\(\int \frac {d+e x+f x^2}{(a+b x^2+c x^4)^2} \, dx\) [34]

Optimal result

Integrand size = 25, antiderivative size = 368 \[ \int \frac {d+e x+f x^2}{\left (a+b x^2+c x^4\right )^2} \, dx=-\frac {e \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {x \left (b^2 d-2 a c d-a b f+c (b d-2 a f) x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\sqrt {c} \left (b d-2 a f+\frac {b^2 d-12 a c d+4 a b f}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a \left (b^2-4 a c\right ) \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (b d-2 a f-\frac {b^2 d-12 a c d+4 a b f}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}}}+\frac {2 c e \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \] Output:

-1/2*e*(2*c*x^2+b)/(-4*a*c+b^2)/(c*x^4+b*x^2+a)+1/2*x*(b^2*d-2*a*c*d-a*b*f 
+c*(-2*a*f+b*d)*x^2)/a/(-4*a*c+b^2)/(c*x^4+b*x^2+a)+1/4*c^(1/2)*(b*d-2*a*f 
+(4*a*b*f-12*a*c*d+b^2*d)/(-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)+1/4*c^(1/2)*(b*d-2*a*f-(4*a*b*f-12*a*c*d+b^2*d)/(-4*a*c+b^2)^(1/2))*a 
rctan(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)+2*c*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 = 1.25 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.08 \[ \int \frac {d+e x+f x^2}{\left (a+b x^2+c x^4\right )^2} \, dx=\frac {1}{4} \left (\frac {2 a b (e+f x)-2 b d x \left (b+c x^2\right )+4 a c x (d+x (e+f x))}{a \left (-b^2+4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\sqrt {2} \sqrt {c} \left (b^2 d+b \left (\sqrt {b^2-4 a c} d+4 a f\right )-2 a \left (6 c d+\sqrt {b^2-4 a c} f\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{a \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \sqrt {c} \left (-b^2 d+12 a c d+b \sqrt {b^2-4 a c} d-4 a b f-2 a \sqrt {b^2-4 a c} f\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{a \left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}}-\frac {4 c e \log \left (-b+\sqrt {b^2-4 a c}-2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac {4 c e \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}\right ) \] Input:

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

Output:

((2*a*b*(e + f*x) - 2*b*d*x*(b + c*x^2) + 4*a*c*x*(d + x*(e + f*x)))/(a*(- 
b^2 + 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*Sqrt[c]*(b^2*d + b*(Sqrt[b^2 
- 4*a*c]*d + 4*a*f) - 2*a*(6*c*d + Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*S 
qrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sq 
rt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(-(b^2*d) + 12*a*c*d + b*Sqrt[b^2 - 4 
*a*c]*d - 4*a*b*f - 2*a*Sqrt[b^2 - 4*a*c]*f)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sq 
rt[b + Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a 
*c]]) - (4*c*e*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(3/2) 
+ (4*c*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 = 0.68 (sec) , antiderivative size = 356, normalized size of antiderivative = 0.97, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, 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.

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

\(\Big \downarrow \) 2202

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1432

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

\(\Big \downarrow \) 1086

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

\(\Big \downarrow \) 1083

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 1492

\(\displaystyle -\frac {\int -\frac {d b^2+a f b+c (b d-2 a f) x^2-6 a c d}{c x^4+b x^2+a}dx}{2 a \left (b^2-4 a c\right )}+\frac {1}{2} e \left (\frac {4 c \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {b+2 c x^2}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )+\frac {x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {d b^2+a f b+c (b d-2 a f) x^2-6 a c d}{c x^4+b x^2+a}dx}{2 a \left (b^2-4 a c\right )}+\frac {1}{2} e \left (\frac {4 c \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {b+2 c x^2}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )+\frac {x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\)

