\(\int \frac {d f+e f x}{(a+b (d+e x)^2+c (d+e x)^4)^3} \, dx\) [286]

Optimal result

Integrand size = 31, antiderivative size = 153 \[ \int \frac {d f+e f x}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=-\frac {f \left (b+2 c (d+e x)^2\right )}{4 \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {3 c f \left (b+2 c (d+e x)^2\right )}{2 \left (b^2-4 a c\right )^2 e \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {6 c^2 f \text {arctanh}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} e} \] Output:

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

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.97 \[ \int \frac {d f+e f x}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\frac {f \left (\frac {\left (b^2-4 a c\right ) \left (-b-2 c (d+e x)^2\right )}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {6 c \left (b+2 c (d+e x)^2\right )}{a+b (d+e x)^2+c (d+e x)^4}+\frac {24 c^2 \arctan \left (\frac {b+2 c (d+e x)^2}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}\right )}{4 \left (b^2-4 a c\right )^2 e} \] Input:

Integrate[(d*f + e*f*x)/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {1462, 1432, 1086, 1086, 1083, 219}

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.

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

\(\Big \downarrow \) 1462

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

\(\Big \downarrow \) 1432

\(\displaystyle \frac {f \int \frac {1}{\left (c (d+e x)^4+b (d+e x)^2+a\right )^3}d(d+e x)^2}{2 e}\)

\(\Big \downarrow \) 1086

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

\(\Big \downarrow \) 1086

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

\(\Big \downarrow \) 1083

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

\(\Big \downarrow \) 219

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

Input:

Int[(d*f + e*f*x)/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3,x]
 

Output:

(f*(-1/2*(b + 2*c*(d + e*x)^2)/((b^2 - 4*a*c)*(a + b*(d + e*x)^2 + c*(d + 
e*x)^4)^2) - (3*c*(-((b + 2*c*(d + e*x)^2)/((b^2 - 4*a*c)*(a + b*(d + e*x) 
^2 + c*(d + e*x)^4))) + (4*c*ArcTanh[(b + 2*c*(d + e*x)^2)/Sqrt[b^2 - 4*a* 
c]])/(b^2 - 4*a*c)^(3/2)))/(b^2 - 4*a*c)))/(2*e)
 

Defintions of rubi rules used

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 1462

Int[(u_)^(m_.)*((a_.) + (b_.)*(v_)^2 + (c_.)*(v_)^4)^(p_.), x_Symbol] :> Si 
mp[u^m/(Coefficient[v, x, 1]*v^m)   Subst[Int[x^m*(a + b*x^2 + c*x^(2*2))^p 
, x], x, v], x] /; FreeQ[{a, b, c, m, p}, x] && LinearPairQ[u, v, x]
 

Maple [C] (verified)

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

Time = 0.25 (sec) , antiderivative size = 543, normalized size of antiderivative = 3.55

method	result	size
default	\(f \left (\frac {\frac {3 c^{3} e^{5} x^{6}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {18 e^{4} d \,c^{3} x^{5}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {9 c^{2} e^{3} \left (10 c \,d^{2}+b \right ) x^{4}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {6 c^{2} d \,e^{2} \left (10 c \,d^{2}+3 b \right ) x^{3}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {c e \left (45 c^{2} d^{4}+27 b c \,d^{2}+5 a c +b^{2}\right ) x^{2}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {2 c d \left (9 c^{2} d^{4}+9 b c \,d^{2}+5 a c +b^{2}\right ) x}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {12 c^{3} d^{6}+18 b \,c^{2} d^{4}+20 a \,c^{2} d^{2}+4 b^{2} c \,d^{2}+10 a b c -b^{3}}{4 e \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,e^{4} x^{4}+4 c d \,e^{3} x^{3}+6 c \,d^{2} e^{2} x^{2}+4 c \,d^{3} e x +b \,e^{2} x^{2}+c \,d^{4}+2 b d e x +b \,d^{2}+a \right )^{2}}+\frac {3 c^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,e^{4} \textit {\_Z}^{4}+4 c d \,e^{3} \textit {\_Z}^{3}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 b d e \right ) \textit {\_Z} +c \,d^{4}+b \,d^{2}+a \right )}{\sum }\frac {\left (\textit {\_R} e +d \right ) \ln \left (x -\textit {\_R} \right )}{2 c \,e^{3} \textit {\_R}^{3}+6 c d \,e^{2} \textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 c \,d^{3}+e b \textit {\_R} +b d}\right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) e}\right )\)	\(543\)
risch	\(\frac {\frac {3 c^{3} e^{5} f \,x^{6}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {18 f \,e^{4} d \,c^{3} x^{5}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {9 c^{2} e^{3} f \left (10 c \,d^{2}+b \right ) x^{4}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {6 c^{2} d \,e^{2} f \left (10 c \,d^{2}+3 b \right ) x^{3}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {c e f \left (45 c^{2} d^{4}+27 b c \,d^{2}+5 a c +b^{2}\right ) x^{2}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {2 c d f \left (9 c^{2} d^{4}+9 b c \,d^{2}+5 a c +b^{2}\right ) x}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {f \left (12 c^{3} d^{6}+18 b \,c^{2} d^{4}+20 a \,c^{2} d^{2}+4 b^{2} c \,d^{2}+10 a b c -b^{3}\right )}{4 e \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,e^{4} x^{4}+4 c d \,e^{3} x^{3}+6 c \,d^{2} e^{2} x^{2}+4 c \,d^{3} e x +b \,e^{2} x^{2}+c \,d^{4}+2 b d e x +b \,d^{2}+a \right )^{2}}-\frac {3 f \,c^{2} \ln \left (\left (\left (-4 a c +b^{2}\right )^{\frac {5}{2}} e^{2}-16 a^{2} c^{2} e^{2} b +8 a c \,e^{2} b^{3}-e^{2} b^{5}\right ) x^{2}+\left (2 \left (-4 a c +b^{2}\right )^{\frac {5}{2}} d e -32 a^{2} b \,c^{2} d e +16 a \,b^{3} c d e -2 b^{5} d e \right ) x +\left (-4 a c +b^{2}\right )^{\frac {5}{2}} d^{2}-16 a^{2} b \,c^{2} d^{2}+8 a \,b^{3} c \,d^{2}-b^{5} d^{2}-32 c^{2} a^{3}+16 c \,a^{2} b^{2}-2 a \,b^{4}\right )}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}} e}+\frac {3 f \,c^{2} \ln \left (\left (\left (-4 a c +b^{2}\right )^{\frac {5}{2}} e^{2}+16 a^{2} c^{2} e^{2} b -8 a c \,e^{2} b^{3}+e^{2} b^{5}\right ) x^{2}+\left (2 \left (-4 a c +b^{2}\right )^{\frac {5}{2}} d e +32 a^{2} b \,c^{2} d e -16 a \,b^{3} c d e +2 b^{5} d e \right ) x +\left (-4 a c +b^{2}\right )^{\frac {5}{2}} d^{2}+16 a^{2} b \,c^{2} d^{2}-8 a \,b^{3} c \,d^{2}+b^{5} d^{2}+32 c^{2} a^{3}-16 c \,a^{2} b^{2}+2 a \,b^{4}\right )}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}} e}\)	\(756\)

Input:

int((e*f*x+d*f)/(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x,method=_RETURNVERBOSE)
 

Output:

f*((3*c^3*e^5/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6+18*e^4*d*c^3/(16*a^2*c^2-8*a* 
b^2*c+b^4)*x^5+9/2*c^2*e^3*(10*c*d^2+b)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4+6*c 
^2*d*e^2*(10*c*d^2+3*b)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3+c*e*(45*c^2*d^4+27* 
b*c*d^2+5*a*c+b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2+2*c*d*(9*c^2*d^4+9*b*c*d 
^2+5*a*c+b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x+1/4/e*(12*c^3*d^6+18*b*c^2*d^4+ 
20*a*c^2*d^2+4*b^2*c*d^2+10*a*b*c-b^3)/(16*a^2*c^2-8*a*b^2*c+b^4))/(c*e^4* 
x^4+4*c*d*e^3*x^3+6*c*d^2*e^2*x^2+4*c*d^3*e*x+b*e^2*x^2+c*d^4+2*b*d*e*x+b* 
d^2+a)^2+3*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)/e*sum((_R*e+d)/(2*_R^3*c*e^3+6*_ 
R^2*c*d*e^2+6*_R*c*d^2*e+2*c*d^3+_R*b*e+b*d)*ln(x-_R),_R=RootOf(c*e^4*_Z^4 
+4*c*d*e^3*_Z^3+(6*c*d^2*e^2+b*e^2)*_Z^2+(4*c*d^3*e+2*b*d*e)*_Z+c*d^4+b*d^ 
2+a)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1808 vs. \(2 (145) = 290\).

Time = 0.48 (sec) , antiderivative size = 3748, normalized size of antiderivative = 24.50 \[ \int \frac {d f+e f x}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((e*f*x+d*f)/(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1707 vs. \(2 (139) = 278\).

Time = 6.90 (sec) , antiderivative size = 1707, normalized size of antiderivative = 11.16 \[ \int \frac {d f+e f x}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((e*f*x+d*f)/(a+b*(e*x+d)**2+c*(e*x+d)**4)**3,x)
 

Output:

-3*c**2*f*sqrt(-1/(4*a*c - b**2)**5)*log(2*d*x/e + x**2 + (-192*a**3*c**5* 
f*sqrt(-1/(4*a*c - b**2)**5) + 144*a**2*b**2*c**4*f*sqrt(-1/(4*a*c - b**2) 
**5) - 36*a*b**4*c**3*f*sqrt(-1/(4*a*c - b**2)**5) + 3*b**6*c**2*f*sqrt(-1 
/(4*a*c - b**2)**5) + 3*b*c**2*f + 6*c**3*d**2*f)/(6*c**3*e**2*f))/e + 3*c 
**2*f*sqrt(-1/(4*a*c - b**2)**5)*log(2*d*x/e + x**2 + (192*a**3*c**5*f*sqr 
t(-1/(4*a*c - b**2)**5) - 144*a**2*b**2*c**4*f*sqrt(-1/(4*a*c - b**2)**5) 
+ 36*a*b**4*c**3*f*sqrt(-1/(4*a*c - b**2)**5) - 3*b**6*c**2*f*sqrt(-1/(4*a 
*c - b**2)**5) + 3*b*c**2*f + 6*c**3*d**2*f)/(6*c**3*e**2*f))/e + (10*a*b* 
c*f + 20*a*c**2*d**2*f - b**3*f + 4*b**2*c*d**2*f + 18*b*c**2*d**4*f + 12* 
c**3*d**6*f + 72*c**3*d*e**5*f*x**5 + 12*c**3*e**6*f*x**6 + x**4*(18*b*c** 
2*e**4*f + 180*c**3*d**2*e**4*f) + x**3*(72*b*c**2*d*e**3*f + 240*c**3*d** 
3*e**3*f) + x**2*(20*a*c**2*e**2*f + 4*b**2*c*e**2*f + 108*b*c**2*d**2*e** 
2*f + 180*c**3*d**4*e**2*f) + x*(40*a*c**2*d*e*f + 8*b**2*c*d*e*f + 72*b*c 
**2*d**3*e*f + 72*c**3*d**5*e*f))/(64*a**4*c**2*e - 32*a**3*b**2*c*e + 128 
*a**3*b*c**2*d**2*e + 128*a**3*c**3*d**4*e + 4*a**2*b**4*e - 64*a**2*b**3* 
c*d**2*e + 128*a**2*b*c**3*d**6*e + 64*a**2*c**4*d**8*e + 8*a*b**5*d**2*e 
- 24*a*b**4*c*d**4*e - 64*a*b**3*c**2*d**6*e - 32*a*b**2*c**3*d**8*e + 4*b 
**6*d**4*e + 8*b**5*c*d**6*e + 4*b**4*c**2*d**8*e + x**8*(64*a**2*c**4*e** 
9 - 32*a*b**2*c**3*e**9 + 4*b**4*c**2*e**9) + x**7*(512*a**2*c**4*d*e**8 - 
 256*a*b**2*c**3*d*e**8 + 32*b**4*c**2*d*e**8) + x**6*(128*a**2*b*c**3*...
 

Maxima [F]

\[ \int \frac {d f+e f x}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\int { \frac {e f x + d f}{{\left ({\left (e x + d\right )}^{4} c + {\left (e x + d\right )}^{2} b + a\right )}^{3}} \,d x } \] Input:

integrate((e*f*x+d*f)/(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x, algorithm="maxima")
 

Output:

6*c^2*f*integrate((e*x + d)/(c*e^4*x^4 + 4*c*d*e^3*x^3 + c*d^4 + (6*c*d^2 
+ b)*e^2*x^2 + b*d^2 + 2*(2*c*d^3 + b*d)*e*x + a), x)/(b^4 - 8*a*b^2*c + 1 
6*a^2*c^2) + 1/4*(12*c^3*e^6*f*x^6 + 72*c^3*d*e^5*f*x^5 + 18*(10*c^3*d^2 + 
 b*c^2)*e^4*f*x^4 + 24*(10*c^3*d^3 + 3*b*c^2*d)*e^3*f*x^3 + 4*(45*c^3*d^4 
+ 27*b*c^2*d^2 + b^2*c + 5*a*c^2)*e^2*f*x^2 + 8*(9*c^3*d^5 + 9*b*c^2*d^3 + 
 (b^2*c + 5*a*c^2)*d)*e*f*x + (12*c^3*d^6 + 18*b*c^2*d^4 - b^3 + 10*a*b*c 
+ 4*(b^2*c + 5*a*c^2)*d^2)*f)/((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*e^9*x^ 
8 + 8*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d*e^8*x^7 + 2*(b^5*c - 8*a*b^3* 
c^2 + 16*a^2*b*c^3 + 14*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^2)*e^7*x^6 
+ 4*(14*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^3 + 3*(b^5*c - 8*a*b^3*c^2 
+ 16*a^2*b*c^3)*d)*e^6*x^5 + (b^6 - 6*a*b^4*c + 32*a^3*c^3 + 70*(b^4*c^2 - 
 8*a*b^2*c^3 + 16*a^2*c^4)*d^4 + 30*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d 
^2)*e^5*x^4 + 4*(14*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^5 + 10*(b^5*c - 
 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*d)*e^4*x 
^3 + 2*(14*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^6 + a*b^5 - 8*a^2*b^3*c 
+ 16*a^3*b*c^2 + 15*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^4 + 3*(b^6 - 6* 
a*b^4*c + 32*a^3*c^3)*d^2)*e^3*x^2 + 4*(2*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2* 
c^4)*d^7 + 3*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^5 + (b^6 - 6*a*b^4*c + 
 32*a^3*c^3)*d^3 + (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d)*e^2*x + ((b^4*c 
^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^8 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*...
                                                                                    
                                                                                    
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 429 vs. \(2 (145) = 290\).

Time = 0.17 (sec) , antiderivative size = 429, normalized size of antiderivative = 2.80 \[ \int \frac {d f+e f x}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\frac {6 \, c^{2} f \arctan \left (\frac {2 \, c d^{2} f + 2 \, {\left (e f x^{2} + 2 \, d f x\right )} c e + b f}{\sqrt {-b^{2} + 4 \, a c} f}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c} e} + \frac {12 \, c^{3} d^{6} f^{5} + 36 \, {\left (e f x^{2} + 2 \, d f x\right )} c^{3} d^{4} e f^{4} + 36 \, {\left (e f x^{2} + 2 \, d f x\right )}^{2} c^{3} d^{2} e^{2} f^{3} + 18 \, b c^{2} d^{4} f^{5} + 12 \, {\left (e f x^{2} + 2 \, d f x\right )}^{3} c^{3} e^{3} f^{2} + 36 \, {\left (e f x^{2} + 2 \, d f x\right )} b c^{2} d^{2} e f^{4} + 18 \, {\left (e f x^{2} + 2 \, d f x\right )}^{2} b c^{2} e^{2} f^{3} + 4 \, b^{2} c d^{2} f^{5} + 20 \, a c^{2} d^{2} f^{5} + 4 \, {\left (e f x^{2} + 2 \, d f x\right )} b^{2} c e f^{4} + 20 \, {\left (e f x^{2} + 2 \, d f x\right )} a c^{2} e f^{4} - b^{3} f^{5} + 10 \, a b c f^{5}}{4 \, {\left (c d^{4} f^{2} + 2 \, {\left (e f x^{2} + 2 \, d f x\right )} c d^{2} e f + {\left (e f x^{2} + 2 \, d f x\right )}^{2} c e^{2} + b d^{2} f^{2} + {\left (e f x^{2} + 2 \, d f x\right )} b e f + a f^{2}\right )}^{2} {\left (b^{4} e - 8 \, a b^{2} c e + 16 \, a^{2} c^{2} e\right )}} \] Input:

integrate((e*f*x+d*f)/(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x, algorithm="giac")
 

Output:

6*c^2*f*arctan((2*c*d^2*f + 2*(e*f*x^2 + 2*d*f*x)*c*e + b*f)/(sqrt(-b^2 + 
4*a*c)*f))/((b^4 - 8*a*b^2*c + 16*a^2*c^2)*sqrt(-b^2 + 4*a*c)*e) + 1/4*(12 
*c^3*d^6*f^5 + 36*(e*f*x^2 + 2*d*f*x)*c^3*d^4*e*f^4 + 36*(e*f*x^2 + 2*d*f* 
x)^2*c^3*d^2*e^2*f^3 + 18*b*c^2*d^4*f^5 + 12*(e*f*x^2 + 2*d*f*x)^3*c^3*e^3 
*f^2 + 36*(e*f*x^2 + 2*d*f*x)*b*c^2*d^2*e*f^4 + 18*(e*f*x^2 + 2*d*f*x)^2*b 
*c^2*e^2*f^3 + 4*b^2*c*d^2*f^5 + 20*a*c^2*d^2*f^5 + 4*(e*f*x^2 + 2*d*f*x)* 
b^2*c*e*f^4 + 20*(e*f*x^2 + 2*d*f*x)*a*c^2*e*f^4 - b^3*f^5 + 10*a*b*c*f^5) 
/((c*d^4*f^2 + 2*(e*f*x^2 + 2*d*f*x)*c*d^2*e*f + (e*f*x^2 + 2*d*f*x)^2*c*e 
^2 + b*d^2*f^2 + (e*f*x^2 + 2*d*f*x)*b*e*f + a*f^2)^2*(b^4*e - 8*a*b^2*c*e 
 + 16*a^2*c^2*e))
 

Mupad [B] (verification not implemented)

Time = 13.28 (sec) , antiderivative size = 1199, normalized size of antiderivative = 7.84 \[ \int \frac {d f+e f x}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx =\text {Too large to display} \] Input:

int((d*f + e*f*x)/(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3,x)
 

Output:

((x^2*(5*a*c^2*e*f + b^2*c*e*f + 45*c^3*d^4*e*f + 27*b*c^2*d^2*e*f))/(b^4 
+ 16*a^2*c^2 - 8*a*b^2*c) + (12*c^3*d^6*f - b^3*f + 20*a*c^2*d^2*f + 4*b^2 
*c*d^2*f + 18*b*c^2*d^4*f + 10*a*b*c*f)/(4*e*(b^4 + 16*a^2*c^2 - 8*a*b^2*c 
)) + (9*x^4*(10*c^3*d^2*e^3*f + b*c^2*e^3*f))/(2*(b^4 + 16*a^2*c^2 - 8*a*b 
^2*c)) + (2*d*x*(9*c^3*d^4*f + 5*a*c^2*f + b^2*c*f + 9*b*c^2*d^2*f))/(b^4 
+ 16*a^2*c^2 - 8*a*b^2*c) + (6*d*x^3*(10*c^3*d^2*e^2*f + 3*b*c^2*e^2*f))/( 
b^4 + 16*a^2*c^2 - 8*a*b^2*c) + (3*c^3*e^5*f*x^6)/(b^4 + 16*a^2*c^2 - 8*a* 
b^2*c) + (18*c^3*d*e^4*f*x^5)/(b^4 + 16*a^2*c^2 - 8*a*b^2*c))/(x^2*(6*b^2* 
d^2*e^2 + 28*c^2*d^6*e^2 + 2*a*b*e^2 + 12*a*c*d^2*e^2 + 30*b*c*d^4*e^2) + 
x^6*(28*c^2*d^2*e^6 + 2*b*c*e^6) + x*(4*b^2*d^3*e + 8*c^2*d^7*e + 8*a*c*d^ 
3*e + 12*b*c*d^5*e + 4*a*b*d*e) + x^3*(4*b^2*d*e^3 + 56*c^2*d^5*e^3 + 8*a* 
c*d*e^3 + 40*b*c*d^3*e^3) + x^5*(56*c^2*d^3*e^5 + 12*b*c*d*e^5) + x^4*(b^2 
*e^4 + 70*c^2*d^4*e^4 + 2*a*c*e^4 + 30*b*c*d^2*e^4) + a^2 + b^2*d^4 + c^2* 
d^8 + c^2*e^8*x^8 + 2*a*b*d^2 + 2*a*c*d^4 + 2*b*c*d^6 + 8*c^2*d*e^7*x^7) + 
 (6*c^2*f*atan(((b^4*(4*a*c - b^2)^5 + 16*a^2*c^2*(4*a*c - b^2)^5 - 8*a*b^ 
2*c*(4*a*c - b^2)^5)*(x^2*((36*c^6*e^8*f^2)/(a*(4*a*c - b^2)^(9/2)*(b^4 + 
16*a^2*c^2 - 8*a*b^2*c)) + (36*b*c^4*f^2*(b^5*c^2*e^10 - 8*a*b^3*c^3*e^10 
+ 16*a^2*b*c^4*e^10))/(a*e^2*(4*a*c - b^2)^(15/2)*(b^4 + 16*a^2*c^2 - 8*a* 
b^2*c))) + x*((72*c^6*d*e^7*f^2)/(a*(4*a*c - b^2)^(9/2)*(b^4 + 16*a^2*c^2 
- 8*a*b^2*c)) + (72*b*c^4*f^2*(b^5*c^2*d*e^9 - 8*a*b^3*c^3*d*e^9 + 16*a...
 

Reduce [F]

\[ \int \frac {d f+e f x}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\text {too large to display} \] Input:

int((e*f*x+d*f)/(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x)
 

Output:

(f*(24*int(x**5/(a**3*b**2 + 8*a**3*b*c*d**2 + 12*a**3*c**2*d**4 + 3*a**2* 
b**3*d**2 + 6*a**2*b**3*d*e*x + 3*a**2*b**3*e**2*x**2 + 27*a**2*b**2*c*d** 
4 + 60*a**2*b**2*c*d**3*e*x + 42*a**2*b**2*c*d**2*e**2*x**2 + 12*a**2*b**2 
*c*d*e**3*x**3 + 3*a**2*b**2*c*e**4*x**4 + 60*a**2*b*c**2*d**6 + 168*a**2* 
b*c**2*d**5*e*x + 180*a**2*b*c**2*d**4*e**2*x**2 + 96*a**2*b*c**2*d**3*e** 
3*x**3 + 24*a**2*b*c**2*d**2*e**4*x**4 + 36*a**2*c**3*d**8 + 144*a**2*c**3 
*d**7*e*x + 216*a**2*c**3*d**6*e**2*x**2 + 144*a**2*c**3*d**5*e**3*x**3 + 
36*a**2*c**3*d**4*e**4*x**4 + 3*a*b**4*d**4 + 12*a*b**4*d**3*e*x + 18*a*b* 
*4*d**2*e**2*x**2 + 12*a*b**4*d*e**3*x**3 + 3*a*b**4*e**4*x**4 + 30*a*b**3 
*c*d**6 + 132*a*b**3*c*d**5*e*x + 234*a*b**3*c*d**4*e**2*x**2 + 216*a*b**3 
*c*d**3*e**3*x**3 + 114*a*b**3*c*d**2*e**4*x**4 + 36*a*b**3*c*d*e**5*x**5 
+ 6*a*b**3*c*e**6*x**6 + 87*a*b**2*c**2*d**8 + 456*a*b**2*c**2*d**7*e*x + 
1020*a*b**2*c**2*d**6*e**2*x**2 + 1272*a*b**2*c**2*d**5*e**3*x**3 + 966*a* 
b**2*c**2*d**4*e**4*x**4 + 456*a*b**2*c**2*d**3*e**5*x**5 + 132*a*b**2*c** 
2*d**2*e**6*x**6 + 24*a*b**2*c**2*d*e**7*x**7 + 3*a*b**2*c**2*e**8*x**8 + 
96*a*b*c**3*d**10 + 624*a*b*c**3*d**9*e*x + 1752*a*b*c**3*d**8*e**2*x**2 + 
 2784*a*b*c**3*d**7*e**3*x**3 + 2760*a*b*c**3*d**6*e**4*x**4 + 1776*a*b*c* 
*3*d**5*e**5*x**5 + 744*a*b*c**3*d**4*e**6*x**6 + 192*a*b*c**3*d**3*e**7*x 
**7 + 24*a*b*c**3*d**2*e**8*x**8 + 36*a*c**4*d**12 + 288*a*c**4*d**11*e*x 
+ 1008*a*c**4*d**10*e**2*x**2 + 2016*a*c**4*d**9*e**3*x**3 + 2520*a*c**...