\(\int \frac {1}{(d f+e f x) (a+b (d+e x)^2+c (d+e x)^4)^3} \, dx\) [287]

Optimal result

Integrand size = 33, antiderivative size = 270 \[ \int \frac {1}{(d f+e f x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\frac {b^2-2 a c+b c (d+e x)^2}{4 a \left (b^2-4 a c\right ) e f \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) (d+e x)^2}{4 a^2 \left (b^2-4 a c\right )^2 e f \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {b \left (b^4-10 a b^2 c+30 a^2 c^2\right ) \text {arctanh}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a^3 \left (b^2-4 a c\right )^{5/2} e f}+\frac {\log (d+e x)}{a^3 e f}-\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^3 e f} \] Output:

1/4*(b^2-2*a*c+b*c*(e*x+d)^2)/a/(-4*a*c+b^2)/e/f/(a+b*(e*x+d)^2+c*(e*x+d)^ 
4)^2+1/4*(2*b^4-15*a*b^2*c+16*c^2*a^2+2*b*c*(-7*a*c+b^2)*(e*x+d)^2)/a^2/(- 
4*a*c+b^2)^2/e/f/(a+b*(e*x+d)^2+c*(e*x+d)^4)+1/2*b*(30*a^2*c^2-10*a*b^2*c+ 
b^4)*arctanh((b+2*c*(e*x+d)^2)/(-4*a*c+b^2)^(1/2))/a^3/(-4*a*c+b^2)^(5/2)/ 
e/f+ln(e*x+d)/a^3/e/f-1/4*ln(a+b*(e*x+d)^2+c*(e*x+d)^4)/a^3/e/f
 

Mathematica [A] (verified)

Time = 4.28 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.46 \[ \int \frac {1}{(d f+e f x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\frac {\frac {a^2 \left (-b^2+2 a c-b c (d+e x)^2\right )}{\left (-b^2+4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {a \left (2 b^4-15 a b^2 c+16 a^2 c^2+2 b^3 c (d+e x)^2-14 a b c^2 (d+e x)^2\right )}{\left (b^2-4 a c\right )^2 \left (a+(d+e x)^2 \left (b+c (d+e x)^2\right )\right )}+4 \log (d+e x)-\frac {\left (b^5-10 a b^3 c+30 a^2 b c^2+b^4 \sqrt {b^2-4 a c}-8 a b^2 c \sqrt {b^2-4 a c}+16 a^2 c^2 \sqrt {b^2-4 a c}\right ) \log \left (b-\sqrt {b^2-4 a c}+2 c (d+e x)^2\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac {\left (b^5-10 a b^3 c+30 a^2 b c^2-b^4 \sqrt {b^2-4 a c}+8 a b^2 c \sqrt {b^2-4 a c}-16 a^2 c^2 \sqrt {b^2-4 a c}\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c (d+e x)^2\right )}{\left (b^2-4 a c\right )^{5/2}}}{4 a^3 e f} \] Input:

Integrate[1/((d*f + e*f*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3),x]
 

Output:

((a^2*(-b^2 + 2*a*c - b*c*(d + e*x)^2))/((-b^2 + 4*a*c)*(a + b*(d + e*x)^2 
 + c*(d + e*x)^4)^2) + (a*(2*b^4 - 15*a*b^2*c + 16*a^2*c^2 + 2*b^3*c*(d + 
e*x)^2 - 14*a*b*c^2*(d + e*x)^2))/((b^2 - 4*a*c)^2*(a + (d + e*x)^2*(b + c 
*(d + e*x)^2))) + 4*Log[d + e*x] - ((b^5 - 10*a*b^3*c + 30*a^2*b*c^2 + b^4 
*Sqrt[b^2 - 4*a*c] - 8*a*b^2*c*Sqrt[b^2 - 4*a*c] + 16*a^2*c^2*Sqrt[b^2 - 4 
*a*c])*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*(d + e*x)^2])/(b^2 - 4*a*c)^(5/2) + 
 ((b^5 - 10*a*b^3*c + 30*a^2*b*c^2 - b^4*Sqrt[b^2 - 4*a*c] + 8*a*b^2*c*Sqr 
t[b^2 - 4*a*c] - 16*a^2*c^2*Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 
 2*c*(d + e*x)^2])/(b^2 - 4*a*c)^(5/2))/(4*a^3*e*f)
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.11, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {1462, 1434, 1165, 25, 1235, 27, 1200, 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.

