\(\int \frac {1}{(d f+e f x)^2 (a+b (d+e x)^2+c (d+e x)^4)^3} \, dx\) [288]

Optimal result

Integrand size = 33, antiderivative size = 499 \[ \int \frac {1}{(d f+e f x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=-\frac {3 \left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )}{8 a^3 \left (b^2-4 a c\right )^2 e f^2 (d+e x)}+\frac {b^2-2 a c+b c (d+e x)^2}{4 a \left (b^2-4 a c\right ) e f^2 (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {5 b^4-35 a b^2 c+36 a^2 c^2+b c \left (5 b^2-32 a c\right ) (d+e x)^2}{8 a^2 \left (b^2-4 a c\right )^2 e f^2 (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {3 \sqrt {c} \left (\left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )+\frac {b \left (5 b^4-47 a b^2 c+124 a^2 c^2\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a^3 \left (b^2-4 a c\right )^2 \sqrt {b-\sqrt {b^2-4 a c}} e f^2}-\frac {3 \sqrt {c} \left (\left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )-\frac {5 b^5-47 a b^3 c+124 a^2 b c^2}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a^3 \left (b^2-4 a c\right )^2 \sqrt {b+\sqrt {b^2-4 a c}} e f^2} \] Output:

-3/8*(-12*a*c+5*b^2)*(-5*a*c+b^2)/a^3/(-4*a*c+b^2)^2/e/f^2/(e*x+d)+1/4*(b^ 
2-2*a*c+b*c*(e*x+d)^2)/a/(-4*a*c+b^2)/e/f^2/(e*x+d)/(a+b*(e*x+d)^2+c*(e*x+ 
d)^4)^2+1/8*(5*b^4-35*a*b^2*c+36*c^2*a^2+b*c*(-32*a*c+5*b^2)*(e*x+d)^2)/a^ 
2/(-4*a*c+b^2)^2/e/f^2/(e*x+d)/(a+b*(e*x+d)^2+c*(e*x+d)^4)-3/16*c^(1/2)*(( 
-12*a*c+5*b^2)*(-5*a*c+b^2)+b*(124*a^2*c^2-47*a*b^2*c+5*b^4)/(-4*a*c+b^2)^ 
(1/2))*arctan(2^(1/2)*c^(1/2)*(e*x+d)/(b-(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2 
)/a^3/(-4*a*c+b^2)^2/(b-(-4*a*c+b^2)^(1/2))^(1/2)/e/f^2-3/16*c^(1/2)*((-12 
*a*c+5*b^2)*(-5*a*c+b^2)-(124*a^2*b*c^2-47*a*b^3*c+5*b^5)/(-4*a*c+b^2)^(1/ 
2))*arctan(2^(1/2)*c^(1/2)*(e*x+d)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)/a 
^3/(-4*a*c+b^2)^2/(b+(-4*a*c+b^2)^(1/2))^(1/2)/e/f^2
 

Mathematica [A] (verified)

Time = 6.26 (sec) , antiderivative size = 575, normalized size of antiderivative = 1.15 \[ \int \frac {1}{(d f+e f x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=-\frac {1}{a^3 e f^2 (d+e x)}+\frac {b^3 (d+e x)-3 a b c (d+e x)+b^2 c (d+e x)^3-2 a c^2 (d+e x)^3}{4 a^2 \left (-b^2+4 a c\right ) e f^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {-7 b^5 (d+e x)+52 a b^3 c (d+e x)-84 a^2 b c^2 (d+e x)-7 b^4 c (d+e x)^3+47 a b^2 c^2 (d+e x)^3-52 a^2 c^3 (d+e x)^3}{8 a^3 \left (-b^2+4 a c\right )^2 e f^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {3 \sqrt {c} \left (5 b^5-47 a b^3 c+124 a^2 b c^2+5 b^4 \sqrt {b^2-4 a c}-37 a b^2 c \sqrt {b^2-4 a c}+60 a^2 c^2 \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}} e f^2}-\frac {3 \sqrt {c} \left (-5 b^5+47 a b^3 c-124 a^2 b c^2+5 b^4 \sqrt {b^2-4 a c}-37 a b^2 c \sqrt {b^2-4 a c}+60 a^2 c^2 \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt {b+\sqrt {b^2-4 a c}} e f^2} \] Input:

