\(\int \frac {1}{(d+e x^2)^2 (a+c x^4)} \, dx\) [317]

Optimal result

Integrand size = 19, antiderivative size = 336 \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (a+c x^4\right )} \, dx=\frac {e^2 x}{2 d \left (c d^2+a e^2\right ) \left (d+e x^2\right )}+\frac {e^{3/2} \left (5 c d^2+a e^2\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \left (c d^2+a e^2\right )^2}-\frac {c^{3/4} \left (c d^2-2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^2}+\frac {c^{3/4} \left (c d^2-2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^2}+\frac {c^{3/4} \left (c d^2+2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x}{\sqrt {a}+\sqrt {c} x^2}\right )}{2 \sqrt {2} a^{3/4} \left (c d^2+a e^2\right )^2} \] Output:

1/2*e^2*x/d/(a*e^2+c*d^2)/(e*x^2+d)+1/2*e^(3/2)*(a*e^2+5*c*d^2)*arctan(e^( 
1/2)*x/d^(1/2))/d^(3/2)/(a*e^2+c*d^2)^2+1/4*c^(3/4)*(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^(3/4)/(a*e^2+ 
c*d^2)^2+1/4*c^(3/4)*(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^(3/4)/(a*e^2+c*d^2)^2+1/4*c^(3/4)*(c*d^2+2*a^ 
(1/2)*c^(1/2)*d*e-a*e^2)*arctanh(2^(1/2)*a^(1/4)*c^(1/4)*x/(a^(1/2)+c^(1/2 
)*x^2))*2^(1/2)/a^(3/4)/(a*e^2+c*d^2)^2
 

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (a+c x^4\right )} \, dx=\frac {\frac {4 e^2 \left (c d^2+a e^2\right ) x}{d \left (d+e x^2\right )}+\frac {4 e^{3/2} \left (5 c d^2+a e^2\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2}}+\frac {2 \sqrt {2} c^{3/4} \left (-c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{3/4}}-\frac {2 \sqrt {2} c^{3/4} \left (-c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{3/4}}+\frac {\sqrt {2} c^{3/4} \left (-c d^2-2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{3/4}}+\frac {\sqrt {2} c^{3/4} \left (c d^2+2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{3/4}}}{8 \left (c d^2+a e^2\right )^2} \] Input:

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

Output:

((4*e^2*(c*d^2 + a*e^2)*x)/(d*(d + e*x^2)) + (4*e^(3/2)*(5*c*d^2 + a*e^2)* 
ArcTan[(Sqrt[e]*x)/Sqrt[d]])/d^(3/2) + (2*Sqrt[2]*c^(3/4)*(-(c*d^2) + 2*Sq 
rt[a]*Sqrt[c]*d*e + a*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(3/4 
) - (2*Sqrt[2]*c^(3/4)*(-(c*d^2) + 2*Sqrt[a]*Sqrt[c]*d*e + a*e^2)*ArcTan[1 
 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(3/4) + (Sqrt[2]*c^(3/4)*(-(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 + Sqr 
t[c]*x^2])/a^(3/4) + (Sqrt[2]*c^(3/4)*(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])/a^(3/4))/(8*(c 
*d^2 + a*e^2)^2)
 

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.35, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1485, 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.

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

\(\Big \downarrow \) 1485

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(e^2*x)/(2*d*(c*d^2 + a*e^2)*(d + e*x^2)) + (2*c*Sqrt[d]*e^(3/2)*ArcTan[(S 
qrt[e]*x)/Sqrt[d]])/(c*d^2 + a*e^2)^2 + (e^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d 
]])/(2*d^(3/2)*(c*d^2 + a*e^2)) - (c^(3/4)*(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]*a^(3/4)*(c*d^ 
2 + a*e^2)^2) + (c^(3/4)*(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]*a^(3/4)*(c*d^2 + a*e^2)^2) - (c 
^(3/4)*(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])/(4*Sqrt[2]*a^(3/4)*(c*d^2 + a*e^2)^2) + (c^(3 
/4)*(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])/(4*Sqrt[2]*a^(3/4)*(c*d^2 + a*e^2)^2)
 