\(\Big \downarrow \) 1480

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

(x*(b^2*d - 2*a*c*d - a*b*f + c*(b*d - 2*a*f)*x^2))/(2*a*(b^2 - 4*a*c)*(a 
+ b*x^2 + c*x^4)) + ((Sqrt[c]*(b*d - 2*a*f + (b^2*d - 12*a*c*d + 4*a*b*f)/ 
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]]) + (Sqrt[c]*(b*d - 2*a*f - (b^2*d - 
 12*a*c*d + 4*a*b*f)/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)) + (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)))/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 = 0.23 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.63

method	result	size
risch	\(\frac {\frac {c \left (2 a f -b d \right ) x^{3}}{2 \left (4 a c -b^{2}\right ) a}+\frac {c \,x^{2} e}{4 a c -b^{2}}+\frac {\left (a b f +2 d a c -b^{2} d \right ) x}{2 a \left (4 a c -b^{2}\right )}+\frac {b e}{8 a c -2 b^{2}}}{c \,x^{4}+b \,x^{2}+a}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\left (\frac {c \left (2 a f -b d \right ) \textit {\_R}^{2}}{\left (4 a c -b^{2}\right ) a}+\frac {4 \textit {\_R} c e}{4 a c -b^{2}}-\frac {a b f -6 d a c +b^{2} d}{\left (4 a c -b^{2}\right ) a}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3} c +\textit {\_R} b}\right )}{4}\)	\(232\)
default	\(16 c^{2} \left (\frac {\frac {\frac {\left (4 \sqrt {-4 a c +b^{2}}\, a c d -\sqrt {-4 a c +b^{2}}\, b^{2} d +8 a^{2} c f -2 a \,b^{2} f -4 a b c d +b^{3} d \right ) x}{16 a c}+\frac {e \left (4 a c -b^{2}\right )}{8 c}}{x^{2}+\frac {\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}+\frac {2 \sqrt {-4 a c +b^{2}}\, a e \ln \left (2 c \,x^{2}+\sqrt {-4 a c +b^{2}}+b \right )+\frac {\left (-4 \sqrt {-4 a c +b^{2}}\, a b f +12 \sqrt {-4 a c +b^{2}}\, a c d -\sqrt {-4 a c +b^{2}}\, b^{2} d +8 a^{2} c f -2 a \,b^{2} f -4 a b c d +b^{3} d \right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}}{8 a}}{4 c \left (4 a c -b^{2}\right )^{2}}-\frac {\frac {-\frac {\left (-4 \sqrt {-4 a c +b^{2}}\, a c d +\sqrt {-4 a c +b^{2}}\, b^{2} d +8 a^{2} c f -2 a \,b^{2} f -4 a b c d +b^{3} d \right ) x}{16 a c}-\frac {e \left (4 a c -b^{2}\right )}{8 c}}{x^{2}+\frac {b}{2 c}-\frac {\sqrt {-4 a c +b^{2}}}{2 c}}+\frac {2 \sqrt {-4 a c +b^{2}}\, a e \ln \left (-2 c \,x^{2}+\sqrt {-4 a c +b^{2}}-b \right )+\frac {\left (4 \sqrt {-4 a c +b^{2}}\, a b f -12 \sqrt {-4 a c +b^{2}}\, a c d +\sqrt {-4 a c +b^{2}}\, b^{2} d +8 a^{2} c f -2 a \,b^{2} f -4 a b c d +b^{3} d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}}{8 a}}{4 c \left (4 a c -b^{2}\right )^{2}}\right )\)	\(579\)

Input:

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

Output:

(1/2*c*(2*a*f-b*d)/(4*a*c-b^2)/a*x^3+c/(4*a*c-b^2)*x^2*e+1/2*(a*b*f+2*a*c* 
d-b^2*d)/a/(4*a*c-b^2)*x+1/2/(4*a*c-b^2)*b*e)/(c*x^4+b*x^2+a)+1/4*sum((c*( 
2*a*f-b*d)/(4*a*c-b^2)/a*_R^2+4/(4*a*c-b^2)*_R*c*e-(a*b*f-6*a*c*d+b^2*d)/( 
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. \[ \int \frac {d+e x+f x^2}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-1/2*(2*a*c*e*x^2 - (b*c*d - 2*a*c*f)*x^3 + a*b*e + (a*b*f - (b^2 - 2*a*c) 
*d)*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((4*a*c*e*x - a*b*f - (b*c*d - 2*a*c*f)*x^2 - (b^2 - 
 6*a*c)*d)/(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. 5159 vs. \(2 (320) = 640\).