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

Output:

-(1/(a^3*e*f^2*(d + e*x))) + (b^3*(d + e*x) - 3*a*b*c*(d + e*x) + b^2*c*(d 
 + e*x)^3 - 2*a*c^2*(d + e*x)^3)/(4*a^2*(-b^2 + 4*a*c)*e*f^2*(a + b*(d + e 
*x)^2 + c*(d + e*x)^4)^2) + (-7*b^5*(d + e*x) + 52*a*b^3*c*(d + e*x) - 84* 
a^2*b*c^2*(d + e*x) - 7*b^4*c*(d + e*x)^3 + 47*a*b^2*c^2*(d + e*x)^3 - 52* 
a^2*c^3*(d + e*x)^3)/(8*a^3*(-b^2 + 4*a*c)^2*e*f^2*(a + b*(d + e*x)^2 + c* 
(d + e*x)^4)) - (3*Sqrt[c]*(5*b^5 - 47*a*b^3*c + 124*a^2*b*c^2 + 5*b^4*Sqr 
t[b^2 - 4*a*c] - 37*a*b^2*c*Sqrt[b^2 - 4*a*c] + 60*a^2*c^2*Sqrt[b^2 - 4*a* 
c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(8*Sq 
rt[2]*a^3*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]*e*f^2) - (3*Sqrt 
[c]*(-5*b^5 + 47*a*b^3*c - 124*a^2*b*c^2 + 5*b^4*Sqrt[b^2 - 4*a*c] - 37*a* 
b^2*c*Sqrt[b^2 - 4*a*c] + 60*a^2*c^2*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sq 
rt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^3*(b^2 - 4*a*c 
)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]*e*f^2)
 

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 474, normalized size of antiderivative = 0.95, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {1462, 1441, 25, 1600, 27, 1604, 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.

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

\(\Big \downarrow \) 1462

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

\(\Big \downarrow \) 1441

\(\displaystyle \frac {\frac {-2 a c+b^2+b c (d+e x)^2}{4 a \left (b^2-4 a c\right ) (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}-\frac {\int -\frac {5 b^2+7 c (d+e x)^2 b-18 a c}{(d+e x)^2 \left (c (d+e x)^4+b (d+e x)^2+a\right )^2}d(d+e x)}{4 a \left (b^2-4 a c\right )}}{e f^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {5 b^2+7 c (d+e x)^2 b-18 a c}{(d+e x)^2 \left (c (d+e x)^4+b (d+e x)^2+a\right )^2}d(d+e x)}{4 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c (d+e x)^2}{4 a \left (b^2-4 a c\right ) (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}}{e f^2}\)

\(\Big \downarrow \) 1600

\(\displaystyle \frac {\frac {\frac {36 a^2 c^2+b c \left (5 b^2-32 a c\right ) (d+e x)^2-35 a b^2 c+5 b^4}{2 a \left (b^2-4 a c\right ) (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {\int -\frac {3 \left (b c \left (5 b^2-32 a c\right ) (d+e x)^2+\left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )\right )}{(d+e x)^2 \left (c (d+e x)^4+b (d+e x)^2+a\right )}d(d+e x)}{2 a \left (b^2-4 a c\right )}}{4 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c (d+e x)^2}{4 a \left (b^2-4 a c\right ) (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}}{e f^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {3 \int \frac {b c \left (5 b^2-32 a c\right ) (d+e x)^2+\left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )}{(d+e x)^2 \left (c (d+e x)^4+b (d+e x)^2+a\right )}d(d+e x)}{2 a \left (b^2-4 a c\right )}+\frac {36 a^2 c^2+b c \left (5 b^2-32 a c\right ) (d+e x)^2-35 a b^2 c+5 b^4}{2 a \left (b^2-4 a c\right ) (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )}}{4 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c (d+e x)^2}{4 a \left (b^2-4 a c\right ) (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}}{e f^2}\)