Defintions of rubi rules used

rule 1485

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (c_.)*(x_)^4), x_Symbol] :> Int[Expa 
ndIntegrand[(d + e*x^2)^q/(a + c*x^4), x], x] /; FreeQ[{a, c, d, e}, x] && 
NeQ[c*d^2 + a*e^2, 0] && IntegerQ[q]
 

rule 2009

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

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.91

method	result	size
default	\(\frac {e^{2} \left (\frac {\left (a \,e^{2}+c \,d^{2}\right ) x}{2 d \left (e \,x^{2}+d \right )}+\frac {\left (a \,e^{2}+5 c \,d^{2}\right ) \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 d \sqrt {d e}}\right )}{\left (a \,e^{2}+c \,d^{2}\right )^{2}}-\frac {c \left (\frac {\left (a \,e^{2}-c \,d^{2}\right ) \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {e d \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{\left (a \,e^{2}+c \,d^{2}\right )^{2}}\)	\(307\)
risch	\(\text {Expression too large to display}\)	\(2526\)

Input:

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

Output:

e^2/(a*e^2+c*d^2)^2*(1/2*(a*e^2+c*d^2)/d*x/(e*x^2+d)+1/2*(a*e^2+5*c*d^2)/d 
/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2)))-c/(a*e^2+c*d^2)^2*(1/8*(a*e^2-c*d^2) 
*(1/c*a)^(1/4)/a*2^(1/2)*(ln((x^2+(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2))/( 
x^2-(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2)))+2*arctan(2^(1/2)/(1/c*a)^(1/4) 
*x+1)+2*arctan(2^(1/2)/(1/c*a)^(1/4)*x-1))+1/4*e*d/(1/c*a)^(1/4)*2^(1/2)*( 
ln((x^2-(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2))/(x^2+(1/c*a)^(1/4)*x*2^(1/2 
)+(1/c*a)^(1/2)))+2*arctan(2^(1/2)/(1/c*a)^(1/4)*x+1)+2*arctan(2^(1/2)/(1/ 
c*a)^(1/4)*x-1)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4193 vs. \(2 (265) = 530\).

Time = 8.63 (sec) , antiderivative size = 8409, normalized size of antiderivative = 25.03 \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (a+c x^4\right )} \, dx=\text {Too large to display} \] Input:

integrate(1/(e*x^2+d)^2/(c*x^4+a),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (a+c x^4\right )} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 531 vs. \(2 (265) = 530\).

Time = 0.12 (sec) , antiderivative size = 531, normalized size of antiderivative = 1.58 \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (a+c x^4\right )} \, dx=\frac {e^{2} x}{2 \, {\left (c d^{3} + a d e^{2}\right )} {\left (e x^{2} + d\right )}} + \frac {{\left (\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 )}{2 \, {\left (\sqrt {2} a c^{3} d^{4} + 2 \, \sqrt {2} a^{2} c^{2} d^{2} e^{2} + \sqrt {2} a^{3} c e^{4}\right )}} + \frac {{\left (\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 )}{2 \, {\left (\sqrt {2} a c^{3} d^{4} + 2 \, \sqrt {2} a^{2} c^{2} d^{2} e^{2} + \sqrt {2} a^{3} c e^{4}\right )}} + \frac {{\left (\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 )}{4 \, {\left (\sqrt {2} a c^{3} d^{4} + 2 \, \sqrt {2} a^{2} c^{2} d^{2} e^{2} + \sqrt {2} a^{3} c e^{4}\right )}} - \frac {{\left (\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 )}{4 \, {\left (\sqrt {2} a c^{3} d^{4} + 2 \, \sqrt {2} a^{2} c^{2} d^{2} e^{2} + \sqrt {2} a^{3} c e^{4}\right )}} + \frac {{\left (5 \, c d^{2} e^{2} + a e^{4}\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \, {\left (c^{2} d^{5} + 2 \, a c d^{3} e^{2} + a^{2} d e^{4}\right )} \sqrt {d e}} \] Input:

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

Output:

1/2*e^2*x/((c*d^3 + a*d*e^2)*(e*x^2 + d)) + 1/2*((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))/(sqrt(2)*a*c^3*d^4 + 2*sqrt(2)*a^2*c^2*d^2*e 
^2 + sqrt(2)*a^3*c*e^4) + 1/2*((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))/(sqrt(2)*a*c^3*d^4 + 2*sqrt(2)*a^2*c^2*d^2*e^2 + sqrt(2)*a^3*c 
*e^4) + 1/4*((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))/(sqrt(2)*a*c^3*d^4 + 
2*sqrt(2)*a^2*c^2*d^2*e^2 + sqrt(2)*a^3*c*e^4) - 1/4*((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))/(sqrt(2)*a*c^3*d^4 + 2*sqrt(2)*a^2*c^2*d^2*e^2 + sqrt( 
2)*a^3*c*e^4) + 1/2*(5*c*d^2*e^2 + a*e^4)*arctan(e*x/sqrt(d*e))/((c^2*d^5 
+ 2*a*c*d^3*e^2 + a^2*d*e^4)*sqrt(d*e))
 

Mupad [B] (verification not implemented)

Time = 19.32 (sec) , antiderivative size = 16369, normalized size of antiderivative = 48.72 \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (a+c x^4\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

(e^2*x)/(2*d*(d + e*x^2)*(a*e^2 + c*d^2)) - atan(((((((256*a^8*c^4*d*e^16 
- 128*a*c^11*d^15*e^2 + 256*a^2*c^10*d^13*e^4 + 3456*a^3*c^9*d^11*e^6 + 89 
60*a^4*c^8*d^9*e^8 + 10880*a^5*c^7*d^7*e^10 + 6912*a^6*c^6*d^5*e^12 + 2176 
*a^7*c^5*d^3*e^14)/(2*(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c* 
d^4*e^6 + 6*a^2*c^2*d^6*e^4)) + (x*((a^2*e^4*(-a^3*c^3)^(1/2) + c^2*d^4*(- 
a^3*c^3)^(1/2) + 4*a^2*c^3*d^3*e - 4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c 
^3)^(1/2))/(16*(a^7*e^8 + a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*a^4*c^3*d^6*e^ 
2 + 6*a^5*c^2*d^4*e^4)))^(1/2)*(512*a^2*c^11*d^16*e^3 + 2560*a^3*c^10*d^14 
*e^5 + 4608*a^4*c^9*d^12*e^7 + 2560*a^5*c^8*d^10*e^9 - 2560*a^6*c^7*d^8*e^ 
11 - 4608*a^7*c^6*d^6*e^13 - 2560*a^8*c^5*d^4*e^15 - 512*a^9*c^4*d^2*e^17) 
)/(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 + 6*a^2*c^2* 
d^6*e^4))*((a^2*e^4*(-a^3*c^3)^(1/2) + c^2*d^4*(-a^3*c^3)^(1/2) + 4*a^2*c^ 
3*d^3*e - 4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^3)^(1/2))/(16*(a^7*e^8 + 
 a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*a^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4)))^ 
(1/2) + (x*(32*a^6*c^5*d*e^14 - 48*a*c^10*d^11*e^4 - 16*c^11*d^13*e^2 + 10 
24*a^2*c^9*d^9*e^6 + 2208*a^3*c^8*d^7*e^8 + 1264*a^4*c^7*d^5*e^10 + 144*a^ 
5*c^6*d^3*e^12))/(c^4*d^10 + a^4*d^2*e^8 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e 
^6 + 6*a^2*c^2*d^6*e^4))*((a^2*e^4*(-a^3*c^3)^(1/2) + c^2*d^4*(-a^3*c^3)^( 
1/2) + 4*a^2*c^3*d^3*e - 4*a^3*c^2*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^3)^(1/2)) 
/(16*(a^7*e^8 + a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*a^4*c^3*d^6*e^2 + 6*a...
 

Reduce [B] (verification not implemented)

Time = 16.34 (sec) , antiderivative size = 1155, normalized size of antiderivative = 3.44 \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (a+c x^4\right )} \, dx =\text {Too large to display} \] Input:

int(1/(e*x^2+d)^2/(c*x^4+a),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)))*c*d**4*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* 
d**3*e**2*x**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*d**3*e**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*d**2*e**3*x**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)))*c*d**5 - 
 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)))*c*d**4*e*x**2 - 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*d**4*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*d**3*e**2*x**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*d**3*e**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*d**2*e**3 
*x**2 + 2*c**(3/4)*a**(1/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) + 2*sq 
rt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*c*d**5 + 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) 
))*c*d**4*e*x**2 + 4*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a**2...