Integrand size = 22, antiderivative size = 345 \[ \int \frac {x^6}{\left (d+e x^2\right ) \left (a+c x^4\right )} \, dx=\frac {x}{c e}-\frac {d^{5/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{3/2} \left (c d^2+a e^2\right )}+\frac {a^{3/4} \left (\sqrt {c} d+\sqrt {a} e\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} c^{5/4} \left (c d^2+a e^2\right )}-\frac {a^{3/4} \left (\sqrt {c} d+\sqrt {a} e\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} c^{5/4} \left (c d^2+a e^2\right )}-\frac {a^{3/4} \left (\sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} c^{5/4} \left (c d^2+a e^2\right )}+\frac {a^{3/4} \left (\sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} c^{5/4} \left (c d^2+a e^2\right )} \]
x/c/e-d^(5/2)*arctan(x*e^(1/2)/d^(1/2))/e^(3/2)/(a*e^2+c*d^2)-1/8*a^(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))/ c^(5/4)/(a*e^2+c*d^2)*2^(1/2)+1/8*a^(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))/c^(5/4)/(a*e^2+c*d^2)*2^(1/2)-1/4 *a^(3/4)*arctan(-1+c^(1/4)*x*2^(1/2)/a^(1/4))*(e*a^(1/2)+d*c^(1/2))/c^(5/4 )/(a*e^2+c*d^2)*2^(1/2)-1/4*a^(3/4)*arctan(1+c^(1/4)*x*2^(1/2)/a^(1/4))*(e *a^(1/2)+d*c^(1/2))/c^(5/4)/(a*e^2+c*d^2)*2^(1/2)
Time = 0.17 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.08 \[ \int \frac {x^6}{\left (d+e x^2\right ) \left (a+c x^4\right )} \, dx=\frac {x}{c e}-\frac {d^{5/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{3/2} \left (c d^2+a e^2\right )}-\frac {\left (a^{3/4} c d+a^{5/4} \sqrt {c} e\right ) \arctan \left (\frac {-\sqrt {2} \sqrt [4]{a}+2 \sqrt [4]{c} x}{\sqrt {2} \sqrt [4]{a}}\right )}{2 \sqrt {2} c^{7/4} \left (c d^2+a e^2\right )}-\frac {\left (a^{3/4} c d+a^{5/4} \sqrt {c} e\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a}+2 \sqrt [4]{c} x}{\sqrt {2} \sqrt [4]{a}}\right )}{2 \sqrt {2} c^{7/4} \left (c d^2+a e^2\right )}-\frac {\left (a^{3/4} c d-a^{5/4} \sqrt {c} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} c^{7/4} \left (c d^2+a e^2\right )}+\frac {\left (a^{3/4} c d-a^{5/4} \sqrt {c} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} c^{7/4} \left (c d^2+a e^2\right )} \]
x/(c*e) - (d^(5/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(e^(3/2)*(c*d^2 + a*e^2)) - ((a^(3/4)*c*d + a^(5/4)*Sqrt[c]*e)*ArcTan[(-(Sqrt[2]*a^(1/4)) + 2*c^(1/4 )*x)/(Sqrt[2]*a^(1/4))])/(2*Sqrt[2]*c^(7/4)*(c*d^2 + a*e^2)) - ((a^(3/4)*c *d + a^(5/4)*Sqrt[c]*e)*ArcTan[(Sqrt[2]*a^(1/4) + 2*c^(1/4)*x)/(Sqrt[2]*a^ (1/4))])/(2*Sqrt[2]*c^(7/4)*(c*d^2 + a*e^2)) - ((a^(3/4)*c*d - a^(5/4)*Sqr t[c]*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2] *c^(7/4)*(c*d^2 + a*e^2)) + ((a^(3/4)*c*d - a^(5/4)*Sqrt[c]*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*c^(7/4)*(c*d^2 + a *e^2))
Time = 0.47 (sec) , antiderivative size = 345, 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.
