\(\int \frac {(d+e x^2)^2}{(a+c x^4)^2} \, dx\) [319]

Optimal result

Integrand size = 19, antiderivative size = 249 \[ \int \frac {\left (d+e x^2\right )^2}{\left (a+c x^4\right )^2} \, dx=\frac {x \left (d^2-\frac {a e^2}{c}+2 d e x^2\right )}{4 a \left (a+c x^4\right )}-\frac {\left (3 c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{5/4}}+\frac {\left (3 c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{5/4}}+\frac {\left (3 c d^2-2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x}{\sqrt {a}+\sqrt {c} x^2}\right )}{8 \sqrt {2} a^{7/4} c^{5/4}} \] Output:

1/4*x*(d^2-a*e^2/c+2*d*e*x^2)/a/(c*x^4+a)+1/16*(3*c*d^2+2*a^(1/2)*c^(1/2)* 
d*e+a*e^2)*arctan(-1+2^(1/2)*c^(1/4)*x/a^(1/4))*2^(1/2)/a^(7/4)/c^(5/4)+1/ 
16*(3*c*d^2+2*a^(1/2)*c^(1/2)*d*e+a*e^2)*arctan(1+2^(1/2)*c^(1/4)*x/a^(1/4 
))*2^(1/2)/a^(7/4)/c^(5/4)+1/16*(3*c*d^2-2*a^(1/2)*c^(1/2)*d*e+a*e^2)*arct 
anh(2^(1/2)*a^(1/4)*c^(1/4)*x/(a^(1/2)+c^(1/2)*x^2))*2^(1/2)/a^(7/4)/c^(5/ 
4)
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.18 \[ \int \frac {\left (d+e x^2\right )^2}{\left (a+c x^4\right )^2} \, dx=\frac {-\frac {8 a^{3/4} \sqrt [4]{c} \left (a e^2 x-c d x \left (d+2 e x^2\right )\right )}{a+c x^4}-2 \sqrt {2} \left (3 c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+2 \sqrt {2} \left (3 c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )-\sqrt {2} \left (3 c d^2-2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )+\sqrt {2} \left (3 c d^2-2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{32 a^{7/4} c^{5/4}} \] Input:

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

Output:

((-8*a^(3/4)*c^(1/4)*(a*e^2*x - c*d*x*(d + 2*e*x^2)))/(a + c*x^4) - 2*Sqrt 
[2]*(3*c*d^2 + 2*Sqrt[a]*Sqrt[c]*d*e + a*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)* 
x)/a^(1/4)] + 2*Sqrt[2]*(3*c*d^2 + 2*Sqrt[a]*Sqrt[c]*d*e + a*e^2)*ArcTan[1 
 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)] - Sqrt[2]*(3*c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e 
 + a*e^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2] + Sqrt[2] 
*(3*c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e + a*e^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c 
^(1/4)*x + Sqrt[c]*x^2])/(32*a^(7/4)*c^(5/4))
 

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.31, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.684, Rules used = {1519, 25, 1493, 27, 1482, 27, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

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

\(\Big \downarrow \) 1519

\(\displaystyle -\frac {\int -\frac {3 c d^2+6 c e x^2 d+a e^2}{\left (c x^4+a\right )^2}dx}{3 c}-\frac {e^2 x}{3 c \left (a+c x^4\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {3 c d^2+6 c e x^2 d+a e^2}{\left (c x^4+a\right )^2}dx}{3 c}-\frac {e^2 x}{3 c \left (a+c x^4\right )}\)

\(\Big \downarrow \) 1493

\(\displaystyle \frac {\frac {x \left (a e^2+3 c d^2+6 c d e x^2\right )}{4 a \left (a+c x^4\right )}-\frac {\int -\frac {3 \left (3 c d^2+2 c e x^2 d+a e^2\right )}{c x^4+a}dx}{4 a}}{3 c}-\frac {e^2 x}{3 c \left (a+c x^4\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3 \int \frac {3 c d^2+2 c e x^2 d+a e^2}{c x^4+a}dx}{4 a}+\frac {x \left (a e^2+3 c d^2+6 c d e x^2\right )}{4 a \left (a+c x^4\right )}}{3 c}-\frac {e^2 x}{3 c \left (a+c x^4\right )}\)

