Integrand size = 22, antiderivative size = 348 \[ \int \frac {1}{x^2 \left (d+e x^2\right ) \left (a+c x^4\right )} \, dx=-\frac {1}{a d x}-\frac {e^{5/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2} \left (c d^2+a e^2\right )}+\frac {c^{3/4} \left (\sqrt {c} d+\sqrt {a} e\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{5/4} \left (c d^2+a e^2\right )}-\frac {c^{3/4} \left (\sqrt {c} d+\sqrt {a} e\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{5/4} \left (c d^2+a e^2\right )}-\frac {c^{3/4} \left (\sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{5/4} \left (c d^2+a e^2\right )}+\frac {c^{3/4} \left (\sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{5/4} \left (c d^2+a e^2\right )} \]
-1/a/d/x-e^(5/2)*arctan(x*e^(1/2)/d^(1/2))/d^(3/2)/(a*e^2+c*d^2)-1/8*c^(3/ 4)*ln(-a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(-e*a^(1/2)+d*c^(1/2 ))/a^(5/4)/(a*e^2+c*d^2)*2^(1/2)+1/8*c^(3/4)*ln(a^(1/4)*c^(1/4)*x*2^(1/2)+ a^(1/2)+x^2*c^(1/2))*(-e*a^(1/2)+d*c^(1/2))/a^(5/4)/(a*e^2+c*d^2)*2^(1/2)- 1/4*c^(3/4)*arctan(-1+c^(1/4)*x*2^(1/2)/a^(1/4))*(e*a^(1/2)+d*c^(1/2))/a^( 5/4)/(a*e^2+c*d^2)*2^(1/2)-1/4*c^(3/4)*arctan(1+c^(1/4)*x*2^(1/2)/a^(1/4)) *(e*a^(1/2)+d*c^(1/2))/a^(5/4)/(a*e^2+c*d^2)*2^(1/2)
Time = 0.18 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.12 \[ \int \frac {1}{x^2 \left (d+e x^2\right ) \left (a+c x^4\right )} \, dx=\frac {-8 a^{5/4} e^{5/2} x \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )-\sqrt {d} \left (8 \sqrt [4]{a} c d^2+8 a^{5/4} e^2-2 \sqrt {2} c^{3/4} d \left (\sqrt {c} d+\sqrt {a} e\right ) x \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+2 \sqrt {2} c^{3/4} d \left (\sqrt {c} d+\sqrt {a} e\right ) x \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+\sqrt {2} c^{5/4} d^2 x \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )-\sqrt {2} \sqrt {a} c^{3/4} d e x \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )-\sqrt {2} c^{5/4} d^2 x \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )+\sqrt {2} \sqrt {a} c^{3/4} d e x \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )\right )}{8 a^{5/4} d^{3/2} \left (c d^2+a e^2\right ) x} \]
(-8*a^(5/4)*e^(5/2)*x*ArcTan[(Sqrt[e]*x)/Sqrt[d]] - Sqrt[d]*(8*a^(1/4)*c*d ^2 + 8*a^(5/4)*e^2 - 2*Sqrt[2]*c^(3/4)*d*(Sqrt[c]*d + Sqrt[a]*e)*x*ArcTan[ 1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + 2*Sqrt[2]*c^(3/4)*d*(Sqrt[c]*d + Sqrt[a ]*e)*x*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + Sqrt[2]*c^(5/4)*d^2*x*Log [Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2] - Sqrt[2]*Sqrt[a]*c^(3 /4)*d*e*x*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2] - Sqrt[2] *c^(5/4)*d^2*x*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2] + Sq rt[2]*Sqrt[a]*c^(3/4)*d*e*x*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt [c]*x^2]))/(8*a^(5/4)*d^(3/2)*(c*d^2 + a*e^2)*x)
Time = 0.48 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1611, 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.