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

\(\Big \downarrow \) 1462

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

\(\Big \downarrow \) 1434

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

\(\Big \downarrow \) 1165

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1235

\(\displaystyle \frac {\frac {\frac {16 a^2 c^2+2 b c \left (b^2-7 a c\right ) (d+e x)^2-15 a b^2 c+2 b^4}{a \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {\int -\frac {2 \left (\left (b^2-4 a c\right )^2+b c \left (b^2-7 a c\right ) (d+e x)^2\right )}{(d+e x)^2 \left (c (d+e x)^4+b (d+e x)^2+a\right )}d(d+e x)^2}{a \left (b^2-4 a c\right )}}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}}{2 e f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {2 \int \frac {\left (b^2-4 a c\right )^2+b c \left (b^2-7 a c\right ) (d+e x)^2}{(d+e x)^2 \left (c (d+e x)^4+b (d+e x)^2+a\right )}d(d+e x)^2}{a \left (b^2-4 a c\right )}+\frac {16 a^2 c^2+2 b c \left (b^2-7 a c\right ) (d+e x)^2-15 a b^2 c+2 b^4}{a \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )}}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}}{2 e f}\)

\(\Big \downarrow \) 1200

\(\displaystyle \frac {\frac {\frac {2 \int \left (\frac {\left (4 a c-b^2\right )^2}{a (d+e x)^2}+\frac {-c \left (b^2-4 a c\right )^2 (d+e x)^2-b \left (b^4-9 a c b^2+23 a^2 c^2\right )}{a \left (c (d+e x)^4+b (d+e x)^2+a\right )}\right )d(d+e x)^2}{a \left (b^2-4 a c\right )}+\frac {16 a^2 c^2+2 b c \left (b^2-7 a c\right ) (d+e x)^2-15 a b^2 c+2 b^4}{a \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )}}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}}{2 e f}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {\frac {2 \left (\frac {b \left (30 a^2 c^2-10 a b^2 c+b^4\right ) \text {arctanh}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{a \sqrt {b^2-4 a c}}+\frac {\left (b^2-4 a c\right )^2 \log \left ((d+e x)^2\right )}{a}-\frac {\left (b^2-4 a c\right )^2 \log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{2 a}\right )}{a \left (b^2-4 a c\right )}+\frac {16 a^2 c^2+2 b c \left (b^2-7 a c\right ) (d+e x)^2-15 a b^2 c+2 b^4}{a \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )}}{2 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c (d+e x)^2}{2 a \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}}{2 e f}\)

Input:

Int[1/((d*f + e*f*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3),x]
 

Output:

((b^2 - 2*a*c + b*c*(d + e*x)^2)/(2*a*(b^2 - 4*a*c)*(a + b*(d + e*x)^2 + c 
*(d + e*x)^4)^2) + ((2*b^4 - 15*a*b^2*c + 16*a^2*c^2 + 2*b*c*(b^2 - 7*a*c) 
*(d + e*x)^2)/(a*(b^2 - 4*a*c)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) + (2*( 
(b*(b^4 - 10*a*b^2*c + 30*a^2*c^2)*ArcTanh[(b + 2*c*(d + e*x)^2)/Sqrt[b^2 
- 4*a*c]])/(a*Sqrt[b^2 - 4*a*c]) + ((b^2 - 4*a*c)^2*Log[(d + e*x)^2])/a - 
((b^2 - 4*a*c)^2*Log[a + b*(d + e*x)^2 + c*(d + e*x)^4])/(2*a)))/(a*(b^2 - 
 4*a*c)))/(2*a*(b^2 - 4*a*c)))/(2*e*f)
 

