\(\int \frac {d+e x}{(a+b (d+e x)^2+c (d+e x)^4)^3} \, dx\) [633]

Optimal result

Integrand size = 28, antiderivative size = 152 \[ \int \frac {d+e x}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\frac {-b-2 c (d+e x)^2}{4 \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {3 c \left (b+2 c (d+e x)^2\right )}{2 \left (b^2-4 a c\right )^2 e \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {6 c^2 \text {arctanh}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {1156, 1121, 628, 632, 212} \[ \int \frac {d+e x}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=-\frac {6 c^2 \text {arctanh}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{e \left (b^2-4 a c\right )^{5/2}}+\frac {3 c \left (b+2 c (d+e x)^2\right )}{2 e \left (b^2-4 a c\right )^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {b+2 c (d+e x)^2}{4 e \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \]

[In]

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

[Out]

-1/4*(b + 2*c*(d + e*x)^2)/((b^2 - 4*a*c)*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2) + (3*c*(b + 2*c*(d + e*x)^2
))/(2*(b^2 - 4*a*c)^2*e*(a + b*(d + e*x)^2 + c*(d + e*x)^4)) - (6*c^2*ArcTanh[(b + 2*c*(d + e*x)^2)/Sqrt[b^2 -
4*a*c]])/((b^2 - 4*a*c)^(5/2)*e)

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 628

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] - Dist[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 1121

Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(a + b*x + c*x^2)^p, x],
x, x^2], x] /; FreeQ[{a, b, c, p}, x]

Rule 1156

Int[(u_)^(m_.)*((a_.) + (b_.)*(v_)^2 + (c_.)*(v_)^4)^(p_.), x_Symbol] :> Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x}{\left (a+b x^2+c x^4\right )^3} \, dx,x,d+e x\right )}{e} \\ & = \frac {\text {Subst}\left (\int \frac {1}{\left (a+b x+c x^2\right )^3} \, dx,x,(d+e x)^2\right )}{2 e} \\ & = -\frac {b+2 c (d+e x)^2}{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) \text {Subst}\left (\int \frac {1}{\left (a+b x+c x^2\right )^2} \, dx,x,(d+e x)^2\right )}{2 \left (b^2-4 a c\right ) e} \\ & = -\frac {b+2 c (d+e x)^2}{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 \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 {\left (3 c^2\right ) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{\left (b^2-4 a c\right )^2 e} \\ & = -\frac {b+2 c (d+e x)^2}{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 \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 {\left (6 c^2\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c (d+e x)^2\right )}{\left (b^2-4 a c\right )^2 e} \\ & = -\frac {b+2 c (d+e x)^2}{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 \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 \tanh ^{-1}\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.97 \[ \int \frac {d+e x}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\frac {\frac {\left (b^2-4 a c\right ) \left (-b-2 c (d+e x)^2\right )}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {6 c \left (b+2 c (d+e x)^2\right )}{a+b (d+e x)^2+c (d+e x)^4}+\frac {24 c^2 \arctan \left (\frac {b+2 c (d+e x)^2}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}}{4 \left (b^2-4 a c\right )^2 e} \]

[In]

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

[Out]

(((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)

Maple [C] (verified)

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

Time = 0.35 (sec) , antiderivative size = 541, normalized size of antiderivative = 3.56