\(\Big \downarrow \) 1482

\(\displaystyle \frac {\frac {3 \left (\frac {\left (-2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \int \frac {\sqrt {c} \left (\sqrt {a}-\sqrt {c} x^2\right )}{c x^4+a}dx}{2 \sqrt {a} \sqrt {c}}+\frac {\left (2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \int \frac {\sqrt {c} \left (\sqrt {c} x^2+\sqrt {a}\right )}{c x^4+a}dx}{2 \sqrt {a} \sqrt {c}}\right )}{4 a}+\frac {x \left (a e^2+3 c d^2+6 c d e x^2\right )}{4 a \left (a+c x^4\right )}}{3 c}-\frac {e^2 x}{3 c \left (a+c x^4\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3 \left (\frac {\left (-2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {a}}+\frac {\left (2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{c x^4+a}dx}{2 \sqrt {a}}\right )}{4 a}+\frac {x \left (a e^2+3 c d^2+6 c d e x^2\right )}{4 a \left (a+c x^4\right )}}{3 c}-\frac {e^2 x}{3 c \left (a+c x^4\right )}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {\frac {3 \left (\frac {\left (2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \left (\frac {\int \frac {1}{x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}\right )}{2 \sqrt {a}}+\frac {\left (-2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {a}}\right )}{4 a}+\frac {x \left (a e^2+3 c d^2+6 c d e x^2\right )}{4 a \left (a+c x^4\right )}}{3 c}-\frac {e^2 x}{3 c \left (a+c x^4\right )}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {\frac {3 \left (\frac {\left (-2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {a}}+\frac {\left (2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \left (\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )^2-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )^2-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {a}}\right )}{4 a}+\frac {x \left (a e^2+3 c d^2+6 c d e x^2\right )}{4 a \left (a+c x^4\right )}}{3 c}-\frac {e^2 x}{3 c \left (a+c x^4\right )}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\frac {3 \left (\frac {\left (-2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {a}}+\frac {\left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right ) \left (2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right )}{2 \sqrt {a}}\right )}{4 a}+\frac {x \left (a e^2+3 c d^2+6 c d e x^2\right )}{4 a \left (a+c x^4\right )}}{3 c}-\frac {e^2 x}{3 c \left (a+c x^4\right )}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {\frac {3 \left (\frac {\left (-2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} x}{\sqrt [4]{c} \left (x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} x+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {a}}+\frac {\left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right ) \left (2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right )}{2 \sqrt {a}}\right )}{4 a}+\frac {x \left (a e^2+3 c d^2+6 c d e x^2\right )}{4 a \left (a+c x^4\right )}}{3 c}-\frac {e^2 x}{3 c \left (a+c x^4\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {3 \left (\frac {\left (-2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} x}{\sqrt [4]{c} \left (x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} x+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {a}}+\frac {\left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right ) \left (2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right )}{2 \sqrt {a}}\right )}{4 a}+\frac {x \left (a e^2+3 c d^2+6 c d e x^2\right )}{4 a \left (a+c x^4\right )}}{3 c}-\frac {e^2 x}{3 c \left (a+c x^4\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3 \left (\frac {\left (-2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} x}{x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{c} x+\sqrt [4]{a}}{x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt [4]{a} \sqrt {c}}\right )}{2 \sqrt {a}}+\frac {\left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right ) \left (2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right )}{2 \sqrt {a}}\right )}{4 a}+\frac {x \left (a e^2+3 c d^2+6 c d e x^2\right )}{4 a \left (a+c x^4\right )}}{3 c}-\frac {e^2 x}{3 c \left (a+c x^4\right )}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\frac {3 \left (\frac {\left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right ) \left (2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right )}{2 \sqrt {a}}+\frac {\left (-2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {a}}\right )}{4 a}+\frac {x \left (a e^2+3 c d^2+6 c d e x^2\right )}{4 a \left (a+c x^4\right )}}{3 c}-\frac {e^2 x}{3 c \left (a+c x^4\right )}\)