\(\Big \downarrow \) 1604

\(\displaystyle \frac {\frac {\frac {3 \left (-\frac {\int \frac {c \left (5 b^2-12 a c\right ) \left (b^2-5 a c\right ) (d+e x)^2+b \left (5 b^4-42 a c b^2+92 a^2 c^2\right )}{c (d+e x)^4+b (d+e x)^2+a}d(d+e x)}{a}-\frac {\left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )}{a (d+e x)}\right )}{2 a \left (b^2-4 a c\right )}+\frac {36 a^2 c^2+b c \left (5 b^2-32 a c\right ) (d+e x)^2-35 a b^2 c+5 b^4}{2 a \left (b^2-4 a c\right ) (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )}}{4 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c (d+e x)^2}{4 a \left (b^2-4 a c\right ) (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}}{e f^2}\)

\(\Big \downarrow \) 1480

\(\displaystyle \frac {\frac {\frac {3 \left (-\frac {\frac {1}{2} c \left (\frac {b \left (124 a^2 c^2-47 a b^2 c+5 b^4\right )}{\sqrt {b^2-4 a c}}+\left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )\right ) \int \frac {1}{c (d+e x)^2+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}d(d+e x)+\frac {1}{2} c \left (\left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )-\frac {b \left (124 a^2 c^2-47 a b^2 c+5 b^4\right )}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{c (d+e x)^2+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}d(d+e x)}{a}-\frac {\left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )}{a (d+e x)}\right )}{2 a \left (b^2-4 a c\right )}+\frac {36 a^2 c^2+b c \left (5 b^2-32 a c\right ) (d+e x)^2-35 a b^2 c+5 b^4}{2 a \left (b^2-4 a c\right ) (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )}}{4 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c (d+e x)^2}{4 a \left (b^2-4 a c\right ) (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}}{e f^2}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\frac {3 \left (-\frac {\frac {\sqrt {c} \left (\frac {b \left (124 a^2 c^2-47 a b^2 c+5 b^4\right )}{\sqrt {b^2-4 a c}}+\left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (\left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )-\frac {b \left (124 a^2 c^2-47 a b^2 c+5 b^4\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b}}}{a}-\frac {\left (5 b^2-12 a c\right ) \left (b^2-5 a c\right )}{a (d+e x)}\right )}{2 a \left (b^2-4 a c\right )}+\frac {36 a^2 c^2+b c \left (5 b^2-32 a c\right ) (d+e x)^2-35 a b^2 c+5 b^4}{2 a \left (b^2-4 a c\right ) (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )}}{4 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c (d+e x)^2}{4 a \left (b^2-4 a c\right ) (d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}}{e f^2}\)

Input:

Int[1/((d*f + e*f*x)^2*(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)/(4*a*(b^2 - 4*a*c)*(d + e*x)*(a + b*(d + 
e*x)^2 + c*(d + e*x)^4)^2) + ((5*b^4 - 35*a*b^2*c + 36*a^2*c^2 + b*c*(5*b^ 
2 - 32*a*c)*(d + e*x)^2)/(2*a*(b^2 - 4*a*c)*(d + e*x)*(a + b*(d + e*x)^2 + 
 c*(d + e*x)^4)) + (3*(-(((5*b^2 - 12*a*c)*(b^2 - 5*a*c))/(a*(d + e*x))) - 
 ((Sqrt[c]*((5*b^2 - 12*a*c)*(b^2 - 5*a*c) + (b*(5*b^4 - 47*a*b^2*c + 124* 
a^2*c^2))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - S 
qrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[c]*((5*b 
^2 - 12*a*c)*(b^2 - 5*a*c) - (b*(5*b^4 - 47*a*b^2*c + 124*a^2*c^2))/Sqrt[b 
^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c] 
]])/(Sqrt[2]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/a))/(2*a*(b^2 - 4*a*c)))/(4*a*( 
b^2 - 4*a*c)))/(e*f^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 1441

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

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 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 1600

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*( 
x_)^4)^(p_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(a + b*x^2 + c*x^4)^(p + 1) 
*((d*(b^2 - 2*a*c) - a*b*e + (b*d - 2*a*e)*c*x^2)/(2*a*f*(p + 1)*(b^2 - 4*a 
*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c))   Int[(f*x)^m*(a + b*x^2 + c 
*x^4)^(p + 1)*Simp[d*(b^2*(m + 2*(p + 1) + 1) - 2*a*c*(m + 4*(p + 1) + 1)) 
- a*b*e*(m + 1) + c*(m + 2*(2*p + 3) + 1)*(b*d - 2*a*e)*x^2, x], x], x] /; 
FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && Int 
egerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
 

rule 1604

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

Maple [C] (verified)