| \(\displaystyle \int \frac {1}{x^2 \left (a+c x^4\right ) \left (d+e x^2\right )} \, dx\) |
| \(\Big \downarrow \) 1611 |
| \(\displaystyle \int \left (-\frac {c \left (a e+c d x^2\right )}{a \left (a+c x^4\right ) \left (a e^2+c d^2\right )}-\frac {e^3}{d \left (d+e x^2\right ) \left (a e^2+c d^2\right )}+\frac {1}{a d x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {c^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) \left (\sqrt {a} e+\sqrt {c} d\right )}{2 \sqrt {2} a^{5/4} \left (a e^2+c d^2\right )}-\frac {c^{3/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {a} e+\sqrt {c} d\right )}{2 \sqrt {2} a^{5/4} \left (a e^2+c d^2\right )}-\frac {c^{3/4} \left (\sqrt {c} d-\sqrt {a} e\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{5/4} \left (a e^2+c d^2\right )}+\frac {c^{3/4} \left (\sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{5/4} \left (a e^2+c d^2\right )}-\frac {e^{5/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2} \left (a e^2+c d^2\right )}-\frac {1}{a d x}\) |
-(1/(a*d*x)) - (e^(5/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(d^(3/2)*(c*d^2 + a*e ^2)) + (c^(3/4)*(Sqrt[c]*d + Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^( 1/4)])/(2*Sqrt[2]*a^(5/4)*(c*d^2 + a*e^2)) - (c^(3/4)*(Sqrt[c]*d + Sqrt[a] *e)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(5/4)*(c*d^2 + a *e^2)) - (c^(3/4)*(Sqrt[c]*d - Sqrt[a]*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^ (1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(5/4)*(c*d^2 + a*e^2)) + (c^(3/4)*(Sq rt[c]*d - Sqrt[a]*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2 ])/(4*Sqrt[2]*a^(5/4)*(c*d^2 + a*e^2))
3.3.42.3.1 Defintions of rubi rules used
Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.))/((a_) + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*((d + e*x^2)^q/(a + c*x^4)), x], x] /; FreeQ[{a, c, d, e, f, m}, x] && IntegerQ[q] && IntegerQ[m]
Time = 0.40 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.76
| method | result | size |
| default | \(-\frac {c \left (\frac {e \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8}+\frac {d \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{\left (a \,e^{2}+c \,d^{2}\right ) a}-\frac {e^{3} \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{d \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {e d}}-\frac {1}{a d x}\) | \(266\) |