Input:

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

Output:

-1/3*(e^2*x)/(c*(a + c*x^4)) + ((x*(3*c*d^2 + a*e^2 + 6*c*d*e*x^2))/(4*a*( 
a + c*x^4)) + (3*(((3*c*d^2 + 2*Sqrt[a]*Sqrt[c]*d*e + a*e^2)*(-(ArcTan[1 - 
 (Sqrt[2]*c^(1/4)*x)/a^(1/4)]/(Sqrt[2]*a^(1/4)*c^(1/4))) + ArcTan[1 + (Sqr 
t[2]*c^(1/4)*x)/a^(1/4)]/(Sqrt[2]*a^(1/4)*c^(1/4))))/(2*Sqrt[a]) + ((3*c*d 
^2 - 2*Sqrt[a]*Sqrt[c]*d*e + a*e^2)*(-1/2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^ 
(1/4)*x + Sqrt[c]*x^2]/(Sqrt[2]*a^(1/4)*c^(1/4)) + Log[Sqrt[a] + Sqrt[2]*a 
^(1/4)*c^(1/4)*x + Sqrt[c]*x^2]/(2*Sqrt[2]*a^(1/4)*c^(1/4))))/(2*Sqrt[a])) 
)/(4*a))/(3*c)
 

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 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

rule 1482

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
a*c, 2]}, Simp[(d*q + a*e)/(2*a*c)   Int[(q + c*x^2)/(a + c*x^4), x], x] + 
Simp[(d*q - a*e)/(2*a*c)   Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ[{a 
, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(- 
a)*c]
 

rule 1493

Int[((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x 
)*(d + e*x^2)*((a + c*x^4)^(p + 1)/(4*a*(p + 1))), x] + Simp[1/(4*a*(p + 1) 
)   Int[Simp[d*(4*p + 5) + e*(4*p + 7)*x^2, x]*(a + c*x^4)^(p + 1), x], x] 
/; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && Integer 
Q[2*p]
 

rule 1519

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Sim 
p[e^q*x^(2*q - 3)*((a + c*x^4)^(p + 1)/(c*(4*p + 2*q + 1))), x] + Simp[1/(c 
*(4*p + 2*q + 1))   Int[(a + c*x^4)^p*ExpandToSum[c*(4*p + 2*q + 1)*(d + e* 
x^2)^q - a*(2*q - 3)*e^q*x^(2*q - 4) - c*(4*p + 2*q + 1)*e^q*x^(2*q), x], x 
], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[q, 1]
 

Maple [C] (verified)

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

Time = 0.11 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.39

method	result	size
risch	\(\frac {\frac {d e \,x^{3}}{2 a}-\frac {\left (a \,e^{2}-c \,d^{2}\right ) x}{4 a c}}{c \,x^{4}+a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+a \right )}{\sum }\frac {\left (2 d e \,\textit {\_R}^{2}+\frac {a \,e^{2}+3 c \,d^{2}}{c}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{16 a c}\)	\(97\)
default	\(\frac {\frac {d e \,x^{3}}{2 a}-\frac {\left (a \,e^{2}-c \,d^{2}\right ) x}{4 a c}}{c \,x^{4}+a}+\frac {\frac {\left (a \,e^{2}+3 c \,d^{2}\right ) \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {e d \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{4 a c}\)	\(266\)

Input:

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

Output:

(1/2*d*e/a*x^3-1/4*(a*e^2-c*d^2)/a/c*x)/(c*x^4+a)+1/16/a/c*sum((2*d*e*_R^2 
+1/c*(a*e^2+3*c*d^2))/_R^3*ln(x-_R),_R=RootOf(_Z^4*c+a))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1596 vs. \(2 (188) = 376\).