| \(\displaystyle \int \frac {x^6}{\left (a+c x^4\right ) \left (d+e x^2\right )} \, dx\) |
| \(\Big \downarrow \) 1611 |
| \(\displaystyle \int \left (-\frac {a \left (a e+c d x^2\right )}{c \left (a+c x^4\right ) \left (a e^2+c d^2\right )}-\frac {d^3}{e \left (d+e x^2\right ) \left (a e^2+c d^2\right )}+\frac {1}{c e}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) \left (\sqrt {a} e+\sqrt {c} d\right )}{2 \sqrt {2} c^{5/4} \left (a e^2+c d^2\right )}-\frac {a^{3/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {a} e+\sqrt {c} d\right )}{2 \sqrt {2} c^{5/4} \left (a e^2+c d^2\right )}-\frac {a^{3/4} \left (\sqrt {c} d-\sqrt {a} e\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} c^{5/4} \left (a e^2+c d^2\right )}+\frac {a^{3/4} \left (\sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} c^{5/4} \left (a e^2+c d^2\right )}-\frac {d^{5/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{3/2} \left (a e^2+c d^2\right )}+\frac {x}{c e}\) |
x/(c*e) - (d^(5/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(e^(3/2)*(c*d^2 + a*e^2)) + (a^(3/4)*(Sqrt[c]*d + Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] )/(2*Sqrt[2]*c^(5/4)*(c*d^2 + a*e^2)) - (a^(3/4)*(Sqrt[c]*d + Sqrt[a]*e)*A rcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*c^(5/4)*(c*d^2 + a*e^2) ) - (a^(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]*c^(5/4)*(c*d^2 + a*e^2)) + (a^(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]*c^(5/4)*(c*d^2 + a*e^2))
3.3.38.3.1 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 = 0.41 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.76
| method | result | size |
| default | \(\frac {x}{c e}-\frac {a \left (\frac {e \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8}+\frac {d \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{\left (a \,e^{2}+c \,d^{2}\right ) c}-\frac {d^{3} \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{e \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {e d}}\) | \(263\) |
| risch | \(\frac {x}{c e}+\frac {\sqrt {-e d}\, d^{2} \ln \left (\left (-16 \left (-e d \right )^{\frac {5}{2}} a \,c^{5} d^{8} e^{2}+16 \left (-e d \right )^{\frac {5}{2}} c^{6} d^{10}-14 a^{3} c^{3} d^{5} e^{7} \left (-e d \right )^{\frac {3}{2}}+4 a^{2} c^{4} d^{7} e^{5} \left (-e d \right )^{\frac {3}{2}}+2 a \,c^{5} d^{9} \left (-e d \right )^{\frac {3}{2}} e^{3}+16 c^{6} d^{11} \left (-e d \right )^{\frac {3}{2}} e -\sqrt {-e d}\, a^{6} e^{14}-2 \sqrt {-e d}\, a^{5} c \,d^{2} e^{12}-\sqrt {-e