| risch | \(-\frac {1}{a d x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{7} e^{4}+2 a^{6} c \,d^{2} e^{2}+a^{5} c^{2} d^{4}\right ) \textit {\_Z}^{4}+4 a^{3} c^{2} d e \,\textit {\_Z}^{2}+c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (6 e^{8} d^{3} a^{9}+19 e^{6} d^{5} c \,a^{8}+25 e^{4} d^{7} c^{2} a^{7}+17 e^{2} d^{9} c^{3} a^{6}+5 d^{11} c^{4} a^{5}\right ) \textit {\_R}^{6}+\left (16 a^{7} e^{9}+28 a^{6} c \,d^{2} e^{7}+44 a^{5} c^{2} d^{4} e^{5}+32 a^{4} c^{3} d^{6} e^{3}+16 a^{3} c^{4} d^{8} e \right ) \textit {\_R}^{4}+\left (64 a^{3} c^{2} d \,e^{6}+6 a^{2} c^{3} d^{3} e^{4}+8 a \,c^{4} d^{5} e^{2}+4 c^{5} d^{7}\right ) \textit {\_R}^{2}+16 c^{3} e^{5}\right ) x +\left (4 a^{8} d^{2} e^{8}+4 a^{7} c \,d^{4} e^{6}-3 a^{6} c^{2} d^{6} e^{4}-2 a^{5} c^{3} d^{8} e^{2}+a^{4} c^{4} d^{10}\right ) \textit {\_R}^{5}+\left (4 a^{4} c^{2} d^{3} e^{5}-a^{3} c^{3} d^{5} e^{3}-a^{2} c^{4} d^{7} e \right ) \textit {\_R}^{3}\right )\right )}{4}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{2} d^{3} e^{4}+2 c \,e^{2} a \,d^{5}+c^{2} d^{7}\right ) \textit {\_Z}^{2}+e^{5}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (48 e^{8} d^{3} a^{9}+152 e^{6} d^{5} c \,a^{8}+200 e^{4} d^{7} c^{2} a^{7}+136 e^{2} d^{9} c^{3} a^{6}+40 d^{11} c^{4} a^{5}\right ) \textit {\_R}^{6}+\left (32 a^{7} e^{9}+56 a^{6} c \,d^{2} e^{7}+88 a^{5} c^{2} d^{4} e^{5}+64 a^{4} c^{3} d^{6} e^{3}+32 a^{3} c^{4} d^{8} e \right ) \textit {\_R}^{4}+\left (32 a^{3} c^{2} d \,e^{6}+3 a^{2} c^{3} d^{3} e^{4}+4 a \,c^{4} d^{5} e^{2}+2 c^{5} d^{7}\right ) \textit {\_R}^{2}+2 c^{3} e^{5}\right ) x +\left (16 a^{8} d^{2} e^{8}+16 a^{7} c \,d^{4} e^{6}-12 a^{6} c^{2} d^{6} e^{4}-8 a^{5} c^{3} d^{8} e^{2}+4 a^{4} c^{4} d^{10}\right ) \textit {\_R}^{5}+\left (4 a^{4} c^{2} d^{3} e^{5}-a^{3} c^{3} d^{5} e^{3}-a^{2} c^{4} d^{7} e \right ) \textit {\_R}^{3}\right )\right )}{2}\) | \(731\) |
-c/(a*e^2+c*d^2)/a*(1/8*e*(a/c)^(1/4)*2^(1/2)*(ln((x^2+(a/c)^(1/4)*x*2^(1/ 2)+(a/c)^(1/2))/(x^2-(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2)))+2*arctan(2^(1/2)/ (a/c)^(1/4)*x+1)+2*arctan(2^(1/2)/(a/c)^(1/4)*x-1))+1/8*d/(a/c)^(1/4)*2^(1 /2)*(ln((x^2-(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2))/(x^2+(a/c)^(1/4)*x*2^(1/2) +(a/c)^(1/2)))+2*arctan(2^(1/2)/(a/c)^(1/4)*x+1)+2*arctan(2^(1/2)/(a/c)^(1 /4)*x-1)))-1/d*e^3/(a*e^2+c*d^2)/(e*d)^(1/2)*arctan(e*x/(e*d)^(1/2))-1/a/d /x
Leaf count of result is larger than twice the leaf count of optimal. 2169 vs. \(2 (259) = 518\).
Time = 1.81 (sec) , antiderivative size = 4362, normalized size of antiderivative = 12.53 \[ \int \frac {1}{x^2 \left (d+e x^2\right ) \left (a+c x^4\right )} \, dx=\text {Too large to display} \]