Time = 0.50 (sec) , antiderivative size = 1596, normalized size of antiderivative = 6.41 \[ \int \frac {\left (d+e x^2\right )^2}{\left (a+c x^4\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/16*(8*c*d*e*x^3 + (a*c^2*x^4 + a^2*c)*sqrt(-(a^3*c^2*sqrt(-(81*c^4*d^8 + 
 36*a*c^3*d^6*e^2 + 22*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c 
^5)) + 12*c*d^3*e + 4*a*d*e^3)/(a^3*c^2))*log((81*c^4*d^8 + 108*a*c^3*d^6* 
e^2 + 38*a^2*c^2*d^4*e^4 + 12*a^3*c*d^2*e^6 + a^4*e^8)*x + (2*a^6*c^4*d*e* 
sqrt(-(81*c^4*d^8 + 36*a*c^3*d^6*e^2 + 22*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^ 
6 + a^4*e^8)/(a^7*c^5)) + 27*a^2*c^4*d^6 + 15*a^3*c^3*d^4*e^2 + 5*a^4*c^2* 
d^2*e^4 + a^5*c*e^6)*sqrt(-(a^3*c^2*sqrt(-(81*c^4*d^8 + 36*a*c^3*d^6*e^2 + 
 22*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5)) + 12*c*d^3*e + 
 4*a*d*e^3)/(a^3*c^2))) - (a*c^2*x^4 + a^2*c)*sqrt(-(a^3*c^2*sqrt(-(81*c^4 
*d^8 + 36*a*c^3*d^6*e^2 + 22*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8)/ 
(a^7*c^5)) + 12*c*d^3*e + 4*a*d*e^3)/(a^3*c^2))*log((81*c^4*d^8 + 108*a*c^ 
3*d^6*e^2 + 38*a^2*c^2*d^4*e^4 + 12*a^3*c*d^2*e^6 + a^4*e^8)*x - (2*a^6*c^ 
4*d*e*sqrt(-(81*c^4*d^8 + 36*a*c^3*d^6*e^2 + 22*a^2*c^2*d^4*e^4 + 4*a^3*c* 
d^2*e^6 + a^4*e^8)/(a^7*c^5)) + 27*a^2*c^4*d^6 + 15*a^3*c^3*d^4*e^2 + 5*a^ 
4*c^2*d^2*e^4 + a^5*c*e^6)*sqrt(-(a^3*c^2*sqrt(-(81*c^4*d^8 + 36*a*c^3*d^6 
*e^2 + 22*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5)) + 12*c*d 
^3*e + 4*a*d*e^3)/(a^3*c^2))) - (a*c^2*x^4 + a^2*c)*sqrt((a^3*c^2*sqrt(-(8 
1*c^4*d^8 + 36*a*c^3*d^6*e^2 + 22*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4* 
e^8)/(a^7*c^5)) - 12*c*d^3*e - 4*a*d*e^3)/(a^3*c^2))*log((81*c^4*d^8 + 108 
*a*c^3*d^6*e^2 + 38*a^2*c^2*d^4*e^4 + 12*a^3*c*d^2*e^6 + a^4*e^8)*x + (...
 

Sympy [A] (verification not implemented)

Time = 0.92 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.10 \[ \int \frac {\left (d+e x^2\right )^2}{\left (a+c x^4\right )^2} \, dx=\operatorname {RootSum} {\left (65536 t^{4} a^{7} c^{5} + t^{2} \cdot \left (2048 a^{5} c^{3} d e^{3} + 6144 a^{4} c^{4} d^{3} e\right ) + a^{4} e^{8} + 20 a^{3} c d^{2} e^{6} + 118 a^{2} c^{2} d^{4} e^{4} + 180 a c^{3} d^{6} e^{2} + 81 c^{4} d^{8}, \left ( t \mapsto t \log {\left (x + \frac {- 8192 t^{3} a^{6} c^{4} d e + 16 t a^{5} c e^{6} - 48 t a^{4} c^{2} d^{2} e^{4} - 144 t a^{3} c^{3} d^{4} e^{2} + 432 t a^{2} c^{4} d^{6}}{a^{4} e^{8} + 12 a^{3} c d^{2} e^{6} + 38 a^{2} c^{2} d^{4} e^{4} + 108 a c^{3} d^{6} e^{2} + 81 c^{4} d^{8}} \right )} \right )\right )} + \frac {2 c d e x^{3} + x \left (- a e^{2} + c d^{2}\right )}{4 a^{2} c + 4 a c^{2} x^{4}} \] Input:

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

Output:

RootSum(65536*_t**4*a**7*c**5 + _t**2*(2048*a**5*c**3*d*e**3 + 6144*a**4*c 
**4*d**3*e) + a**4*e**8 + 20*a**3*c*d**2*e**6 + 118*a**2*c**2*d**4*e**4 + 
180*a*c**3*d**6*e**2 + 81*c**4*d**8, Lambda(_t, _t*log(x + (-8192*_t**3*a* 
*6*c**4*d*e + 16*_t*a**5*c*e**6 - 48*_t*a**4*c**2*d**2*e**4 - 144*_t*a**3* 
c**3*d**4*e**2 + 432*_t*a**2*c**4*d**6)/(a**4*e**8 + 12*a**3*c*d**2*e**6 + 
 38*a**2*c**2*d**4*e**4 + 108*a*c**3*d**6*e**2 + 81*c**4*d**8)))) + (2*c*d 
*e*x**3 + x*(-a*e**2 + c*d**2))/(4*a**2*c + 4*a*c**2*x**4)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.30 \[ \int \frac {\left (d+e x^2\right )^2}{\left (a+c x^4\right )^2} \, dx=\frac {2 \, c d e x^{3} + {\left (c d^{2} - a e^{2}\right )} x}{4 \, {\left (a c^{2} x^{4} + a^{2} c\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (3 \, c^{\frac {3}{2}} d^{2} + 2 \, \sqrt {a} c d e + a \sqrt {c} e^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (3 \, c^{\frac {3}{2}} d^{2} + 2 \, \sqrt {a} c d e + a \sqrt {c} e^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (3 \, c^{\frac {3}{2}} d^{2} - 2 \, \sqrt {a} c d e + a \sqrt {c} e^{2}\right )} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (3 \, c^{\frac {3}{2}} d^{2} - 2 \, \sqrt {a} c d e + a \sqrt {c} e^{2}\right )} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}}{32 \, a c} \] Input:

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

Output:

1/4*(2*c*d*e*x^3 + (c*d^2 - a*e^2)*x)/(a*c^2*x^4 + a^2*c) + 1/32*(2*sqrt(2 
)*(3*c^(3/2)*d^2 + 2*sqrt(a)*c*d*e + a*sqrt(c)*e^2)*arctan(1/2*sqrt(2)*(2* 
sqrt(c)*x + sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqrt(a)*sqrt(c)))/(sqrt(a)*sqrt( 
sqrt(a)*sqrt(c))*sqrt(c)) + 2*sqrt(2)*(3*c^(3/2)*d^2 + 2*sqrt(a)*c*d*e + a 
*sqrt(c)*e^2)*arctan(1/2*sqrt(2)*(2*sqrt(c)*x - sqrt(2)*a^(1/4)*c^(1/4))/s 
qrt(sqrt(a)*sqrt(c)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(c))*sqrt(c)) + sqrt(2)*(3 
*c^(3/2)*d^2 - 2*sqrt(a)*c*d*e + a*sqrt(c)*e^2)*log(sqrt(c)*x^2 + sqrt(2)* 
a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(3/4)) - sqrt(2)*(3*c^(3/2)*d^2 - 
2*sqrt(a)*c*d*e + a*sqrt(c)*e^2)*log(sqrt(c)*x^2 - sqrt(2)*a^(1/4)*c^(1/4) 
*x + sqrt(a))/(a^(3/4)*c^(3/4)))/(a*c)
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.41 \[ \int \frac {\left (d+e x^2\right )^2}{\left (a+c x^4\right )^2} \, dx=\frac {2 \, c d e x^{3} + c d^{2} x - a e^{2} x}{4 \, {\left (c x^{4} + a\right )} a c} + \frac {\sqrt {2} {\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} + \left (a c^{3}\right )^{\frac {1}{4}} a c e^{2} + 2 \, \left (a c^{3}\right )^{\frac {3}{4}} d e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{16 \, a^{2} c^{3}} + \frac {\sqrt {2} {\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} + \left (a c^{3}\right )^{\frac {1}{4}} a c e^{2} + 2 \, \left (a c^{3}\right )^{\frac {3}{4}} d e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{16 \, a^{2} c^{3}} + \frac {\sqrt {2} {\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} + \left (a c^{3}\right )^{\frac {1}{4}} a c e^{2} - 2 \, \left (a c^{3}\right )^{\frac {3}{4}} d e\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, a^{2} c^{3}} - \frac {\sqrt {2} {\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} + \left (a c^{3}\right )^{\frac {1}{4}} a c e^{2} - 2 \, \left (a c^{3}\right )^{\frac {3}{4}} d e\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, a^{2} c^{3}} \] Input:

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

Output:

1/4*(2*c*d*e*x^3 + c*d^2*x - a*e^2*x)/((c*x^4 + a)*a*c) + 1/16*sqrt(2)*(3* 
(a*c^3)^(1/4)*c^2*d^2 + (a*c^3)^(1/4)*a*c*e^2 + 2*(a*c^3)^(3/4)*d*e)*arcta 
n(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a^2*c^3) + 1/16*sq 
rt(2)*(3*(a*c^3)^(1/4)*c^2*d^2 + (a*c^3)^(1/4)*a*c*e^2 + 2*(a*c^3)^(3/4)*d 
*e)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a^2*c^3) 
+ 1/32*sqrt(2)*(3*(a*c^3)^(1/4)*c^2*d^2 + (a*c^3)^(1/4)*a*c*e^2 - 2*(a*c^3 
)^(3/4)*d*e)*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^2*c^3) - 1/32 
*sqrt(2)*(3*(a*c^3)^(1/4)*c^2*d^2 + (a*c^3)^(1/4)*a*c*e^2 - 2*(a*c^3)^(3/4 
)*d*e)*log(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^2*c^3)
 

Mupad [B] (verification not implemented)

Time = 0.54 (sec) , antiderivative size = 1565, normalized size of antiderivative = 6.29 \[ \int \frac {\left (d+e x^2\right )^2}{\left (a+c x^4\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