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 1165

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

rule 1200

Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* 
(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* 
x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In 
tegersQ[n]
 

rule 1235

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 
*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a 
+ b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] 
 + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^m 
*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 
 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* 
m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - 
f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] 
)
 

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]
 

rule 2009

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

Maple [C] (verified)

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

Time = 0.38 (sec) , antiderivative size = 970, normalized size of antiderivative = 3.59

method	result	size
default	\(\frac {-\frac {\frac {\frac {c^{2} e^{5} \left (7 a c -b^{2}\right ) a b \,x^{6}}{32 a^{2} c^{2}-16 a \,b^{2} c +2 b^{4}}+\frac {3 \left (7 a c -b^{2}\right ) a b \,c^{2} d \,e^{4} x^{5}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {e^{3} a c \left (-210 a b \,c^{2} d^{2}+30 b^{3} c \,d^{2}+16 a^{2} c^{2}-29 a \,b^{2} c +4 b^{4}\right ) x^{4}}{4 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {c d \,e^{2} a \left (-70 a b \,c^{2} d^{2}+10 b^{3} c \,d^{2}+16 a^{2} c^{2}-29 a \,b^{2} c +4 b^{4}\right ) x^{3}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {e a \left (105 a b \,c^{3} d^{4}-15 b^{3} c^{2} d^{4}-48 a^{2} c^{3} d^{2}+87 a \,b^{2} c^{2} d^{2}-12 b^{4} c \,d^{2}+a^{2} b \,c^{2}+6 a \,b^{3} c -b^{5}\right ) x^{2}}{32 a^{2} c^{2}-16 a \,b^{2} c +2 b^{4}}+\frac {d a \left (21 a b \,c^{3} d^{4}-3 b^{3} c^{2} d^{4}-16 a^{2} c^{3} d^{2}+29 a \,b^{2} c^{2} d^{2}-4 b^{4} c \,d^{2}+a^{2} b \,c^{2}+6 a \,b^{3} c -b^{5}\right ) x}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {a \left (-14 a b \,c^{3} d^{6}+2 b^{3} c^{2} d^{6}+16 a^{2} c^{3} d^{4}-29 a \,b^{2} c^{2} d^{4}+4 b^{4} c \,d^{4}-2 a^{2} b \,c^{2} d^{2}-12 a \,b^{3} c \,d^{2}+2 b^{5} d^{2}+24 c^{2} a^{3}-21 c \,a^{2} b^{2}+3 a \,b^{4}\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 {\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 (c \,e^{3} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \textit {\_R}^{3}+3 c d \,e^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \textit {\_R}^{2}+e \left (48 a^{2} c^{3} d^{2}-24 a \,b^{2} c^{2} d^{2}+3 b^{4} c \,d^{2}+23 a^{2} b \,c^{2}-9 a \,b^{3} c +b^{5}\right ) \textit {\_R} +16 a^{2} c^{3} d^{3}-8 a \,b^{2} c^{2} d^{3}+b^{4} c \,d^{3}+23 a^{2} b \,c^{2} d -9 a \,b^{3} c d +b^{5} 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}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) e}}{a^{3}}+\frac {\ln \left (e x +d \right )}{a^{3} e}}{f}\)	\(970\)
risch	\(\text {Expression too large to display}\)	\(1710\)

Input:

int(1/(e*f*x+d*f)/(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x,method=_RETURNVERBOSE)
 

Output:

1/f*(-1/a^3*((1/2*c^2*e^5*(7*a*c-b^2)*a*b/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6+3 
*(7*a*c-b^2)*a*b*c^2*d*e^4/(16*a^2*c^2-8*a*b^2*c+b^4)*x^5-1/4*e^3*a*c*(-21 
0*a*b*c^2*d^2+30*b^3*c*d^2+16*a^2*c^2-29*a*b^2*c+4*b^4)/(16*a^2*c^2-8*a*b^ 
2*c+b^4)*x^4-c*d*e^2*a*(-70*a*b*c^2*d^2+10*b^3*c*d^2+16*a^2*c^2-29*a*b^2*c 
+4*b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3+1/2*e*a*(105*a*b*c^3*d^4-15*b^3*c^2 
*d^4-48*a^2*c^3*d^2+87*a*b^2*c^2*d^2-12*b^4*c*d^2+a^2*b*c^2+6*a*b^3*c-b^5) 
/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2+d*a*(21*a*b*c^3*d^4-3*b^3*c^2*d^4-16*a^2*c 
^3*d^2+29*a*b^2*c^2*d^2-4*b^4*c*d^2+a^2*b*c^2+6*a*b^3*c-b^5)/(16*a^2*c^2-8 
*a*b^2*c+b^4)*x-1/4/e*a*(-14*a*b*c^3*d^6+2*b^3*c^2*d^6+16*a^2*c^3*d^4-29*a 
*b^2*c^2*d^4+4*b^4*c*d^4-2*a^2*b*c^2*d^2-12*a*b^3*c*d^2+2*b^5*d^2+24*a^3*c 
^2-21*a^2*b^2*c+3*a*b^4)/(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+1/2/( 
16*a^2*c^2-8*a*b^2*c+b^4)/e*sum((c*e^3*(16*a^2*c^2-8*a*b^2*c+b^4)*_R^3+3*c 
*d*e^2*(16*a^2*c^2-8*a*b^2*c+b^4)*_R^2+e*(48*a^2*c^3*d^2-24*a*b^2*c^2*d^2+ 
3*b^4*c*d^2+23*a^2*b*c^2-9*a*b^3*c+b^5)*_R+16*a^2*c^3*d^3-8*a*b^2*c^2*d^3+ 
b^4*c*d^3+23*a^2*b*c^2*d-9*a*b^3*c*d+b^5*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)))+1/a^3/ 
e*ln(e*x+d))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4898 vs. \(2 (258) = 516\).

Time = 1.75 (sec) , antiderivative size = 9926, normalized size of antiderivative = 36.76 \[ \int \frac {1}{(d f+e f x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(1/(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 [F(-1)]

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

integrate(1/(e*f*x+d*f)/(a+b*(e*x+d)**2+c*(e*x+d)**4)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{(d f+e f x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\int { \frac {1}{{\left ({\left (e x + d\right )}^{4} c + {\left (e x + d\right )}^{2} b + a\right )}^{3} {\left (e f x + d f\right )}} \,d x } \] Input:

integrate(1/(e*f*x+d*f)/(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x, algorithm="maxima 
")
 

Output:

1/4*(2*(b^3*c^2 - 7*a*b*c^3)*e^6*x^6 + 12*(b^3*c^2 - 7*a*b*c^3)*d*e^5*x^5 
+ (4*b^4*c - 29*a*b^2*c^2 + 16*a^2*c^3 + 30*(b^3*c^2 - 7*a*b*c^3)*d^2)*e^4 
*x^4 + 2*(b^3*c^2 - 7*a*b*c^3)*d^6 + 4*(10*(b^3*c^2 - 7*a*b*c^3)*d^3 + (4* 
b^4*c - 29*a*b^2*c^2 + 16*a^2*c^3)*d)*e^3*x^3 + 3*a*b^4 - 21*a^2*b^2*c + 2 
4*a^3*c^2 + (4*b^4*c - 29*a*b^2*c^2 + 16*a^2*c^3)*d^4 + 2*(b^5 - 6*a*b^3*c 
 - a^2*b*c^2 + 15*(b^3*c^2 - 7*a*b*c^3)*d^4 + 3*(4*b^4*c - 29*a*b^2*c^2 + 
16*a^2*c^3)*d^2)*e^2*x^2 + 2*(b^5 - 6*a*b^3*c - a^2*b*c^2)*d^2 + 4*(3*(b^3 
*c^2 - 7*a*b*c^3)*d^5 + (4*b^4*c - 29*a*b^2*c^2 + 16*a^2*c^3)*d^3 + (b^5 - 
 6*a*b^3*c - a^2*b*c^2)*d)*e*x)/((a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4 
)*e^9*f*x^8 + 8*(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*d*e^8*f*x^7 + 2 
*(a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3 + 14*(a^2*b^4*c^2 - 8*a^3*b^2*c 
^3 + 16*a^4*c^4)*d^2)*e^7*f*x^6 + 4*(14*(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16* 
a^4*c^4)*d^3 + 3*(a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*d)*e^6*f*x^5 + 
 (a^2*b^6 - 6*a^3*b^4*c + 32*a^5*c^3 + 70*(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 1 
6*a^4*c^4)*d^4 + 30*(a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*d^2)*e^5*f* 
x^4 + 4*(14*(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*d^5 + 10*(a^2*b^5*c 
 - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*d^3 + (a^2*b^6 - 6*a^3*b^4*c + 32*a^5*c^3 
)*d)*e^4*f*x^3 + 2*(a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2 + 14*(a^2*b^4*c^2 
 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*d^6 + 15*(a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^ 
4*b*c^3)*d^4 + 3*(a^2*b^6 - 6*a^3*b^4*c + 32*a^5*c^3)*d^2)*e^3*f*x^2 + ...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1077 vs. \(2 (258) = 516\).