method	result	size
default	\(\frac {\frac {3 c^{3} e^{5} x^{6}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {18 e^{4} c^{3} d \,x^{5}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {9 c^{2} e^{3} \left (10 c \,d^{2}+b \right ) x^{4}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {6 c^{2} d \,e^{2} \left (10 c \,d^{2}+3 b \right ) x^{3}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {c e \left (45 c^{2} d^{4}+27 b c \,d^{2}+5 a c +b^{2}\right ) x^{2}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {2 c d \left (9 c^{2} d^{4}+9 b c \,d^{2}+5 a c +b^{2}\right ) x}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {12 c^{3} d^{6}+18 b \,c^{2} d^{4}+20 a \,c^{2} d^{2}+4 b^{2} c \,d^{2}+10 a b c -b^{3}}{4 e \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4} e^{4}+4 c d \,e^{3} x^{3}+6 c \,d^{2} e^{2} x^{2}+4 c \,d^{3} e x +b \,e^{2} x^{2}+d^{4} c +2 b d e x +b \,d^{2}+a \right )^{2}}+\frac {3 c^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,e^{4} \textit {\_Z}^{4}+4 c d \,e^{3} \textit {\_Z}^{3}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 d^{3} e c +2 b d e \right ) \textit {\_Z} +d^{4} c +b \,d^{2}+a \right )}{\sum }\frac {\left (\textit {\_R} e +d \right ) \ln \left (x -\textit {\_R} \right )}{2 e^{3} c \,\textit {\_R}^{3}+6 c d \,e^{2} \textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 d^{3} c +b e \textit {\_R} +b d}\right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) e}\)	\(541\)
risch	\(\frac {\frac {3 c^{3} e^{5} x^{6}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {18 e^{4} c^{3} d \,x^{5}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {9 c^{2} e^{3} \left (10 c \,d^{2}+b \right ) x^{4}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {6 c^{2} d \,e^{2} \left (10 c \,d^{2}+3 b \right ) x^{3}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {c e \left (45 c^{2} d^{4}+27 b c \,d^{2}+5 a c +b^{2}\right ) x^{2}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {2 c d \left (9 c^{2} d^{4}+9 b c \,d^{2}+5 a c +b^{2}\right ) x}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {12 c^{3} d^{6}+18 b \,c^{2} d^{4}+20 a \,c^{2} d^{2}+4 b^{2} c \,d^{2}+10 a b c -b^{3}}{4 e \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4} e^{4}+4 c d \,e^{3} x^{3}+6 c \,d^{2} e^{2} x^{2}+4 c \,d^{3} e x +b \,e^{2} x^{2}+d^{4} c +2 b d e x +b \,d^{2}+a \right )^{2}}-\frac {3 c^{2} \ln \left (\left (\left (-4 a c +b^{2}\right )^{\frac {5}{2}} e^{2}-16 a^{2} c^{2} e^{2} b +8 e^{2} a c \,b^{3}-b^{5} e^{2}\right ) x^{2}+\left (2 \left (-4 a c +b^{2}\right )^{\frac {5}{2}} d e -32 a^{2} b \,c^{2} d e +16 a \,b^{3} c d e -2 b^{5} d e \right ) x +\left (-4 a c +b^{2}\right )^{\frac {5}{2}} d^{2}-16 a^{2} b \,c^{2} d^{2}+8 a \,b^{3} c \,d^{2}-b^{5} d^{2}-32 a^{3} c^{2}+16 a^{2} b^{2} c -2 b^{4} a \right )}{e \left (-4 a c +b^{2}\right )^{\frac {5}{2}}}+\frac {3 c^{2} \ln \left (\left (\left (-4 a c +b^{2}\right )^{\frac {5}{2}} e^{2}+16 a^{2} c^{2} e^{2} b -8 e^{2} a c \,b^{3}+b^{5} e^{2}\right ) x^{2}+\left (2 \left (-4 a c +b^{2}\right )^{\frac {5}{2}} d e +32 a^{2} b \,c^{2} d e -16 a \,b^{3} c d e +2 b^{5} d e \right ) x +\left (-4 a c +b^{2}\right )^{\frac {5}{2}} d^{2}+16 a^{2} b \,c^{2} d^{2}-8 a \,b^{3} c \,d^{2}+b^{5} d^{2}+32 a^{3} c^{2}-16 a^{2} b^{2} c +2 b^{4} a \right )}{e \left (-4 a c +b^{2}\right )^{\frac {5}{2}}}\)	\(747\)

[In]

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

[Out]

(3*c^3*e^5/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6+18*e^4*c^3*d/(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+d^4*c+b*d^2+a))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1788 vs. \(2 (142) = 284\).