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

Time = 0.40 (sec) , antiderivative size = 1201, normalized size of antiderivative = 2.41

method	result	size
default	\(\text {Expression too large to display}\)	\(1201\)
risch	\(\text {Expression too large to display}\)	\(2710\)

Input:

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

Output:

1/f^2*(-1/a^3*((1/8*c^2*e^6*(52*a^2*c^2-47*a*b^2*c+7*b^4)/(16*a^2*c^2-8*a* 
b^2*c+b^4)*x^7+7/8*c^2*d*e^5*(52*a^2*c^2-47*a*b^2*c+7*b^4)/(16*a^2*c^2-8*a 
*b^2*c+b^4)*x^6+1/8*(1092*a^2*c^3*d^2-987*a*b^2*c^2*d^2+147*b^4*c*d^2+136* 
a^2*b*c^2-99*a*b^3*c+14*b^5)*c*e^4/(16*a^2*c^2-8*a*b^2*c+b^4)*x^5+5/8*c*d* 
e^3*(364*a^2*c^3*d^2-329*a*b^2*c^2*d^2+49*b^4*c*d^2+136*a^2*b*c^2-99*a*b^3 
*c+14*b^5)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4+1/8*e^2*(1820*a^2*c^4*d^4-1645*a 
*b^2*c^3*d^4+245*b^4*c^2*d^4+1360*a^2*b*c^3*d^2-990*a*b^3*c^2*d^2+140*b^5* 
c*d^2+68*a^3*c^3+25*a^2*b^2*c^2-43*a*b^4*c+7*b^6)/(16*a^2*c^2-8*a*b^2*c+b^ 
4)*x^3+1/8*d*e*(1092*a^2*c^4*d^4-987*a*b^2*c^3*d^4+147*b^4*c^2*d^4+1360*a^ 
2*b*c^3*d^2-990*a*b^3*c^2*d^2+140*b^5*c*d^2+204*a^3*c^3+75*a^2*b^2*c^2-129 
*a*b^4*c+21*b^6)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2+1/8*(364*a^2*c^4*d^6-329*a 
*b^2*c^3*d^6+49*b^4*c^2*d^6+680*a^2*b*c^3*d^4-495*a*b^3*c^2*d^4+70*b^5*c*d 
^4+204*a^3*c^3*d^2+75*a^2*b^2*c^2*d^2-129*a*b^4*c*d^2+21*b^6*d^2+108*a^3*b 
*c^2-66*a^2*b^3*c+9*a*b^5)/(16*a^2*c^2-8*a*b^2*c+b^4)*x+1/8*d/e*(52*a^2*c^ 
4*d^6-47*a*b^2*c^3*d^6+7*b^4*c^2*d^6+136*a^2*b*c^3*d^4-99*a*b^3*c^2*d^4+14 
*b^5*c*d^4+68*a^3*c^3*d^2+25*a^2*b^2*c^2*d^2-43*a*b^4*c*d^2+7*b^6*d^2+108* 
a^3*b*c^2-66*a^2*b^3*c+9*a*b^5)/(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+3/16/(16*a^2*c^2-8*a*b^2*c+b^4)/e*sum((c*e^2*(60*a^2*c^2-37*a*b^2*c+5*b^ 
4)*_R^2+2*c*d*e*(60*a^2*c^2-37*a*b^2*c+5*b^4)*_R+60*a^2*c^3*d^2-37*a*b^...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 10518 vs. \(2 (453) = 906\).