[1/4*(2*a*e^2*x*sqrt(-e/d)*log((e*x^2 - 2*d*x*sqrt(-e/d) - d)/(e*x^2 + d)) + (a*c*d^3 + a^2*d*e^2)*x*sqrt(-(2*c^2*d*e + (a^2*c^2*d^4 + 2*a^3*c*d^2*e ^2 + a^4*e^4)*sqrt(-(c^5*d^4 - 2*a*c^4*d^2*e^2 + a^2*c^3*e^4)/(a^5*c^4*d^8 + 4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*e^4 + 4*a^8*c*d^2*e^6 + a^9*e^8)))/(a ^2*c^2*d^4 + 2*a^3*c*d^2*e^2 + a^4*e^4))*log(-(c^3*d^2 - a*c^2*e^2)*x + (a ^2*c^2*d^2*e - a^3*c*e^3 - (a^4*c^2*d^5 + 2*a^5*c*d^3*e^2 + a^6*d*e^4)*sqr t(-(c^5*d^4 - 2*a*c^4*d^2*e^2 + a^2*c^3*e^4)/(a^5*c^4*d^8 + 4*a^6*c^3*d^6* e^2 + 6*a^7*c^2*d^4*e^4 + 4*a^8*c*d^2*e^6 + a^9*e^8)))*sqrt(-(2*c^2*d*e + (a^2*c^2*d^4 + 2*a^3*c*d^2*e^2 + a^4*e^4)*sqrt(-(c^5*d^4 - 2*a*c^4*d^2*e^2 + a^2*c^3*e^4)/(a^5*c^4*d^8 + 4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*e^4 + 4*a ^8*c*d^2*e^6 + a^9*e^8)))/(a^2*c^2*d^4 + 2*a^3*c*d^2*e^2 + a^4*e^4))) - (a *c*d^3 + a^2*d*e^2)*x*sqrt(-(2*c^2*d*e + (a^2*c^2*d^4 + 2*a^3*c*d^2*e^2 + a^4*e^4)*sqrt(-(c^5*d^4 - 2*a*c^4*d^2*e^2 + a^2*c^3*e^4)/(a^5*c^4*d^8 + 4* a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*e^4 + 4*a^8*c*d^2*e^6 + a^9*e^8)))/(a^2*c^ 2*d^4 + 2*a^3*c*d^2*e^2 + a^4*e^4))*log(-(c^3*d^2 - a*c^2*e^2)*x - (a^2*c^ 2*d^2*e - a^3*c*e^3 - (a^4*c^2*d^5 + 2*a^5*c*d^3*e^2 + a^6*d*e^4)*sqrt(-(c ^5*d^4 - 2*a*c^4*d^2*e^2 + a^2*c^3*e^4)/(a^5*c^4*d^8 + 4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*e^4 + 4*a^8*c*d^2*e^6 + a^9*e^8)))*sqrt(-(2*c^2*d*e + (a^2* c^2*d^4 + 2*a^3*c*d^2*e^2 + a^4*e^4)*sqrt(-(c^5*d^4 - 2*a*c^4*d^2*e^2 + a^ 2*c^3*e^4)/(a^5*c^4*d^8 + 4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*e^4 + 4*a^8...
Timed out. \[ \int \frac {1}{x^2 \left (d+e x^2\right ) \left (a+c x^4\right )} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {1}{x^2 \left (d+e x^2\right ) \left (a+c x^4\right )} \, dx=\text {Exception raised: ValueError} \]
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
Time = 0.28 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.01 \[ \int \frac {1}{x^2 \left (d+e x^2\right ) \left (a+c x^4\right )} \, dx=-\frac {e^{3} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{{\left (c d^{3} + a d e^{2}\right )} \sqrt {d e}} - \frac {{\left (\left (a c^{3}\right )^{\frac {1}{4}} a c e + \left (a c^{3}\right )^{\frac {3}{4}} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{2 \, {\left (\sqrt {2} a^{2} c^{2} d^{2} + \sqrt {2} a^{3} c e^{2}\right )}} - \frac {{\left (\left (a c^{3}\right )^{\frac {1}{4}} a c e + \left (a c^{3}\right )^{\frac {3}{4}} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{2 \, {\left (\sqrt {2} a^{2} c^{2} d^{2} + \sqrt {2} a^{3} c e^{2}\right )}} - \frac {{\left (\left (a c^{3}\right )^{\frac {1}{4}} a c e - \left (a c^{3}\right )^{\frac {3}{4}} d\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{4 \, {\left (\sqrt {2} a^{2} c^{2} d^{2} + \sqrt {2} a^{3} c e^{2}\right )}} + \frac {{\left (\left (a c^{3}\right )^{\frac {1}{4}} a c e - \left (a c^{3}\right )^{\frac {3}{4}} d\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{4 \, {\left (\sqrt {2} a^{2} c^{2} d^{2} + \sqrt {2} a^{3} c e^{2}\right )}} - \frac {1}{a d x} \]