Time = 0.43 (sec) , antiderivative size = 3708, normalized size of antiderivative = 24.39 \[ \int \frac {d+e x}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/4*(12*(b^2*c^3 - 4*a*c^4)*e^6*x^6 + 72*(b^2*c^3 - 4*a*c^4)*d*e^5*x^5 + 18*(b^3*c^2 - 4*a*b*c^3 + 10*(b^2*c^
3 - 4*a*c^4)*d^2)*e^4*x^4 + 12*(b^2*c^3 - 4*a*c^4)*d^6 + 24*(10*(b^2*c^3 - 4*a*c^4)*d^3 + 3*(b^3*c^2 - 4*a*b*c
^3)*d)*e^3*x^3 - b^5 + 14*a*b^3*c - 40*a^2*b*c^2 + 18*(b^3*c^2 - 4*a*b*c^3)*d^4 + 4*(b^4*c + a*b^2*c^2 - 20*a^
2*c^3 + 45*(b^2*c^3 - 4*a*c^4)*d^4 + 27*(b^3*c^2 - 4*a*b*c^3)*d^2)*e^2*x^2 + 4*(b^4*c + a*b^2*c^2 - 20*a^2*c^3
)*d^2 + 8*(9*(b^2*c^3 - 4*a*c^4)*d^5 + 9*(b^3*c^2 - 4*a*b*c^3)*d^3 + (b^4*c + a*b^2*c^2 - 20*a^2*c^3)*d)*e*x +
12*(c^4*e^8*x^8 + 8*c^4*d*e^7*x^7 + 2*(14*c^4*d^2 + b*c^3)*e^6*x^6 + c^4*d^8 + 4*(14*c^4*d^3 + 3*b*c^3*d)*e^5
*x^5 + 2*b*c^3*d^6 + (70*c^4*d^4 + 30*b*c^3*d^2 + b^2*c^2 + 2*a*c^3)*e^4*x^4 + 4*(14*c^4*d^5 + 10*b*c^3*d^3 +
(b^2*c^2 + 2*a*c^3)*d)*e^3*x^3 + 2*a*b*c^2*d^2 + (b^2*c^2 + 2*a*c^3)*d^4 + 2*(14*c^4*d^6 + 15*b*c^3*d^4 + a*b*
c^2 + 3*(b^2*c^2 + 2*a*c^3)*d^2)*e^2*x^2 + a^2*c^2 + 4*(2*c^4*d^7 + 3*b*c^3*d^5 + a*b*c^2*d + (b^2*c^2 + 2*a*c
^3)*d^3)*e*x)*sqrt(b^2 - 4*a*c)*log((2*c^2*e^4*x^4 + 8*c^2*d*e^3*x^3 + 2*c^2*d^4 + 2*(6*c^2*d^2 + b*c)*e^2*x^2
+ 2*b*c*d^2 + 4*(2*c^2*d^3 + b*c*d)*e*x + b^2 - 2*a*c - (2*c*e^2*x^2 + 4*c*d*e*x + 2*c*d^2 + b)*sqrt(b^2 - 4*
a*c))/(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)))/((b^6*
c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*e^9*x^8 + 8*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^
3*c^5)*d*e^8*x^7 + 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4 + 14*(b^6*c^2 - 12*a*b^4*c^3 + 48*a
^2*b^2*c^4 - 64*a^3*c^5)*d^2)*e^7*x^6 + 4*(14*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^3 + 3*(
b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d)*e^6*x^5 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3
*b^2*c^3 - 128*a^4*c^4 + 70*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^4 + 30*(b^7*c - 12*a*b^5*
c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^2)*e^5*x^4 + 4*(14*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^
5)*d^5 + 10*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 +
32*a^3*b^2*c^3 - 128*a^4*c^4)*d)*e^4*x^3 + 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3 + 14*(b^6*c
^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^6 + 15*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^
4)*d^4 + 3*(b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^2)*e^3*x^2 + 4*(2*(b^6*c^2 - 1
2*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^7 + 3*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^5
+ (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^3 + (a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*
c^2 - 64*a^4*b*c^3)*d)*e^2*x + ((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^8 + a^2*b^6 - 12*a^3*
b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3 + 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^6 + (b^8 - 10