Time = 1.36 (sec) , antiderivative size = 10518, normalized size of antiderivative = 21.08 \[ \int \frac {1}{(d f+e f x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-1/8*(3*(5*b^4*c^2 - 37*a*b^2*c^3 + 60*a^2*c^4)*e^8*x^8 + 24*(5*b^4*c^2 - 
37*a*b^2*c^3 + 60*a^2*c^4)*d*e^7*x^7 + (30*b^5*c - 227*a*b^3*c^2 + 392*a^2 
*b*c^3 + 84*(5*b^4*c^2 - 37*a*b^2*c^3 + 60*a^2*c^4)*d^2)*e^6*x^6 + 6*(28*( 
5*b^4*c^2 - 37*a*b^2*c^3 + 60*a^2*c^4)*d^3 + (30*b^5*c - 227*a*b^3*c^2 + 3 
92*a^2*b*c^3)*d)*e^5*x^5 + 3*(5*b^4*c^2 - 37*a*b^2*c^3 + 60*a^2*c^4)*d^8 + 
 (15*b^6 - 91*a*b^4*c + 25*a^2*b^2*c^2 + 324*a^3*c^3 + 210*(5*b^4*c^2 - 37 
*a*b^2*c^3 + 60*a^2*c^4)*d^4 + 15*(30*b^5*c - 227*a*b^3*c^2 + 392*a^2*b*c^ 
3)*d^2)*e^4*x^4 + (30*b^5*c - 227*a*b^3*c^2 + 392*a^2*b*c^3)*d^6 + 4*(42*( 
5*b^4*c^2 - 37*a*b^2*c^3 + 60*a^2*c^4)*d^5 + 5*(30*b^5*c - 227*a*b^3*c^2 + 
 392*a^2*b*c^3)*d^3 + (15*b^6 - 91*a*b^4*c + 25*a^2*b^2*c^2 + 324*a^3*c^3) 
*d)*e^3*x^3 + 8*a^2*b^4 - 64*a^3*b^2*c + 128*a^4*c^2 + (15*b^6 - 91*a*b^4* 
c + 25*a^2*b^2*c^2 + 324*a^3*c^3)*d^4 + (84*(5*b^4*c^2 - 37*a*b^2*c^3 + 60 
*a^2*c^4)*d^6 + 25*a*b^5 - 194*a^2*b^3*c + 364*a^3*b*c^2 + 15*(30*b^5*c - 
227*a*b^3*c^2 + 392*a^2*b*c^3)*d^4 + 6*(15*b^6 - 91*a*b^4*c + 25*a^2*b^2*c 
^2 + 324*a^3*c^3)*d^2)*e^2*x^2 + (25*a*b^5 - 194*a^2*b^3*c + 364*a^3*b*c^2 
)*d^2 + 2*(12*(5*b^4*c^2 - 37*a*b^2*c^3 + 60*a^2*c^4)*d^7 + 3*(30*b^5*c - 
227*a*b^3*c^2 + 392*a^2*b*c^3)*d^5 + 2*(15*b^6 - 91*a*b^4*c + 25*a^2*b^2*c 
^2 + 324*a^3*c^3)*d^3 + (25*a*b^5 - 194*a^2*b^3*c + 364*a^3*b*c^2)*d)*e*x) 
/((a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*e^10*f^2*x^9 + 9*(a^3*b^4*c^2 
 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*d*e^9*f^2*x^8 + 2*(a^3*b^5*c - 8*a^4*b^3...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1704 vs. \(2 (453) = 906\).