-e^3*arctan(e*x/sqrt(d*e))/((c*d^3 + a*d*e^2)*sqrt(d*e)) - 1/2*((a*c^3)^(1 /4)*a*c*e + (a*c^3)^(3/4)*d)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4) )/(a/c)^(1/4))/(sqrt(2)*a^2*c^2*d^2 + sqrt(2)*a^3*c*e^2) - 1/2*((a*c^3)^(1 /4)*a*c*e + (a*c^3)^(3/4)*d)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4) )/(a/c)^(1/4))/(sqrt(2)*a^2*c^2*d^2 + sqrt(2)*a^3*c*e^2) - 1/4*((a*c^3)^(1 /4)*a*c*e - (a*c^3)^(3/4)*d)*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/ (sqrt(2)*a^2*c^2*d^2 + sqrt(2)*a^3*c*e^2) + 1/4*((a*c^3)^(1/4)*a*c*e - (a* c^3)^(3/4)*d)*log(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(sqrt(2)*a^2*c^ 2*d^2 + sqrt(2)*a^3*c*e^2) - 1/(a*d*x)
Time = 8.68 (sec) , antiderivative size = 5761, normalized size of antiderivative = 16.55 \[ \int \frac {1}{x^2 \left (d+e x^2\right ) \left (a+c x^4\right )} \, dx=\text {Too large to display} \]
atan(((x*(2*a^7*c^7*d^9*e^5 - 4*a^8*c^6*d^7*e^7) - (-(a*e^2*(-a^5*c^3)^(1/ 2) - c*d^2*(-a^5*c^3)^(1/2) + 2*a^3*c^2*d*e)/(16*(a^7*e^4 + a^5*c^2*d^4 + 2*a^6*c*d^2*e^2)))^(1/2)*(((-(a*e^2*(-a^5*c^3)^(1/2) - c*d^2*(-a^5*c^3)^(1 /2) + 2*a^3*c^2*d*e)/(16*(a^7*e^4 + a^5*c^2*d^4 + 2*a^6*c*d^2*e^2)))^(1/2) *(x*(-(a*e^2*(-a^5*c^3)^(1/2) - c*d^2*(-a^5*c^3)^(1/2) + 2*a^3*c^2*d*e)/(1 6*(a^7*e^4 + a^5*c^2*d^4 + 2*a^6*c*d^2*e^2)))^(1/2)*(512*a^11*c^7*d^15*e^3 + 512*a^12*c^6*d^13*e^5 - 512*a^13*c^5*d^11*e^7 - 512*a^14*c^4*d^9*e^9) - 192*a^10*c^7*d^14*e^3 - 128*a^11*c^6*d^12*e^5 + 320*a^12*c^5*d^10*e^7 + 2 56*a^13*c^4*d^8*e^9) + x*(16*a^8*c^8*d^14*e^2 + 32*a^9*c^7*d^12*e^4 - 112* a^10*c^6*d^10*e^6 + 128*a^11*c^5*d^8*e^8))*(-(a*e^2*(-a^5*c^3)^(1/2) - c*d ^2*(-a^5*c^3)^(1/2) + 2*a^3*c^2*d*e)/(16*(a^7*e^4 + a^5*c^2*d^4 + 2*a^6*c* d^2*e^2)))^(1/2) - 4*a^7*c^8*d^13*e^2 - 4*a^8*c^7*d^11*e^4 + 16*a^10*c^5*d ^7*e^8))*(-(a*e^2*(-a^5*c^3)^(1/2) - c*d^2*(-a^5*c^3)^(1/2) + 2*a^3*c^2*d* e)/(16*(a^7*e^4 + a^5*c^2*d^4 + 2*a^6*c*d^2*e^2)))^(1/2)*1i + (x*(2*a^7*c^ 7*d^9*e^5 - 4*a^8*c^6*d^7*e^7) - (-(a*e^2*(-a^5*c^3)^(1/2) - c*d^2*(-a^5*c ^3)^(1/2) + 2*a^3*c^2*d*e)/(16*(a^7*e^4 + a^5*c^2*d^4 + 2*a^6*c*d^2*e^2))) ^(1/2)*(((-(a*e^2*(-a^5*c^3)^(1/2) - c*d^2*(-a^5*c^3)^(1/2) + 2*a^3*c^2*d* e)/(16*(a^7*e^4 + a^5*c^2*d^4 + 2*a^6*c*d^2*e^2)))^(1/2)*(x*(-(a*e^2*(-a^5 *c^3)^(1/2) - c*d^2*(-a^5*c^3)^(1/2) + 2*a^3*c^2*d*e)/(16*(a^7*e^4 + a^5*c ^2*d^4 + 2*a^6*c*d^2*e^2)))^(1/2)*(512*a^11*c^7*d^15*e^3 + 512*a^12*c^6...