*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^4 + 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*
a^4*b*c^3)*d^2)*e), 1/4*(12*(b^2*c^3 - 4*a*c^4)*e^6*x^6 + 72*(b^2*c^3 - 4*a*c^4)*d*e^5*x^5 + 18*(b^3*c^2 - 4*a
*b*c^3 + 10*(b^2*c^3 - 4*a*c^4)*d^2)*e^4*x^4 + 12*(b^2*c^3 - 4*a*c^4)*d^6 + 24*(10*(b^2*c^3 - 4*a*c^4)*d^3 + 3
*(b^3*c^2 - 4*a*b*c^3)*d)*e^3*x^3 - b^5 + 14*a*b^3*c - 40*a^2*b*c^2 + 18*(b^3*c^2 - 4*a*b*c^3)*d^4 + 4*(b^4*c
+ a*b^2*c^2 - 20*a^2*c^3 + 45*(b^2*c^3 - 4*a*c^4)*d^4 + 27*(b^3*c^2 - 4*a*b*c^3)*d^2)*e^2*x^2 + 4*(b^4*c + a*b
^2*c^2 - 20*a^2*c^3)*d^2 + 8*(9*(b^2*c^3 - 4*a*c^4)*d^5 + 9*(b^3*c^2 - 4*a*b*c^3)*d^3 + (b^4*c + a*b^2*c^2 - 2
0*a^2*c^3)*d)*e*x - 24*(c^4*e^8*x^8 + 8*c^4*d*e^7*x^7 + 2*(14*c^4*d^2 + b*c^3)*e^6*x^6 + c^4*d^8 + 4*(14*c^4*d
^3 + 3*b*c^3*d)*e^5*x^5 + 2*b*c^3*d^6 + (70*c^4*d^4 + 30*b*c^3*d^2 + b^2*c^2 + 2*a*c^3)*e^4*x^4 + 4*(14*c^4*d^
5 + 10*b*c^3*d^3 + (b^2*c^2 + 2*a*c^3)*d)*e^3*x^3 + 2*a*b*c^2*d^2 + (b^2*c^2 + 2*a*c^3)*d^4 + 2*(14*c^4*d^6 +
15*b*c^3*d^4 + a*b*c^2 + 3*(b^2*c^2 + 2*a*c^3)*d^2)*e^2*x^2 + a^2*c^2 + 4*(2*c^4*d^7 + 3*b*c^3*d^5 + a*b*c^2*d
+ (b^2*c^2 + 2*a*c^3)*d^3)*e*x)*sqrt(-b^2 + 4*a*c)*arctan(-(2*c*e^2*x^2 + 4*c*d*e*x + 2*c*d^2 + b)*sqrt(-b^2
+ 4*a*c)/(b^2 - 4*a*c)))/((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*e^9*x^8 + 8*(b^6*c^2 - 12*a*b
^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d*e^8*x^7 + 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4 + 14
*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^2)*e^7*x^6 + 4*(14*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*
b^2*c^4 - 64*a^3*c^5)*d^3 + 3*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d)*e^6*x^5 + (b^8 - 10*a*
b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4 + 70*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c
^5)*d^4 + 30*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^2)*e^5*x^4 + 4*(14*(b^6*c^2 - 12*a*b^4*c
^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^5 + 10*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^3 + (b^8 -
10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d)*e^4*x^3 + 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3
*c^2 - 64*a^4*b*c^3 + 14*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^6 + 15*(b^7*c - 12*a*b^5*c^2
+ 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^4 + 3*(b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d
^2)*e^3*x^2 + 4*(2*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^7 + 3*(b^7*c - 12*a*b^5*c^2 + 48*a
^2*b^3*c^3 - 64*a^3*b*c^4)*d^5 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^3 + (a*b
^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d)*e^2*x + ((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a
^3*c^5)*d^8 + a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3 + 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3
- 64*a^3*b*c^4)*d^6 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d^4 + 2*(a*b^7 - 12*a
^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d^2)*e)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1646 vs. \(2 (136) = 272\).