Time = 0.18 (sec) , antiderivative size = 1704, normalized size of antiderivative = 3.41 \[ \int \frac {1}{(d f+e f x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/8*(7*b^4*c^2/((e*f*x + d*f)*e*f) - 47*a*b^2*c^3/((e*f*x + d*f)*e*f) + 5 
2*a^2*c^4/((e*f*x + d*f)*e*f) + 14*b^5*c*f/((e*f*x + d*f)^3*e) - 99*a*b^3* 
c^2*f/((e*f*x + d*f)^3*e) + 136*a^2*b*c^3*f/((e*f*x + d*f)^3*e) + 7*b^6*f^ 
3/((e*f*x + d*f)^5*e) - 43*a*b^4*c*f^3/((e*f*x + d*f)^5*e) + 25*a^2*b^2*c^ 
2*f^3/((e*f*x + d*f)^5*e) + 68*a^3*c^3*f^3/((e*f*x + d*f)^5*e) + 9*a*b^5*f 
^5/((e*f*x + d*f)^7*e) - 66*a^2*b^3*c*f^5/((e*f*x + d*f)^7*e) + 108*a^3*b* 
c^2*f^5/((e*f*x + d*f)^7*e))/((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*(c + b* 
f^2/(e*f*x + d*f)^2 + a*f^4/(e*f*x + d*f)^4)^2) - 1/((e*f*x + d*f)*a^3*e*f 
) + 3/64*((5*a^6*b^13 - 112*a^7*b^11*c + 1030*a^8*b^9*c^2 - 4928*a^9*b^7*c 
^3 + 12736*a^10*b^5*c^4 - 16384*a^11*b^3*c^5 + 7680*a^12*b*c^6)*sqrt(2*a*b 
 + 2*sqrt(b^2 - 4*a*c)*a)*e^4*f^8 + 2*(5*a^4*b^6*c - 57*a^5*b^4*c^2 + 208* 
a^6*b^2*c^3 - 240*a^7*c^4)*sqrt(2*a*b + 2*sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 
4*a*c)*e^2*f^4*abs(a^3*b^4*e^2*f^4 - 8*a^4*b^2*c*e^2*f^4 + 16*a^5*c^2*e^2* 
f^4) - (a^3*b^4*e^2*f^4 - 8*a^4*b^2*c*e^2*f^4 + 16*a^5*c^2*e^2*f^4)^2*(5*b 
^5 - 42*a*b^3*c + 92*a^2*b*c^2)*sqrt(2*a*b + 2*sqrt(b^2 - 4*a*c)*a))*arcta 
n(2*sqrt(1/2)/((e*f*x + d*f)*e*f*sqrt((a^3*b^5*e^2*f^4 - 8*a^4*b^3*c*e^2*f 
^4 + 16*a^5*b*c^2*e^2*f^4 + sqrt((a^3*b^5*e^2*f^4 - 8*a^4*b^3*c*e^2*f^4 + 
16*a^5*b*c^2*e^2*f^4)^2 - 4*(a^4*b^4*e^4*f^8 - 8*a^5*b^2*c*e^4*f^8 + 16*a^ 
6*c^2*e^4*f^8)*(a^3*b^4*c - 8*a^4*b^2*c^2 + 16*a^5*c^3)))/(a^4*b^4*e^4*f^8 
 - 8*a^5*b^2*c*e^4*f^8 + 16*a^6*c^2*e^4*f^8))))/((a^7*b^6*c - 12*a^8*b^...
 

Mupad [B] (verification not implemented)

Time = 20.64 (sec) , antiderivative size = 20580, normalized size of antiderivative = 41.24 \[ \int \frac {1}{(d f+e f x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

- atan(((-(9*(25*b^21 - 25*b^6*(-(4*a*c - b^2)^15)^(1/2) + 18923520*a^10*b 
*c^10 + 17794*a^2*b^17*c^2 - 188095*a^3*b^15*c^3 + 1299860*a^4*b^13*c^4 - 
6126640*a^5*b^11*c^5 + 19905600*a^6*b^9*c^6 - 43904256*a^7*b^7*c^7 + 62684 
160*a^8*b^5*c^8 - 52039680*a^9*b^3*c^9 + 225*a^3*c^3*(-(4*a*c - b^2)^15)^( 
1/2) - 995*a*b^19*c - 694*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) + 245*a*b^ 
4*c*(-(4*a*c - b^2)^15)^(1/2)))/(512*(a^7*b^20*e^2*f^4 + 1048576*a^17*c^10 
*e^2*f^4 + 720*a^9*b^16*c^2*e^2*f^4 - 7680*a^10*b^14*c^3*e^2*f^4 + 53760*a 
^11*b^12*c^4*e^2*f^4 - 258048*a^12*b^10*c^5*e^2*f^4 + 860160*a^13*b^8*c^6* 
e^2*f^4 - 1966080*a^14*b^6*c^7*e^2*f^4 + 2949120*a^15*b^4*c^8*e^2*f^4 - 26 
21440*a^16*b^2*c^9*e^2*f^4 - 40*a^8*b^18*c*e^2*f^4)))^(1/2)*(x*(2717908992 
00*a^20*c^14*e^12*f^6 - 230400*a^9*b^22*c^3*e^12*f^6 + 9861120*a^10*b^20*c 
^4*e^12*f^6 - 191038464*a^11*b^18*c^5*e^12*f^6 + 2207803392*a^12*b^16*c^6* 
e^12*f^6 - 16878108672*a^13*b^14*c^7*e^12*f^6 + 89374851072*a^14*b^12*c^8* 
e^12*f^6 - 333226967040*a^15*b^10*c^9*e^12*f^6 + 869815812096*a^16*b^8*c^1 
0*e^12*f^6 - 1543847804928*a^17*b^6*c^11*e^12*f^6 + 1747313491968*a^18*b^4 
*c^12*e^12*f^6 - 1101055131648*a^19*b^2*c^13*e^12*f^6) - (-(9*(25*b^21 - 2 
5*b^6*(-(4*a*c - b^2)^15)^(1/2) + 18923520*a^10*b*c^10 + 17794*a^2*b^17*c^ 
2 - 188095*a^3*b^15*c^3 + 1299860*a^4*b^13*c^4 - 6126640*a^5*b^11*c^5 + 19 
905600*a^6*b^9*c^6 - 43904256*a^7*b^7*c^7 + 62684160*a^8*b^5*c^8 - 5203968 
0*a^9*b^3*c^9 + 225*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2) - 995*a*b^19*c - ...
 