Time = 0.22 (sec) , antiderivative size = 1077, normalized size of antiderivative = 3.99 \[ \int \frac {1}{(d f+e f x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx =\text {Too large to display} \] Input:

integrate(1/(e*f*x+d*f)/(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x, algorithm="giac")
 

Output:

-1/4*((a^3*b^7*c*e^3*f - 14*a^4*b^5*c^2*e^3*f + 70*a^5*b^3*c^3*e^3*f - 120 
*a^6*b*c^4*e^3*f)*sqrt(b^2 - 4*a*c)*log(abs(b*e^2*x^2 + sqrt(b^2 - 4*a*c)* 
e^2*x^2 + 2*b*d*e*x + 2*sqrt(b^2 - 4*a*c)*d*e*x + b*d^2 + sqrt(b^2 - 4*a*c 
)*d^2 + 2*a)) - (a^3*b^7*c*e^3*f - 14*a^4*b^5*c^2*e^3*f + 70*a^5*b^3*c^3*e 
^3*f - 120*a^6*b*c^4*e^3*f)*sqrt(b^2 - 4*a*c)*log(abs(-b*e^2*x^2 + sqrt(b^ 
2 - 4*a*c)*e^2*x^2 - 2*b*d*e*x + 2*sqrt(b^2 - 4*a*c)*d*e*x - b*d^2 + sqrt( 
b^2 - 4*a*c)*d^2 - 2*a)))/(a^6*b^8*c*e^4*f^2 - 16*a^7*b^6*c^2*e^4*f^2 + 96 
*a^8*b^4*c^3*e^4*f^2 - 256*a^9*b^2*c^4*e^4*f^2 + 256*a^10*c^5*e^4*f^2) - 1 
/4*log(abs(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 + c*d 
^4 + b*e^2*x^2 + 2*b*d*e*x + b*d^2 + a))/(a^3*e*f) + log(abs(e*x + d))/(a^ 
3*e*f) + 1/4*(2*a*b^3*c^2*d^6 - 14*a^2*b*c^3*d^6 + 4*a*b^4*c*d^4 - 29*a^2* 
b^2*c^2*d^4 + 16*a^3*c^3*d^4 + 2*a*b^5*d^2 - 12*a^2*b^3*c*d^2 - 2*a^3*b*c^ 
2*d^2 + 2*(a*b^3*c^2*e^6 - 7*a^2*b*c^3*e^6)*x^6 + 3*a^2*b^4 - 21*a^3*b^2*c 
 + 24*a^4*c^2 + 12*(a*b^3*c^2*d*e^5 - 7*a^2*b*c^3*d*e^5)*x^5 + (30*a*b^3*c 
^2*d^2*e^4 - 210*a^2*b*c^3*d^2*e^4 + 4*a*b^4*c*e^4 - 29*a^2*b^2*c^2*e^4 + 
16*a^3*c^3*e^4)*x^4 + 4*(10*a*b^3*c^2*d^3*e^3 - 70*a^2*b*c^3*d^3*e^3 + 4*a 
*b^4*c*d*e^3 - 29*a^2*b^2*c^2*d*e^3 + 16*a^3*c^3*d*e^3)*x^3 + 2*(15*a*b^3* 
c^2*d^4*e^2 - 105*a^2*b*c^3*d^4*e^2 + 12*a*b^4*c*d^2*e^2 - 87*a^2*b^2*c^2* 
d^2*e^2 + 48*a^3*c^3*d^2*e^2 + a*b^5*e^2 - 6*a^2*b^3*c*e^2 - a^3*b*c^2*e^2 
)*x^2 + 4*(3*a*b^3*c^2*d^5*e - 21*a^2*b*c^3*d^5*e + 4*a*b^4*c*d^3*e - 2...
 