Time = 6.93 (sec) , antiderivative size = 1646, normalized size of antiderivative = 10.83 \[ \int \frac {d+e x}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-3*c**2*sqrt(-1/(4*a*c - b**2)**5)*log(2*d*x/e + x**2 + (-192*a**3*c**5*sqrt(-1/(4*a*c - b**2)**5) + 144*a**2*
b**2*c**4*sqrt(-1/(4*a*c - b**2)**5) - 36*a*b**4*c**3*sqrt(-1/(4*a*c - b**2)**5) + 3*b**6*c**2*sqrt(-1/(4*a*c
- b**2)**5) + 3*b*c**2 + 6*c**3*d**2)/(6*c**3*e**2))/e + 3*c**2*sqrt(-1/(4*a*c - b**2)**5)*log(2*d*x/e + x**2
+ (192*a**3*c**5*sqrt(-1/(4*a*c - b**2)**5) - 144*a**2*b**2*c**4*sqrt(-1/(4*a*c - b**2)**5) + 36*a*b**4*c**3*s
qrt(-1/(4*a*c - b**2)**5) - 3*b**6*c**2*sqrt(-1/(4*a*c - b**2)**5) + 3*b*c**2 + 6*c**3*d**2)/(6*c**3*e**2))/e
+ (10*a*b*c + 20*a*c**2*d**2 - b**3 + 4*b**2*c*d**2 + 18*b*c**2*d**4 + 12*c**3*d**6 + 72*c**3*d*e**5*x**5 + 12
*c**3*e**6*x**6 + x**4*(18*b*c**2*e**4 + 180*c**3*d**2*e**4) + x**3*(72*b*c**2*d*e**3 + 240*c**3*d**3*e**3) +
x**2*(20*a*c**2*e**2 + 4*b**2*c*e**2 + 108*b*c**2*d**2*e**2 + 180*c**3*d**4*e**2) + x*(40*a*c**2*d*e + 8*b**2*
c*d*e + 72*b*c**2*d**3*e + 72*c**3*d**5*e))/(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*e**7 + 1792*a**2*c**4*d**
2*e**7 - 64*a*b**3*c**2*e**7 - 896*a*b**2*c**3*d**2*e**7 + 8*b**5*c*e**7 + 112*b**4*c**2*d**2*e**7) + x**5*(76
8*a**2*b*c**3*d*e**6 + 3584*a**2*c**4*d**3*e**6 - 384*a*b**3*c**2*d*e**6 - 1792*a*b**2*c**3*d**3*e**6 + 48*b**
5*c*d*e**6 + 224*b**4*c**2*d**3*e**6) + x**4*(128*a**3*c**3*e**5 + 1920*a**2*b*c**3*d**2*e**5 + 4480*a**2*c**4
*d**4*e**5 - 24*a*b**4*c*e**5 - 960*a*b**3*c**2*d**2*e**5 - 2240*a*b**2*c**3*d**4*e**5 + 4*b**6*e**5 + 120*b**
5*c*d**2*e**5 + 280*b**4*c**2*d**4*e**5) + x**3*(512*a**3*c**3*d*e**4 + 2560*a**2*b*c**3*d**3*e**4 + 3584*a**2
*c**4*d**5*e**4 - 96*a*b**4*c*d*e**4 - 1280*a*b**3*c**2*d**3*e**4 - 1792*a*b**2*c**3*d**5*e**4 + 16*b**6*d*e**
4 + 160*b**5*c*d**3*e**4 + 224*b**4*c**2*d**5*e**4) + x**2*(128*a**3*b*c**2*e**3 + 768*a**3*c**3*d**2*e**3 - 6
4*a**2*b**3*c*e**3 + 1920*a**2*b*c**3*d**4*e**3 + 1792*a**2*c**4*d**6*e**3 + 8*a*b**5*e**3 - 144*a*b**4*c*d**2
*e**3 - 960*a*b**3*c**2*d**4*e**3 - 896*a*b**2*c**3*d**6*e**3 + 24*b**6*d**2*e**3 + 120*b**5*c*d**4*e**3 + 112
*b**4*c**2*d**6*e**3) + x*(256*a**3*b*c**2*d*e**2 + 512*a**3*c**3*d**3*e**2 - 128*a**2*b**3*c*d*e**2 + 768*a**
2*b*c**3*d**5*e**2 + 512*a**2*c**4*d**7*e**2 + 16*a*b**5*d*e**2 - 96*a*b**4*c*d**3*e**2 - 384*a*b**3*c**2*d**5
*e**2 - 256*a*b**2*c**3*d**7*e**2 + 16*b**6*d**3*e**2 + 48*b**5*c*d**5*e**2 + 32*b**4*c**2*d**7*e**2))