Reduce [F]

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

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

Output:

int(1/(a**3*d**2 + 2*a**3*d*e*x + a**3*e**2*x**2 + 3*a**2*b*d**4 + 12*a**2 
*b*d**3*e*x + 18*a**2*b*d**2*e**2*x**2 + 12*a**2*b*d*e**3*x**3 + 3*a**2*b* 
e**4*x**4 + 3*a**2*c*d**6 + 18*a**2*c*d**5*e*x + 45*a**2*c*d**4*e**2*x**2 
+ 60*a**2*c*d**3*e**3*x**3 + 45*a**2*c*d**2*e**4*x**4 + 18*a**2*c*d*e**5*x 
**5 + 3*a**2*c*e**6*x**6 + 3*a*b**2*d**6 + 18*a*b**2*d**5*e*x + 45*a*b**2* 
d**4*e**2*x**2 + 60*a*b**2*d**3*e**3*x**3 + 45*a*b**2*d**2*e**4*x**4 + 18* 
a*b**2*d*e**5*x**5 + 3*a*b**2*e**6*x**6 + 6*a*b*c*d**8 + 48*a*b*c*d**7*e*x 
 + 168*a*b*c*d**6*e**2*x**2 + 336*a*b*c*d**5*e**3*x**3 + 420*a*b*c*d**4*e* 
*4*x**4 + 336*a*b*c*d**3*e**5*x**5 + 168*a*b*c*d**2*e**6*x**6 + 48*a*b*c*d 
*e**7*x**7 + 6*a*b*c*e**8*x**8 + 3*a*c**2*d**10 + 30*a*c**2*d**9*e*x + 135 
*a*c**2*d**8*e**2*x**2 + 360*a*c**2*d**7*e**3*x**3 + 630*a*c**2*d**6*e**4* 
x**4 + 756*a*c**2*d**5*e**5*x**5 + 630*a*c**2*d**4*e**6*x**6 + 360*a*c**2* 
d**3*e**7*x**7 + 135*a*c**2*d**2*e**8*x**8 + 30*a*c**2*d*e**9*x**9 + 3*a*c 
**2*e**10*x**10 + b**3*d**8 + 8*b**3*d**7*e*x + 28*b**3*d**6*e**2*x**2 + 5 
6*b**3*d**5*e**3*x**3 + 70*b**3*d**4*e**4*x**4 + 56*b**3*d**3*e**5*x**5 + 
28*b**3*d**2*e**6*x**6 + 8*b**3*d*e**7*x**7 + b**3*e**8*x**8 + 3*b**2*c*d* 
*10 + 30*b**2*c*d**9*e*x + 135*b**2*c*d**8*e**2*x**2 + 360*b**2*c*d**7*e** 
3*x**3 + 630*b**2*c*d**6*e**4*x**4 + 756*b**2*c*d**5*e**5*x**5 + 630*b**2* 
c*d**4*e**6*x**6 + 360*b**2*c*d**3*e**7*x**7 + 135*b**2*c*d**2*e**8*x**8 + 
 30*b**2*c*d*e**9*x**9 + 3*b**2*c*e**10*x**10 + 3*b*c**2*d**12 + 36*b*c...