Mupad [B] (verification not implemented)

Time = 23.77 (sec) , antiderivative size = 22621, normalized size of antiderivative = 83.78 \[ \int \frac {1}{(d f+e f x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\text {Too large to display} \] Input:

int(1/((d*f + e*f*x)*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^3),x)
 

Output:

((x^2*(b^5*e + 48*a^2*c^3*d^2*e + 15*b^3*c^2*d^4*e - 6*a*b^3*c*e - a^2*b*c 
^2*e + 12*b^4*c*d^2*e - 105*a*b*c^3*d^4*e - 87*a*b^2*c^2*d^2*e))/(2*(a^2*b 
^4 + 16*a^4*c^2 - 8*a^3*b^2*c)) + (x^4*(4*b^4*c*e^3 + 16*a^2*c^3*e^3 - 29* 
a*b^2*c^2*e^3 + 30*b^3*c^2*d^2*e^3 - 210*a*b*c^3*d^2*e^3))/(4*(a^2*b^4 + 1 
6*a^4*c^2 - 8*a^3*b^2*c)) + (x^3*(16*a^2*c^3*d*e^2 + 10*b^3*c^2*d^3*e^2 + 
4*b^4*c*d*e^2 - 29*a*b^2*c^2*d*e^2 - 70*a*b*c^3*d^3*e^2))/(a^2*b^4 + 16*a^ 
4*c^2 - 8*a^3*b^2*c) + (3*x^5*(b^3*c^2*d*e^4 - 7*a*b*c^3*d*e^4))/(a^2*b^4 
+ 16*a^4*c^2 - 8*a^3*b^2*c) + (x^6*(b^3*c^2*e^5 - 7*a*b*c^3*e^5))/(2*(a^2* 
b^4 + 16*a^4*c^2 - 8*a^3*b^2*c)) + (x*(b^5*d + 4*b^4*c*d^3 + 16*a^2*c^3*d^ 
3 + 3*b^3*c^2*d^5 - 29*a*b^2*c^2*d^3 - 6*a*b^3*c*d - a^2*b*c^2*d - 21*a*b* 
c^3*d^5))/(a^2*b^4 + 16*a^4*c^2 - 8*a^3*b^2*c) + (3*a*b^4 + 24*a^3*c^2 + 2 
*b^5*d^2 - 21*a^2*b^2*c + 4*b^4*c*d^4 + 16*a^2*c^3*d^4 + 2*b^3*c^2*d^6 - 2 
*a^2*b*c^2*d^2 - 29*a*b^2*c^2*d^4 - 12*a*b^3*c*d^2 - 14*a*b*c^3*d^6)/(4*e* 
(a^2*b^4 + 16*a^4*c^2 - 8*a^3*b^2*c)))/(x^3*(56*c^2*d^5*e^3*f + 4*b^2*d*e^ 
3*f + 40*b*c*d^3*e^3*f + 8*a*c*d*e^3*f) + x^2*(6*b^2*d^2*e^2*f + 28*c^2*d^ 
6*e^2*f + 2*a*b*e^2*f + 12*a*c*d^2*e^2*f + 30*b*c*d^4*e^2*f) + x*(4*b^2*d^ 
3*e*f + 8*c^2*d^7*e*f + 4*a*b*d*e*f + 8*a*c*d^3*e*f + 12*b*c*d^5*e*f) + x^ 
4*(b^2*e^4*f + 70*c^2*d^4*e^4*f + 2*a*c*e^4*f + 30*b*c*d^2*e^4*f) + x^5*(5 
6*c^2*d^3*e^5*f + 12*b*c*d*e^5*f) + a^2*f + x^6*(28*c^2*d^2*e^6*f + 2*b*c* 
e^6*f) + b^2*d^4*f + c^2*d^8*f + c^2*e^8*f*x^8 + 2*a*b*d^2*f + 2*a*c*d^...
 

