (4 a c -b^{2}\right ) a}+\frac {2 c d e \,x^{2}}{4 a c -b^{2}}+\frac {\left (a b \,e^{2}+2 a c \,d^{2}-b^{2} d^{2}\right ) x}{2 a \left (4 a c -b^{2}\right )}+\frac {b d e}{4 a c -b^{2}}}{c \,x^{4}+b \,x^{2}+a}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\left (\frac {c \left (2 a \,e^{2}-b \,d^{2}\right ) \textit {\_R}^{2}}{\left (4 a c -b^{2}\right ) a}+\frac {8 c d e \textit {\_R}}{4 a c -b^{2}}-\frac {a b \,e^{2}-6 a c \,d^{2}+b^{2} d^{2}}{\left (4 a c -b^{2}\right ) a}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3} c +b \textit {\_R}}\right )}{4}\)	\(255\)
default	\(16 c^{2} \left (-\frac {\frac {-\frac {\left (-4 \sqrt {-4 a c +b^{2}}\, a c \,d^{2}+\sqrt {-4 a c +b^{2}}\, b^{2} d^{2}+8 a^{2} c \,e^{2}-2 a \,b^{2} e^{2}-4 a b c \,d^{2}+b^{3} d^{2}\right ) x}{16 a c}-\frac {d e \left (4 a c -b^{2}\right )}{4 c}}{x^{2}+\frac {b}{2 c}-\frac {\sqrt {-4 a c +b^{2}}}{2 c}}+\frac {4 \sqrt {-4 a c +b^{2}}\, a d e \ln \left (-2 c \,x^{2}+\sqrt {-4 a c +b^{2}}-b \right )+\frac {\left (4 \sqrt {-4 a c +b^{2}}\, a b \,e^{2}-12 \sqrt {-4 a c +b^{2}}\, a c \,d^{2}+\sqrt {-4 a c +b^{2}}\, b^{2} d^{2}+8 a^{2} c \,e^{2}-2 a \,b^{2} e^{2}-4 a b c \,d^{2}+b^{3} d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}}{8 a}}{4 c \left (4 a c -b^{2}\right )^{2}}+\frac {\frac {\frac {\left (4 \sqrt {-4 a c +b^{2}}\, a c \,d^{2}-\sqrt {-4 a c +b^{2}}\, b^{2} d^{2}+8 a^{2} c \,e^{2}-2 a \,b^{2} e^{2}-4 a b c \,d^{2}+b^{3} d^{2}\right ) x}{16 a c}+\frac {d e \left (4 a c -b^{2}\right )}{4 c}}{x^{2}+\frac {\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}+\frac {4 \sqrt {-4 a c +b^{2}}\, a d e \ln \left (2 c \,x^{2}+\sqrt {-4 a c +b^{2}}+b \right )+\frac {\left (-4 \sqrt {-4 a c +b^{2}}\, a b \,e^{2}+12 \sqrt {-4 a c +b^{2}}\, a c \,d^{2}-\sqrt {-4 a c +b^{2}}\, b^{2} d^{2}+8 a^{2} c \,e^{2}-2 a \,b^{2} e^{2}-4 a b c \,d^{2}+b^{3} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}}{8 a}}{4 c \left (4 a c -b^{2}\right )^{2}}\right )\)	\(635\)

Input:

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

Output:

(1/2*c*(2*a*e^2-b*d^2)/(4*a*c-b^2)/a*x^3+2*c*d*e/(4*a*c-b^2)*x^2+1/2*(a*b* 
e^2+2*a*c*d^2-b^2*d^2)/a/(4*a*c-b^2)*x+b*d*e/(4*a*c-b^2))/(c*x^4+b*x^2+a)+ 
1/4*sum((c*(2*a*e^2-b*d^2)/(4*a*c-b^2)/a*_R^2+8*c*d*e/(4*a*c-b^2)*_R-(a*b* 
e^2-6*a*c*d^2+b^2*d^2)/(4*a*c-b^2)/a)/(2*_R^3*c+_R*b)*ln(x-_R),_R=RootOf(_ 
Z^4*c+_Z^2*b+a))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-1/2*(4*a*c*d*e*x^2 + 2*a*b*d*e - (b*c*d^2 - 2*a*c*e^2)*x^3 + (a*b*e^2 - ( 
b^2 - 2*a*c)*d^2)*x)/((a*b^2*c - 4*a^2*c^2)*x^4 + a^2*b^2 - 4*a^3*c + (a*b 
^3 - 4*a^2*b*c)*x^2) - 1/2*integrate((8*a*c*d*e*x - a*b*e^2 - (b^2 - 6*a*c 
)*d^2 - (b*c*d^2 - 2*a*c*e^2)*x^2)/(c*x^4 + b*x^2 + a), x)/(a*b^2 - 4*a^2* 
c)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5199 vs. \(2 (361) = 722\).

Time = 1.37 (sec) , antiderivative size = 5199, normalized size of antiderivative = 12.68 \[ \int \frac {(d+e x)^2}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*(b*c*d^2*x^3 - 2*a*c*e^2*x^3 - 4*a*c*d*e*x^2 + b^2*d^2*x - 2*a*c*d^2*x 
 - a*b*e^2*x - 2*a*b*d*e)/((c*x^4 + b*x^2 + a)*(a*b^2 - 4*a^2*c)) + 1/16*( 
(2*b^3*c^2 - 8*a*b*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4 
*a*c)*c)*b^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c) 
*a*b*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c 
 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b*c^2 - 2*(b^ 
2 - 4*a*c)*b*c^2)*(a*b^2 - 4*a^2*c)^2*d^2 - 2*(2*a*b^2*c^2 - 8*a^2*c^3 - s 
qrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2 + 4*sqrt(2) 
*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c + 2*sqrt(2)*sqrt( 
b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c - sqrt(2)*sqrt(b^2 - 4* 
a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*c^2 - 2*(b^2 - 4*a*c)*a*c^2)*(a*b^2 
 - 4*a^2*c)^2*e^2 + 2*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^6 - 14* 
sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c - 2*sqrt(2)*sqrt(b*c + s 
qrt(b^2 - 4*a*c)*c)*a*b^5*c - 2*a*b^6*c + 64*sqrt(2)*sqrt(b*c + sqrt(b^2 - 
 4*a*c)*c)*a^3*b^2*c^2 + 20*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^ 
3*c^2 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c^2 + 28*a^2*b^4*c^2 
 - 96*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*c^3 - 48*sqrt(2)*sqrt(b* 
c + sqrt(b^2 - 4*a*c)*c)*a^3*b*c^3 - 10*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a* 
c)*c)*a^2*b^2*c^3 - 128*a^3*b^2*c^3 + 24*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a 
*c)*c)*a^3*c^4 + 192*a^4*c^4 + 2*(b^2 - 4*a*c)*a*b^4*c - 20*(b^2 - 4*a*...
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 22.38 (sec) , antiderivative size = 4118, normalized size of antiderivative = 10.04 \[ \int \frac {(d+e x)^2}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