Maxima [F]

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

[In]

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

[Out]

6*c^2*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*(12*c^3*e^6*x^6 + 72*c^3*d*e^5*x^5 + 12*c^3*d^6 + 18*(10*
c^3*d^2 + b*c^2)*e^4*x^4 + 18*b*c^2*d^4 + 24*(10*c^3*d^3 + 3*b*c^2*d)*e^3*x^3 + 4*(45*c^3*d^4 + 27*b*c^2*d^2 +
b^2*c + 5*a*c^2)*e^2*x^2 - b^3 + 10*a*b*c + 4*(b^2*c + 5*a*c^2)*d^2 + 8*(9*c^3*d^5 + 9*b*c^2*d^3 + (b^2*c + 5
*a*c^2)*d)*e*x)/((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*c^3)*d^6 + a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*d^
4 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d^2)*e)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 349 vs. \(2 (142) = 284\).

Time = 0.31 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.30 \[ \int \frac {d+e x}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\frac {6 \, c^{2} \arctan \left (\frac {2 \, c d^{2} + 2 \, {\left (e x^{2} + 2 \, d x\right )} c e + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c} e} + \frac {12 \, c^{3} d^{6} + 36 \, {\left (e x^{2} + 2 \, d x\right )} c^{3} d^{4} e + 36 \, {\left (e x^{2} + 2 \, d x\right )}^{2} c^{3} d^{2} e^{2} + 12 \, {\left (e x^{2} + 2 \, d x\right )}^{3} c^{3} e^{3} + 18 \, b c^{2} d^{4} + 36 \, {\left (e x^{2} + 2 \, d x\right )} b c^{2} d^{2} e + 18 \, {\left (e x^{2} + 2 \, d x\right )}^{2} b c^{2} e^{2} + 4 \, b^{2} c d^{2} + 20 \, a c^{2} d^{2} + 4 \, {\left (e x^{2} + 2 \, d x\right )} b^{2} c e + 20 \, {\left (e x^{2} + 2 \, d x\right )} a c^{2} e - b^{3} + 10 \, a b c}{4 \, {\left (c d^{4} + 2 \, {\left (e x^{2} + 2 \, d x\right )} c d^{2} e + {\left (e x^{2} + 2 \, d x\right )}^{2} c e^{2} + b d^{2} + {\left (e x^{2} + 2 \, d x\right )} b e + a\right )}^{2} {\left (b^{4} e - 8 \, a b^{2} c e + 16 \, a^{2} c^{2} e\right )}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.81 (sec) , antiderivative size = 1157, normalized size of antiderivative = 7.61 \[ \int \frac {d+e x}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx=\frac {\frac {-b^3+4\,b^2\,c\,d^2+18\,b\,c^2\,d^4+10\,a\,b\,c+12\,c^3\,d^6+20\,a\,c^2\,d^2}{4\,e\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {x^2\,\left (e\,b^2\,c+27\,e\,b\,c^2\,d^2+45\,e\,c^3\,d^4+5\,a\,e\,c^2\right )}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}+\frac {9\,x^4\,\left (10\,c^3\,d^2\,e^3+b\,c^2\,e^3\right )}{2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {3\,c^3\,e^5\,x^6}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}+\frac {2\,d\,x\,\left (b^2\,c+9\,b\,c^2\,d^2+9\,c^3\,d^4+5\,a\,c^2\right )}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}+\frac {6\,d\,x^3\,\left (10\,c^3\,d^2\,e^2+3\,b\,c^2\,e^2\right )}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}+\frac {18\,c^3\,d\,e^4\,x^5}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}}{x^2\,\left (6\,b^2\,d^2\,e^2+30\,b\,c\,d^4\,e^2+2\,a\,b\,e^2+28\,c^2\,d^6\,e^2+12\,a\,c\,d^2\,e^2\right )+x^6\,\left (28\,c^2\,d^2\,e^6+2\,b\,c\,e^6\right )+x\,\left (4\,e\,b^2\,d^3+12\,e\,b\,c\,d^5+4\,a\,e\,b\,d+8\,e\,c^2\,d^7+8\,a\,e\,c\,d^3\right )+x^3\,\left (4\,b^2\,d\,e^3+40\,b\,c\,d^3\,e^3+56\,c^2\,d^5\,e^3+8\,a\,c\,d\,e^3\right )+x^5\,\left (56\,c^2\,d^3\,e^5+12\,b\,c\,d\,e^5\right )+x^4\,\left (b^2\,e^4+30\,b\,c\,d^2\,e^4+70\,c^2\,d^4\,e^4+2\,a\,c\,e^4\right )+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}+\frac {6\,c^2\,\mathrm {atan}\left (\frac {\left (b^4\,{\left (4\,a\,c-b^2\right )}^5+16\,a^2\,c^2\,{\left (4\,a\,c-b^2\right )}^5-8\,a\,b^2\,c\,{\left (4\,a\,c-b^2\right )}^5\right )\,\left (x\,\left (\frac {72\,c^6\,d\,e^7}{a\,{\left (4\,a\,c-b^2\right )}^{9/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {72\,b\,c^4\,\left (16\,d\,a^2\,b\,c^4\,e^9-8\,d\,a\,b^3\,c^3\,e^9+d\,b^5\,c^2\,e^9\right )}{a\,e^2\,{\left (4\,a\,c-b^2\right )}^{15/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )+x^2\,\left (\frac {36\,c^6\,e^8}{a\,{\left (4\,a\,c-b^2\right )}^{9/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {36\,b\,c^4\,\left (16\,a^2\,b\,c^4\,e^{10}-8\,a\,b^3\,c^3\,e^{10}+b^5\,c^2\,e^{10}\right )}{a\,e^2\,{\left (4\,a\,c-b^2\right )}^{15/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )+\frac {36\,c^6\,d^2\,e^6}{a\,{\left (4\,a\,c-b^2\right )}^{9/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {36\,b\,c^4\,\left (32\,a^3\,c^4\,e^8-16\,a^2\,b^2\,c^3\,e^8+16\,a^2\,b\,c^4\,d^2\,e^8+2\,a\,b^4\,c^2\,e^8-8\,a\,b^3\,c^3\,d^2\,e^8+b^5\,c^2\,d^2\,e^8\right )}{a\,e^2\,{\left (4\,a\,c-b^2\right )}^{15/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )}{72\,c^6\,e^6}\right )}{e\,{\left (4\,a\,c-b^2\right )}^{5/2}} \]