Reduce [F]

\[ \int \frac {1}{(d f+e f x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx =\text {Too large to display} \] Input:

int(1/(e*f*x+d*f)/(a+b*(e*x+d)^2+c*(e*x+d)^4)^3,x)
 

Output:

int(1/(a**3*d + a**3*e*x + 3*a**2*b*d**3 + 9*a**2*b*d**2*e*x + 9*a**2*b*d* 
e**2*x**2 + 3*a**2*b*e**3*x**3 + 3*a**2*c*d**5 + 15*a**2*c*d**4*e*x + 30*a 
**2*c*d**3*e**2*x**2 + 30*a**2*c*d**2*e**3*x**3 + 15*a**2*c*d*e**4*x**4 + 
3*a**2*c*e**5*x**5 + 3*a*b**2*d**5 + 15*a*b**2*d**4*e*x + 30*a*b**2*d**3*e 
**2*x**2 + 30*a*b**2*d**2*e**3*x**3 + 15*a*b**2*d*e**4*x**4 + 3*a*b**2*e** 
5*x**5 + 6*a*b*c*d**7 + 42*a*b*c*d**6*e*x + 126*a*b*c*d**5*e**2*x**2 + 210 
*a*b*c*d**4*e**3*x**3 + 210*a*b*c*d**3*e**4*x**4 + 126*a*b*c*d**2*e**5*x** 
5 + 42*a*b*c*d*e**6*x**6 + 6*a*b*c*e**7*x**7 + 3*a*c**2*d**9 + 27*a*c**2*d 
**8*e*x + 108*a*c**2*d**7*e**2*x**2 + 252*a*c**2*d**6*e**3*x**3 + 378*a*c* 
*2*d**5*e**4*x**4 + 378*a*c**2*d**4*e**5*x**5 + 252*a*c**2*d**3*e**6*x**6 
+ 108*a*c**2*d**2*e**7*x**7 + 27*a*c**2*d*e**8*x**8 + 3*a*c**2*e**9*x**9 + 
 b**3*d**7 + 7*b**3*d**6*e*x + 21*b**3*d**5*e**2*x**2 + 35*b**3*d**4*e**3* 
x**3 + 35*b**3*d**3*e**4*x**4 + 21*b**3*d**2*e**5*x**5 + 7*b**3*d*e**6*x** 
6 + b**3*e**7*x**7 + 3*b**2*c*d**9 + 27*b**2*c*d**8*e*x + 108*b**2*c*d**7* 
e**2*x**2 + 252*b**2*c*d**6*e**3*x**3 + 378*b**2*c*d**5*e**4*x**4 + 378*b* 
*2*c*d**4*e**5*x**5 + 252*b**2*c*d**3*e**6*x**6 + 108*b**2*c*d**2*e**7*x** 
7 + 27*b**2*c*d*e**8*x**8 + 3*b**2*c*e**9*x**9 + 3*b*c**2*d**11 + 33*b*c** 
2*d**10*e*x + 165*b*c**2*d**9*e**2*x**2 + 495*b*c**2*d**8*e**3*x**3 + 990* 
b*c**2*d**7*e**4*x**4 + 1386*b*c**2*d**6*e**5*x**5 + 1386*b*c**2*d**5*e**6 
*x**6 + 990*b*c**2*d**4*e**7*x**7 + 495*b*c**2*d**3*e**8*x**8 + 165*b*c...