Time = 1.24 (sec) , antiderivative size = 5159, normalized size of antiderivative = 14.02 \[ \int \frac {d+e x+f x^2}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*(b*c*d*x^3 - 2*a*c*f*x^3 - 2*a*c*e*x^2 + b^2*d*x - 2*a*c*d*x - a*b*f*x 
 - a*b*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*a*b^2*c^2 - 8*a^2*c^3 - sqrt(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 + sq 
rt(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*f + 
2*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^6 - 14*sqrt(2)*sqrt(b*c + s 
qrt(b^2 - 4*a*c)*c)*a^2*b^4*c - 2*sqrt(2)*sqrt(b*c + sqrt(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*c)*a^2*b^2*c^2 + 48*...
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 18.85 (sec) , antiderivative size = 4707, normalized size of antiderivative = 12.79 \[ \int \frac {d+e x+f x^2}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

symsum(log((5*b^3*c^4*d^3 + 8*a^3*c^4*f^3 - 96*a^2*c^5*d*e^2 + 72*a^2*c^5* 
d^2*f - 3*b^4*c^3*d^2*f + 6*a^2*b^2*c^3*f^3 - 36*a*b*c^5*d^3 + 16*a*b^2*c^ 
4*d*e^2 + 18*a*b^2*c^4*d^2*f + 3*a*b^3*c^3*d*f^2 - 60*a^2*b*c^4*d*f^2 + 16 
*a^2*b*c^4*e^2*f)/(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*f*z^2 + 24576*a^5*b^2*c^4*d*f*z^2 
 - 3072*a^3*b^6*c^2*d*f*z^2 + 2048*a^4*b^4*c^3*d*f*z^2 + 12288*a^6*b*c^4*f 
^2*z^2 + 61440*a^5*b*c^5*d^2*z^2 - 49152*a^6*c^5*d*f*z^2 + 432*a*b^9*c*d^2 
*z^2 - 8192*a^5*b^3*c^3*f^2*z^2 + 1536*a^4*b^5*c^2*f^2*z^2 + 24576*a^5*b^2 
*c^4*e^2*z^2 - 6144*a^4*b^4*c^3*e^2*z^2 + 512*a^3*b^6*c^2*e^2*z^2 - 61440* 
a^4*b^3*c^4*d^2*z^2 + 24064*a^3*b^5*c^3*d^2*z^2 - 4608*a^2*b^7*c^2*d^2*z^2 
 - 32*a*b^10*d*f*z^2 - 32768*a^6*c^5*e^2*z^2 - 16*a^2*b^9*f^2*z^2 - 16*b^1 
1*d^2*z^2 - 4096*a^4*b*c^4*d*e*f*z + 64*a*b^7*c*d*e*f*z + 3072*a^3*b^3*c^3 
*d*e*f*z - 768*a^2*b^5*c^2*d*e*f*z + 32*a^2*b^6*c*e*f^2*z - 672*a*b^6*c^2* 
d^2*e*z + 1536*a^4*b^2*c^3*e*f^2*z - 384*a^3*b^4*c^2*e*f^2*z - 15872*a^3*b 
^2*c^4*d^2*e*z + 4992*a^2*b^4*c^3*d^2*e*z - 2048*a^5*c^4*e*f^2*z + 18432*a 
^4*c^5*d^2*e*z + 32*b^8*c*d^2*e*z - 32*a*b^4*c^2*d*e^2*f + 192*a^2*b^2*c^3 
*d*e^2*f - 192*a^3*b*c^3*e^2*f^2 + 198*a*b^4*c^2*d^2*f^2 + 144*a^2*b^3*c^2 
*d*f^3 - 960*a^2*b*c^4*d^2*e^2 + 240*a*b^3*c^3*d^2*e^2 + 768*a^3*c^4*d*...
 

Reduce [B] (verification not implemented)

Time = 0.57 (sec) , antiderivative size = 4754, normalized size of antiderivative = 12.92 \[ \int \frac {d+e x+f x^2}{\left (a+b x^2+c x^4\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 16*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*e - 16*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqr 
t(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**2* 
b**2*c*e*x**2 - 16*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*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*f - 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*f + 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 - 8*sqrt(a)*sqrt(2*sqrt(c)*s 
qrt(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*f*x**2 - 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**2*b*c**2*f*x**4 - 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*b**4 
*d - 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*b**4*f*x**2 + 16*sqrt(a)* 
sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(...