2*atanh((9*c^3*d^4*x*((9*d^4*(-a^7*c^5)^(1/2))/(256*a^7*c^3) - (3*d^3*e)/( 
64*a^3*c) - (d*e^3)/(64*a^2*c^2) + (e^4*(-a^7*c^5)^(1/2))/(256*a^5*c^5) + 
(d^2*e^2*(-a^7*c^5)^(1/2))/(128*a^6*c^4))^(1/2))/(2*((27*d^6*(-a^7*c^5)^(1 
/2))/(32*a^5) - (c*d^3*e^3)/8 - (a*d*e^5)/16 - (9*c^2*d^5*e)/(16*a) + (e^6 
*(-a^7*c^5)^(1/2))/(32*a^2*c^3) + (5*d^2*e^4*(-a^7*c^5)^(1/2))/(32*a^3*c^2 
) + (15*d^4*e^2*(-a^7*c^5)^(1/2))/(32*a^4*c))) + (c*e^4*x*((9*d^4*(-a^7*c^ 
5)^(1/2))/(256*a^7*c^3) - (3*d^3*e)/(64*a^3*c) - (d*e^3)/(64*a^2*c^2) + (e 
^4*(-a^7*c^5)^(1/2))/(256*a^5*c^5) + (d^2*e^2*(-a^7*c^5)^(1/2))/(128*a^6*c 
^4))^(1/2))/(2*((27*d^6*(-a^7*c^5)^(1/2))/(32*a^7) - (d*e^5)/(16*a) - (c*d 
^3*e^3)/(8*a^2) - (9*c^2*d^5*e)/(16*a^3) + (e^6*(-a^7*c^5)^(1/2))/(32*a^4* 
c^3) + (5*d^2*e^4*(-a^7*c^5)^(1/2))/(32*a^5*c^2) + (15*d^4*e^2*(-a^7*c^5)^ 
(1/2))/(32*a^6*c))) + (c^2*d^2*e^2*x*((9*d^4*(-a^7*c^5)^(1/2))/(256*a^7*c^ 
3) - (3*d^3*e)/(64*a^3*c) - (d*e^3)/(64*a^2*c^2) + (e^4*(-a^7*c^5)^(1/2))/ 
(256*a^5*c^5) + (d^2*e^2*(-a^7*c^5)^(1/2))/(128*a^6*c^4))^(1/2))/((27*d^6* 
(-a^7*c^5)^(1/2))/(32*a^6) - (d*e^5)/16 - (c*d^3*e^3)/(8*a) - (9*c^2*d^5*e 
)/(16*a^2) + (e^6*(-a^7*c^5)^(1/2))/(32*a^3*c^3) + (5*d^2*e^4*(-a^7*c^5)^( 
1/2))/(32*a^4*c^2) + (15*d^4*e^2*(-a^7*c^5)^(1/2))/(32*a^5*c)))*((a^2*e^4* 
(-a^7*c^5)^(1/2) + 9*c^2*d^4*(-a^7*c^5)^(1/2) - 12*a^4*c^4*d^3*e - 4*a^5*c 
^3*d*e^3 + 2*a*c*d^2*e^2*(-a^7*c^5)^(1/2))/(256*a^7*c^5))^(1/2) - 2*atanh( 
(9*c^3*d^4*x*(- (d*e^3)/(64*a^2*c^2) - (3*d^3*e)/(64*a^3*c) - (9*d^4*(-...
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 957, normalized size of antiderivative = 3.84 \[ \int \frac {\left (d+e x^2\right )^2}{\left (a+c x^4\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 4*c**(1/4)*a**(3/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c 
)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a*c*d*e - 4*c**(1/4)*a**(3/4)*sqrt(2)*at 
an((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))* 
c**2*d*e*x**4 - 2*c**(3/4)*a**(1/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2 
) - 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a**2*e**2 - 6*c**(3/4)*a**(1 
/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c)*x)/(c**(1/4)*a**(1 
/4)*sqrt(2)))*a*c*d**2 - 2*c**(3/4)*a**(1/4)*sqrt(2)*atan((c**(1/4)*a**(1/ 
4)*sqrt(2) - 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a*c*e**2*x**4 - 6*c 
**(3/4)*a**(1/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c)*x)/(c 
**(1/4)*a**(1/4)*sqrt(2)))*c**2*d**2*x**4 + 4*c**(1/4)*a**(3/4)*sqrt(2)*at 
an((c**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))* 
a*c*d*e + 4*c**(1/4)*a**(3/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) + 2* 
sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*c**2*d*e*x**4 + 2*c**(3/4)*a**(1/4 
)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(c)*x)/(c**(1/4)*a**(1/4 
)*sqrt(2)))*a**2*e**2 + 6*c**(3/4)*a**(1/4)*sqrt(2)*atan((c**(1/4)*a**(1/4 
)*sqrt(2) + 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a*c*d**2 + 2*c**(3/4 
)*a**(1/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(c)*x)/(c**(1/4 
)*a**(1/4)*sqrt(2)))*a*c*e**2*x**4 + 6*c**(3/4)*a**(1/4)*sqrt(2)*atan((c** 
(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*c**2*d* 
*2*x**4 + 2*c**(1/4)*a**(3/4)*sqrt(2)*log( - c**(1/4)*a**(1/4)*sqrt(2)*...