d}\, a^{4} c^{2} d^{4} e^{10}+2 \sqrt {-e d}\, a^{3} c^{3} d^{6} e^{8}+4 \sqrt {-e d}\, a^{2} c^{4} d^{8} e^{6}+2 \sqrt {-e d}\, a \,c^{5} d^{10} e^{4}\right ) x -a^{6} d \,e^{14}-2 a^{5} c \,d^{3} e^{12}-a^{4} c^{2} d^{5} e^{10}+16 a^{3} c^{3} d^{7} e^{8}-16 a \,c^{5} d^{11} e^{4}\right )}{2 e^{2} \left (a \,e^{2}+c \,d^{2}\right )}-\frac {\sqrt {-e d}\, d^{2} \ln \left (\left (16 \left (-e d \right )^{\frac {5}{2}} a \,c^{5} d^{8} e^{2}-16 \left (-e d \right )^{\frac {5}{2}} c^{6} d^{10}+14 a^{3} c^{3} d^{5} e^{7} \left (-e d \right )^{\frac {3}{2}}-4 a^{2} c^{4} d^{7} e^{5} \left (-e d \right )^{\frac {3}{2}}-2 a \,c^{5} d^{9} \left (-e d \right )^{\frac {3}{2}} e^{3}-16 c^{6} d^{11} \left (-e d \right )^{\frac {3}{2}} e +\sqrt {-e d}\, a^{6} e^{14}+2 \sqrt {-e d}\, a^{5} c \,d^{2} e^{12}+\sqrt {-e d}\, a^{4} c^{2} d^{4} e^{10}-2 \sqrt {-e d}\, a^{3} c^{3} d^{6} e^{8}-4 \sqrt {-e d}\, a^{2} c^{4} d^{8} e^{6}-2 \sqrt {-e d}\, a \,c^{5} d^{10} e^{4}\right ) x -a^{6} d \,e^{14}-2 a^{5} c \,d^{3} e^{12}-a^{4} c^{2} d^{5} e^{10}+16 a^{3} c^{3} d^{7} e^{8}-16 a \,c^{5} d^{11} e^{4}\right )}{2 e^{2} \left (a \,e^{2}+c \,d^{2}\right )}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{2} c \,e^{4}+2 a \,c^{2} d^{2} e^{2}+c^{3} d^{4}\right ) \textit {\_Z}^{4}+4 a^{2} c d \,e^{3} \textit {\_Z}^{2}+e^{4} a^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-2 a^{3} c \,e^{7}-2 a^{2} c^{2} d^{2} e^{5}+2 a \,c^{3} d^{4} e^{3}+2 c^{4} d^{6} e \right ) \textit {\_R}^{5}+\left (-7 a^{3} c d \,e^{6}+2 a^{2} c^{2} d^{3} e^{4}+a \,c^{3} d^{5} e^{2}+8 c^{4} d^{7}\right ) \textit {\_R}^{3}+\left (-2 a^{4} e^{7}+4 a \,c^{3} d^{6} e \right ) \textit {\_R} \right ) x +\left (-a^{3} c d \,e^{6}-2 a^{2} c^{2} d^{3} e^{4}-a \,c^{3} d^{5} e^{2}\right ) \textit {\_R}^{4}+\left (a^{3} c \,d^{2} e^{5}-15 a^{2} c^{2} d^{4} e^{3}+12 a \,c^{3} d^{6} e \right ) \textit {\_R}^{2}-4 a^{3} c \,d^{3} e^{4}+4 a^{2} c^{2} d^{5} e^{2}\right )}{4 e c}\) | \(921\) |
x/c/e-a/(a*e^2+c*d^2)/c*(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/e*d^3/(a*e^2+c*d^2)/(e*d)^(1/2)*arctan(e*x/(e*d)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 2167 vs. \(2 (256) = 512\).
Time = 1.08 (sec) , antiderivative size = 4354, normalized size of antiderivative = 12.62 \[ \int \frac {x^6}{\left (d+e x^2\right ) \left (a+c x^4\right )} \, dx=\text {Too large to display} \]