symsum(log((8*a^3*c^4*e^6 + 5*b^3*c^4*d^6 + 6*a^2*b^2*c^3*e^6 - 312*a^2*c^ 
5*d^4*e^2 - 3*b^4*c^3*d^4*e^2 - 36*a*b*c^5*d^6 + 82*a*b^2*c^4*d^4*e^2 + 3* 
a*b^3*c^3*d^2*e^4 + 4*a^2*b*c^4*d^2*e^4)/(8*(a^2*b^6 - 64*a^5*c^3 - 12*a^3 
*b^4*c + 48*a^4*b^2*c^2)) - root(1572864*a^8*b^2*c^5*z^4 - 983040*a^7*b^4* 
c^4*z^4 + 327680*a^6*b^6*c^3*z^4 - 61440*a^5*b^8*c^2*z^4 + 6144*a^4*b^10*c 
*z^4 - 1048576*a^9*c^6*z^4 - 256*a^3*b^12*z^4 + 576*a^2*b^8*c*d^2*e^2*z^2 
+ 122880*a^5*b^2*c^4*d^2*e^2*z^2 - 22528*a^4*b^4*c^3*d^2*e^2*z^2 - 1024*a^ 
3*b^6*c^2*d^2*e^2*z^2 - 32*a*b^10*d^2*e^2*z^2 + 12288*a^6*b*c^4*e^4*z^2 + 
61440*a^5*b*c^5*d^4*z^2 + 432*a*b^9*c*d^4*z^2 - 180224*a^6*c^5*d^2*e^2*z^2 
 - 8192*a^5*b^3*c^3*e^4*z^2 + 1536*a^4*b^5*c^2*e^4*z^2 - 61440*a^4*b^3*c^4 
*d^4*z^2 + 24064*a^3*b^5*c^3*d^4*z^2 - 4608*a^2*b^7*c^2*d^4*z^2 - 16*a^2*b 
^9*e^4*z^2 - 16*b^11*d^4*z^2 - 1344*a*b^6*c^2*d^5*e*z + 128*a*b^7*c*d^3*e^ 
3*z + 64*a^2*b^6*c*d*e^5*z + 6144*a^3*b^3*c^3*d^3*e^3*z - 1536*a^2*b^5*c^2 
*d^3*e^3*z - 31744*a^3*b^2*c^4*d^5*e*z + 9984*a^2*b^4*c^3*d^5*e*z - 8192*a 
^4*b*c^4*d^3*e^3*z + 3072*a^4*b^2*c^3*d*e^5*z - 768*a^3*b^4*c^2*d*e^5*z + 
36864*a^4*c^5*d^5*e*z - 4096*a^5*c^4*d*e^5*z + 64*b^8*c*d^5*e*z - 1824*a^2 
*b*c^4*d^6*e^2 - 544*a^3*b*c^3*d^2*e^6 + 464*a*b^3*c^3*d^6*e^2 + 70*a*b^4* 
c^2*d^4*e^4 - 18*a*b^5*c*d^2*e^6 - 192*a^2*b^2*c^3*d^4*e^4 + 80*a^2*b^3*c^ 
2*d^2*e^6 - 34*b^5*c^2*d^6*e^2 - 1312*a^3*c^4*d^4*e^4 - 24*a^3*b^2*c^2*e^8 
 - 9*b^6*c*d^4*e^4 - 9*a^2*b^4*c*e^8 + 360*a*b^2*c^4*d^8 - 25*b^4*c^3*d...
 

Reduce [B] (verification not implemented)

Time = 0.55 (sec) , antiderivative size = 4951, normalized size of antiderivative = 12.08 \[ \int \frac {(d+e x)^2}{\left (a+b x^2+c x^4\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 32*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt( 
2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**3*b* 
c*d*e - 32*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((s 
qrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a** 
2*b**2*c*d*e*x**2 - 32*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) 
- b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt( 
a) + b))*a**2*b*c**2*d*e*x**4 - 8*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan 
((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))* 
a**3*b*c*e**2 - 2*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c) 
*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**2*b**3*e**2 + 
 16*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) 
- 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**2*b**2*c*d**2 - 8*sqrt(a)*s 
qrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x 
)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**2*b**2*c*e**2*x**2 - 8*sqrt(a)*sqrt(2*sq 
rt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2 
*sqrt(c)*sqrt(a) + b))*a**2*b*c**2*e**2*x**4 - 2*sqrt(a)*sqrt(2*sqrt(c)*sq 
rt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c) 
*sqrt(a) + b))*a*b**4*d**2 - 2*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((s 
qrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a*b 
**4*e**2*x**2 + 16*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqr...