\(\int \frac {(d f+e f x)^3}{(a+b (d+e x)^2+c (d+e x)^4)^3} \, dx\) [284]

Optimal result

Integrand size = 33, antiderivative size = 159 \[ \int \frac {(d f+e f x)^3}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\frac {f^3 \left (2 a+b (d+e x)^2\right )}{4 \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}-\frac {3 b f^3 \left (b+2 c (d+e x)^2\right )}{4 \left (b^2-4 a c\right )^2 e \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {3 b c f^3 \text {arctanh}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} e} \] Output:

1/4*f^3*(2*a+b*(e*x+d)^2)/(-4*a*c+b^2)/e/(a+b*(e*x+d)^2+c*(e*x+d)^4)^2-3/4 
*b*f^3*(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)+3*b* 
c*f^3*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 = 0.21 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.94 \[ \int \frac {(d f+e f x)^3}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\frac {f^3 \left (-\frac {3 b \left (b+2 c (d+e x)^2\right )}{a+b (d+e x)^2+c (d+e x)^4}+\frac {\left (b^2-4 a c\right ) \left (2 a+b (d+e x)^2\right )}{\left (a+(d+e x)^2 \left (b+c (d+e x)^2\right )\right )^2}-\frac {12 b c \arctan \left (\frac {b+2 c (d+e x)^2}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}\right )}{4 \left (b^2-4 a c\right )^2 e} \] Input:

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

Output:

(f^3*((-3*b*(b + 2*c*(d + e*x)^2))/(a + b*(d + e*x)^2 + c*(d + e*x)^4) + ( 
(b^2 - 4*a*c)*(2*a + b*(d + e*x)^2))/(a + (d + e*x)^2*(b + c*(d + e*x)^2)) 
^2 - (12*b*c*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 = 0.32 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1462, 1434, 1159, 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.

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

\(\Big \downarrow \) 1462

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

\(\Big \downarrow \) 1434

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

\(\Big \downarrow \) 1159

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

\(\Big \downarrow \) 1086

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

\(\Big \downarrow \) 1083

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(f^3*((2*a + b*(d + e*x)^2)/(2*(b^2 - 4*a*c)*(a + b*(d + e*x)^2 + c*(d + e 
*x)^4)^2) + (3*b*(-((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)))/(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 1159

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

rule 1434

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

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 = 0.26 (sec) , antiderivative size = 548, normalized size of antiderivative = 3.45

method	result	size
default	\(f^{3} \left (\frac {-\frac {3 c^{2} e^{5} b \,x^{6}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {9 e^{4} d b \,c^{2} x^{5}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {9 b c \,e^{3} \left (10 c \,d^{2}+b \right ) x^{4}}{4 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {3 c d \,e^{2} b \left (10 c \,d^{2}+3 b \right ) x^{3}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {b e \left (45 c^{2} d^{4}+27 b c \,d^{2}+5 a c +b^{2}\right ) x^{2}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {d b \left (9 c^{2} d^{4}+9 b c \,d^{2}+5 a c +b^{2}\right ) x}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {6 b \,c^{2} d^{6}+9 b^{2} c \,d^{4}+10 a b c \,d^{2}+2 b^{3} d^{2}+8 a^{2} c +b^{2} a}{4 e \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,e^{4} x^{4}+4 c d \,e^{3} x^{3}+6 c \,d^{2} e^{2} x^{2}+4 c \,d^{3} e x +b \,e^{2} x^{2}+c \,d^{4}+2 b d e x +b \,d^{2}+a \right )^{2}}+\frac {3 b c \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,e^{4} \textit {\_Z}^{4}+4 c d \,e^{3} \textit {\_Z}^{3}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 b d e \right ) \textit {\_Z} +c \,d^{4}+b \,d^{2}+a \right )}{\sum }\frac {\left (-\textit {\_R} e -d \right ) \ln \left (x -\textit {\_R} \right )}{2 c \,e^{3} \textit {\_R}^{3}+6 c d \,e^{2} \textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 c \,d^{3}+e b \textit {\_R} +b d}\right )}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) e}\right )\)	\(548\)
risch	\(\frac {-\frac {3 c^{2} e^{5} b \,f^{3} x^{6}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {9 f^{3} e^{4} d b \,c^{2} x^{5}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {9 b c \,e^{3} f^{3} \left (10 c \,d^{2}+b \right ) x^{4}}{4 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {3 c d \,e^{2} b \,f^{3} \left (10 c \,d^{2}+3 b \right ) x^{3}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {b e \,f^{3} \left (45 c^{2} d^{4}+27 b c \,d^{2}+5 a c +b^{2}\right ) x^{2}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {d b \,f^{3} \left (9 c^{2} d^{4}+9 b c \,d^{2}+5 a c +b^{2}\right ) x}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {f^{3} \left (6 b \,c^{2} d^{6}+9 b^{2} c \,d^{4}+10 a b c \,d^{2}+2 b^{3} d^{2}+8 a^{2} c +b^{2} a \right )}{4 e \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,e^{4} x^{4}+4 c d \,e^{3} x^{3}+6 c \,d^{2} e^{2} x^{2}+4 c \,d^{3} e x +b \,e^{2} x^{2}+c \,d^{4}+2 b d e x +b \,d^{2}+a \right )^{2}}-\frac {3 f^{3} c b \ln \left (\left (-\left (-4 a c +b^{2}\right )^{\frac {5}{2}} e^{2}-16 a^{2} c^{2} e^{2} b +8 a c \,e^{2} b^{3}-e^{2} b^{5}\right ) x^{2}+\left (-2 \left (-4 a c +b^{2}\right )^{\frac {5}{2}} d e -32 a^{2} b \,c^{2} d e +16 a \,b^{3} c d e -2 b^{5} d e \right ) x -\left (-4 a c +b^{2}\right )^{\frac {5}{2}} d^{2}-16 a^{2} b \,c^{2} d^{2}+8 a \,b^{3} c \,d^{2}-b^{5} d^{2}-32 c^{2} a^{3}+16 c \,a^{2} b^{2}-2 a \,b^{4}\right )}{2 \left (-4 a c +b^{2}\right )^{\frac {5}{2}} e}+\frac {3 f^{3} c b \ln \left (\left (-\left (-4 a c +b^{2}\right )^{\frac {5}{2}} e^{2}+16 a^{2} c^{2} e^{2} b -8 a c \,e^{2} b^{3}+e^{2} b^{5}\right ) x^{2}+\left (-2 \left (-4 a c +b^{2}\right )^{\frac {5}{2}} d e +32 a^{2} b \,c^{2} d e -16 a \,b^{3} c d e +2 b^{5} d e \right ) x -\left (-4 a c +b^{2}\right )^{\frac {5}{2}} d^{2}+16 a^{2} b \,c^{2} d^{2}-8 a \,b^{3} c \,d^{2}+b^{5} d^{2}+32 c^{2} a^{3}-16 c \,a^{2} b^{2}+2 a \,b^{4}\right )}{2 \left (-4 a c +b^{2}\right )^{\frac {5}{2}} e}\)	\(777\)

Input:

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

Output:

f^3*((-3/2*c^2*e^5*b/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6-9*e^4*d*b*c^2/(16*a^2* 
c^2-8*a*b^2*c+b^4)*x^5-9/4*b*c*e^3*(10*c*d^2+b)/(16*a^2*c^2-8*a*b^2*c+b^4) 
*x^4-3*c*d*e^2*b*(10*c*d^2+3*b)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3-1/2*b*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-d*b*(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*(6*b*c^2*d^6+9* 
b^2*c*d^4+10*a*b*c*d^2+2*b^3*d^2+8*a^2*c+a*b^2)/(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+3/2*b*c/(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=RootO 
f(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. 1856 vs. \(2 (151) = 302\).

Time = 0.26 (sec) , antiderivative size = 3843, normalized size of antiderivative = 24.17 \[ \int \frac {(d f+e f x)^3}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((e*f*x+d*f)^3/(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. 1794 vs. \(2 (144) = 288\).

Time = 7.54 (sec) , antiderivative size = 1794, normalized size of antiderivative = 11.28 \[ \int \frac {(d f+e f x)^3}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

3*b*c*f**3*sqrt(-1/(4*a*c - b**2)**5)*log(2*d*x/e + x**2 + (-192*a**3*b*c* 
*4*f**3*sqrt(-1/(4*a*c - b**2)**5) + 144*a**2*b**3*c**3*f**3*sqrt(-1/(4*a* 
c - b**2)**5) - 36*a*b**5*c**2*f**3*sqrt(-1/(4*a*c - b**2)**5) + 3*b**7*c* 
f**3*sqrt(-1/(4*a*c - b**2)**5) + 3*b**2*c*f**3 + 6*b*c**2*d**2*f**3)/(6*b 
*c**2*e**2*f**3))/(2*e) - 3*b*c*f**3*sqrt(-1/(4*a*c - b**2)**5)*log(2*d*x/ 
e + x**2 + (192*a**3*b*c**4*f**3*sqrt(-1/(4*a*c - b**2)**5) - 144*a**2*b** 
3*c**3*f**3*sqrt(-1/(4*a*c - b**2)**5) + 36*a*b**5*c**2*f**3*sqrt(-1/(4*a* 
c - b**2)**5) - 3*b**7*c*f**3*sqrt(-1/(4*a*c - b**2)**5) + 3*b**2*c*f**3 + 
 6*b*c**2*d**2*f**3)/(6*b*c**2*e**2*f**3))/(2*e) + (-8*a**2*c*f**3 - a*b** 
2*f**3 - 10*a*b*c*d**2*f**3 - 2*b**3*d**2*f**3 - 9*b**2*c*d**4*f**3 - 6*b* 
c**2*d**6*f**3 - 36*b*c**2*d*e**5*f**3*x**5 - 6*b*c**2*e**6*f**3*x**6 + x* 
*4*(-9*b**2*c*e**4*f**3 - 90*b*c**2*d**2*e**4*f**3) + x**3*(-36*b**2*c*d*e 
**3*f**3 - 120*b*c**2*d**3*e**3*f**3) + x**2*(-10*a*b*c*e**2*f**3 - 2*b**3 
*e**2*f**3 - 54*b**2*c*d**2*e**2*f**3 - 90*b*c**2*d**4*e**2*f**3) + x*(-20 
*a*b*c*d*e*f**3 - 4*b**3*d*e*f**3 - 36*b**2*c*d**3*e*f**3 - 36*b*c**2*d**5 
*e*f**3))/(64*a**4*c**2*e - 32*a**3*b**2*c*e + 128*a**3*b*c**2*d**2*e + 12 
8*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 - 6 
4*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*...
 

Maxima [F]

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

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

Output:

-3*b*c*f^3*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 
+ 16*a^2*c^2) - 1/4*(6*b*c^2*e^6*f^3*x^6 + 36*b*c^2*d*e^5*f^3*x^5 + 9*(10* 
b*c^2*d^2 + b^2*c)*e^4*f^3*x^4 + 12*(10*b*c^2*d^3 + 3*b^2*c*d)*e^3*f^3*x^3 
 + 2*(45*b*c^2*d^4 + 27*b^2*c*d^2 + b^3 + 5*a*b*c)*e^2*f^3*x^2 + 4*(9*b*c^ 
2*d^5 + 9*b^2*c*d^3 + (b^3 + 5*a*b*c)*d)*e*f^3*x + (6*b*c^2*d^6 + 9*b^2*c* 
d^4 + a*b^2 + 8*a^2*c + 2*(b^3 + 5*a*b*c)*d^2)*f^3)/((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...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 431 vs. \(2 (151) = 302\).

Time = 0.17 (sec) , antiderivative size = 431, normalized size of antiderivative = 2.71 \[ \int \frac {(d f+e f x)^3}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=-\frac {3 \, b c f^{3} \arctan \left (\frac {2 \, c d^{2} f + 2 \, {\left (e f x^{2} + 2 \, d f x\right )} c e + b f}{\sqrt {-b^{2} + 4 \, a c} f}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c} e} - \frac {6 \, b c^{2} d^{6} f^{7} + 18 \, {\left (e f x^{2} + 2 \, d f x\right )} b c^{2} d^{4} e f^{6} + 18 \, {\left (e f x^{2} + 2 \, d f x\right )}^{2} b c^{2} d^{2} e^{2} f^{5} + 9 \, b^{2} c d^{4} f^{7} + 6 \, {\left (e f x^{2} + 2 \, d f x\right )}^{3} b c^{2} e^{3} f^{4} + 18 \, {\left (e f x^{2} + 2 \, d f x\right )} b^{2} c d^{2} e f^{6} + 9 \, {\left (e f x^{2} + 2 \, d f x\right )}^{2} b^{2} c e^{2} f^{5} + 2 \, b^{3} d^{2} f^{7} + 10 \, a b c d^{2} f^{7} + 2 \, {\left (e f x^{2} + 2 \, d f x\right )} b^{3} e f^{6} + 10 \, {\left (e f x^{2} + 2 \, d f x\right )} a b c e f^{6} + a b^{2} f^{7} + 8 \, a^{2} c f^{7}}{4 \, {\left (c d^{4} f^{2} + 2 \, {\left (e f x^{2} + 2 \, d f x\right )} c d^{2} e f + {\left (e f x^{2} + 2 \, d f x\right )}^{2} c e^{2} + b d^{2} f^{2} + {\left (e f x^{2} + 2 \, d f x\right )} b e f + a f^{2}\right )}^{2} {\left (b^{4} e - 8 \, a b^{2} c e + 16 \, a^{2} c^{2} e\right )}} \] Input:

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

Output:

-3*b*c*f^3*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* 
(6*b*c^2*d^6*f^7 + 18*(e*f*x^2 + 2*d*f*x)*b*c^2*d^4*e*f^6 + 18*(e*f*x^2 + 
2*d*f*x)^2*b*c^2*d^2*e^2*f^5 + 9*b^2*c*d^4*f^7 + 6*(e*f*x^2 + 2*d*f*x)^3*b 
*c^2*e^3*f^4 + 18*(e*f*x^2 + 2*d*f*x)*b^2*c*d^2*e*f^6 + 9*(e*f*x^2 + 2*d*f 
*x)^2*b^2*c*e^2*f^5 + 2*b^3*d^2*f^7 + 10*a*b*c*d^2*f^7 + 2*(e*f*x^2 + 2*d* 
f*x)*b^3*e*f^6 + 10*(e*f*x^2 + 2*d*f*x)*a*b*c*e*f^6 + a*b^2*f^7 + 8*a^2*c* 
f^7)/((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 = 12.22 (sec) , antiderivative size = 1267, normalized size of antiderivative = 7.97 \[ \int \frac {(d f+e f x)^3}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

- ((a*b^2*f^3 + 8*a^2*c*f^3 + 2*b^3*d^2*f^3 + 9*b^2*c*d^4*f^3 + 6*b*c^2*d^ 
6*f^3 + 10*a*b*c*d^2*f^3)/(4*e*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x^2*(b^3 
*e*f^3 + 27*b^2*c*d^2*e*f^3 + 45*b*c^2*d^4*e*f^3 + 5*a*b*c*e*f^3))/(2*(b^4 
 + 16*a^2*c^2 - 8*a*b^2*c)) + (9*x^4*(b^2*c*e^3*f^3 + 10*b*c^2*d^2*e^3*f^3 
))/(4*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (3*d*x^3*(3*b^2*c*e^2*f^3 + 10*b*c 
^2*d^2*e^2*f^3))/(b^4 + 16*a^2*c^2 - 8*a*b^2*c) + (d*x*(b^3*f^3 + 9*b^2*c* 
d^2*f^3 + 9*b*c^2*d^4*f^3 + 5*a*b*c*f^3))/(b^4 + 16*a^2*c^2 - 8*a*b^2*c) + 
 (3*b*c^2*e^5*f^3*x^6)/(2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (9*b*c^2*d*e^4 
*f^3*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) - (3*b*c*f^3*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*((9*b^2*c^4*e^8*f^6)/(a*(4*a*c - b^2)^(9/2)*(b^4 + 16*a^2*c^2 - 8* 
a*b^2*c)) + (9*b^3*c^2*f^6*(2*b^5*c^2*e^10 - 16*a*b^3*c^3*e^10 + 32*a^2*b* 
c^4*e^10))/(2*a*e^2*(4*a*c - b^2)^(15/2)*(b^4 + 16*a^2*c^2 - 8*a*b^2*c))) 
+ x*((9*b^3*c^2*f^6*(2*b^5*c^2*d*e^9 - 16*a*b^3*c^3*d*e^9 + 32*a^2*b*c^...
 

Reduce [F]

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

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

Output:

(f**3*(4*int(x**3/(a**4*b + 2*a**4*c*d**2 + 4*a**3*b**2*d**2 + 6*a**3*b**2 
*d*e*x + 3*a**3*b**2*e**2*x**2 + 12*a**3*b*c*d**4 + 24*a**3*b*c*d**3*e*x + 
 24*a**3*b*c*d**2*e**2*x**2 + 12*a**3*b*c*d*e**3*x**3 + 3*a**3*b*c*e**4*x* 
*4 + 8*a**3*c**2*d**6 + 24*a**3*c**2*d**5*e*x + 36*a**3*c**2*d**4*e**2*x** 
2 + 24*a**3*c**2*d**3*e**3*x**3 + 6*a**3*c**2*d**2*e**4*x**4 + 6*a**2*b**3 
*d**4 + 18*a**2*b**3*d**3*e*x + 21*a**2*b**3*d**2*e**2*x**2 + 12*a**2*b**3 
*d*e**3*x**3 + 3*a**2*b**3*e**4*x**4 + 24*a**2*b**2*c*d**6 + 90*a**2*b**2* 
c*d**5*e*x + 153*a**2*b**2*c*d**4*e**2*x**2 + 156*a**2*b**2*c*d**3*e**3*x* 
*3 + 99*a**2*b**2*c*d**2*e**4*x**4 + 36*a**2*b**2*c*d*e**5*x**5 + 6*a**2*b 
**2*c*e**6*x**6 + 30*a**2*b*c**2*d**8 + 144*a**2*b*c**2*d**7*e*x + 324*a** 
2*b*c**2*d**6*e**2*x**2 + 444*a**2*b*c**2*d**5*e**3*x**3 + 399*a**2*b*c**2 
*d**4*e**4*x**4 + 240*a**2*b*c**2*d**3*e**5*x**5 + 96*a**2*b*c**2*d**2*e** 
6*x**6 + 24*a**2*b*c**2*d*e**7*x**7 + 3*a**2*b*c**2*e**8*x**8 + 12*a**2*c* 
*3*d**10 + 72*a**2*c**3*d**9*e*x + 204*a**2*c**3*d**8*e**2*x**2 + 360*a**2 
*c**3*d**7*e**3*x**3 + 426*a**2*c**3*d**6*e**4*x**4 + 336*a**2*c**3*d**5*e 
**5*x**5 + 168*a**2*c**3*d**4*e**6*x**6 + 48*a**2*c**3*d**3*e**7*x**7 + 6* 
a**2*c**3*d**2*e**8*x**8 + 4*a*b**4*d**6 + 18*a*b**4*d**5*e*x + 33*a*b**4* 
d**4*e**2*x**2 + 32*a*b**4*d**3*e**3*x**3 + 18*a*b**4*d**2*e**4*x**4 + 6*a 
*b**4*d*e**5*x**5 + a*b**4*e**6*x**6 + 20*a*b**3*c*d**8 + 108*a*b**3*c*d** 
7*e*x + 258*a*b**3*c*d**6*e**2*x**2 + 364*a*b**3*c*d**5*e**3*x**3 + 339...