[In]

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

[Out]

((12*c^3*d^6 - b^3 + 20*a*c^2*d^2 + 4*b^2*c*d^2 + 18*b*c^2*d^4 + 10*a*b*c)/(4*e*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)
) + (x^2*(45*c^3*d^4*e + 5*a*c^2*e + b^2*c*e + 27*b*c^2*d^2*e))/(b^4 + 16*a^2*c^2 - 8*a*b^2*c) + (9*x^4*(b*c^2
*e^3 + 10*c^3*d^2*e^3))/(2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (3*c^3*e^5*x^6)/(b^4 + 16*a^2*c^2 - 8*a*b^2*c) +
(2*d*x*(5*a*c^2 + b^2*c + 9*c^3*d^4 + 9*b*c^2*d^2))/(b^4 + 16*a^2*c^2 - 8*a*b^2*c) + (6*d*x^3*(3*b*c^2*e^2 + 1
0*c^3*d^2*e^2))/(b^4 + 16*a^2*c^2 - 8*a*b^2*c) + (18*c^3*d*e^4*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*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*((72*c^6
*d*e^7)/(a*(4*a*c - b^2)^(9/2)*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (72*b*c^4*(b^5*c^2*d*e^9 - 8*a*b^3*c^3*d*e^9
+ 16*a^2*b*c^4*d*e^9))/(a*e^2*(4*a*c - b^2)^(15/2)*(b^4 + 16*a^2*c^2 - 8*a*b^2*c))) + x^2*((36*c^6*e^8)/(a*(4*
a*c - b^2)^(9/2)*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (36*b*c^4*(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))) + (36*c^6*d^2*e^6)/(a*(4*a*c - b^2)^(9/2)*(
b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (36*b*c^4*(32*a^3*c^4*e^8 + 2*a*b^4*c^2*e^8 - 16*a^2*b^2*c^3*e^8 + b^5*c^2*d^
2*e^8 - 8*a*b^3*c^3*d^2*e^8 + 16*a^2*b*c^4*d^2*e^8))/(a*e^2*(4*a*c - b^2)^(15/2)*(b^4 + 16*a^2*c^2 - 8*a*b^2*c
))))/(72*c^6*e^6)))/(e*(4*a*c - b^2)^(5/2))