[1/4*(2*c*d^2*sqrt(-d/e)*log((e*x^2 - 2*e*x*sqrt(-d/e) - d)/(e*x^2 + d)) + (c^2*d^2*e + a*c*e^3)*sqrt(-(2*a^2*d*e + (c^4*d^4 + 2*a*c^3*d^2*e^2 + a^2 *c^2*e^4)*sqrt(-(a^3*c^2*d^4 - 2*a^4*c*d^2*e^2 + a^5*e^4)/(c^9*d^8 + 4*a*c ^8*d^6*e^2 + 6*a^2*c^7*d^4*e^4 + 4*a^3*c^6*d^2*e^6 + a^4*c^5*e^8)))/(c^4*d ^4 + 2*a*c^3*d^2*e^2 + a^2*c^2*e^4))*log(-(a^2*c*d^2 - a^3*e^2)*x + (a^2*c ^2*d^2*e - a^3*c*e^3 - (c^6*d^5 + 2*a*c^5*d^3*e^2 + a^2*c^4*d*e^4)*sqrt(-( a^3*c^2*d^4 - 2*a^4*c*d^2*e^2 + a^5*e^4)/(c^9*d^8 + 4*a*c^8*d^6*e^2 + 6*a^ 2*c^7*d^4*e^4 + 4*a^3*c^6*d^2*e^6 + a^4*c^5*e^8)))*sqrt(-(2*a^2*d*e + (c^4 *d^4 + 2*a*c^3*d^2*e^2 + a^2*c^2*e^4)*sqrt(-(a^3*c^2*d^4 - 2*a^4*c*d^2*e^2 + a^5*e^4)/(c^9*d^8 + 4*a*c^8*d^6*e^2 + 6*a^2*c^7*d^4*e^4 + 4*a^3*c^6*d^2 *e^6 + a^4*c^5*e^8)))/(c^4*d^4 + 2*a*c^3*d^2*e^2 + a^2*c^2*e^4))) - (c^2*d ^2*e + a*c*e^3)*sqrt(-(2*a^2*d*e + (c^4*d^4 + 2*a*c^3*d^2*e^2 + a^2*c^2*e^ 4)*sqrt(-(a^3*c^2*d^4 - 2*a^4*c*d^2*e^2 + a^5*e^4)/(c^9*d^8 + 4*a*c^8*d^6* e^2 + 6*a^2*c^7*d^4*e^4 + 4*a^3*c^6*d^2*e^6 + a^4*c^5*e^8)))/(c^4*d^4 + 2* a*c^3*d^2*e^2 + a^2*c^2*e^4))*log(-(a^2*c*d^2 - a^3*e^2)*x - (a^2*c^2*d^2* e - a^3*c*e^3 - (c^6*d^5 + 2*a*c^5*d^3*e^2 + a^2*c^4*d*e^4)*sqrt(-(a^3*c^2 *d^4 - 2*a^4*c*d^2*e^2 + a^5*e^4)/(c^9*d^8 + 4*a*c^8*d^6*e^2 + 6*a^2*c^7*d ^4*e^4 + 4*a^3*c^6*d^2*e^6 + a^4*c^5*e^8)))*sqrt(-(2*a^2*d*e + (c^4*d^4 + 2*a*c^3*d^2*e^2 + a^2*c^2*e^4)*sqrt(-(a^3*c^2*d^4 - 2*a^4*c*d^2*e^2 + a^5* e^4)/(c^9*d^8 + 4*a*c^8*d^6*e^2 + 6*a^2*c^7*d^4*e^4 + 4*a^3*c^6*d^2*e^6...
Timed out. \[ \int \frac {x^6}{\left (d+e x^2\right ) \left (a+c x^4\right )} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {x^6}{\left (d+e x^2\right ) \left (a+c x^4\right )} \, dx=\text {Exception raised: ValueError} \]
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 = 338, normalized size of antiderivative = 0.98 \[ \int \frac {x^6}{\left (d+e x^2\right ) \left (a+c x^4\right )} \, dx=-\frac {d^{3} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{{\left (c d^{2} e + a e^{3}\right )} \sqrt {d e}} - \frac {{\left (\left (a c^{3}\right )^{\frac {1}{4}} a c e + \left (a c^{3}\right )^{\frac {3}{4}} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{2 \, {\left (\sqrt {2} c^{4} d^{2} + \sqrt {2} a c^{3} e^{2}\right )}} - \frac {{\left (\left (a c^{3}\right )^{\frac {1}{4}} a c e + \left (a c^{3}\right )^{\frac {3}{4}} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{2 \, {\left (\sqrt {2} c^{4} d^{2} + \sqrt {2} a c^{3} e^{2}\right )}} - \frac {{\left (\left (a c^{3}\right )^{\frac {1}{4}} a c e - \left (a c^{3}\right )^{\frac {3}{4}} d\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{4 \, {\left (\sqrt {2} c^{4} d^{2} + \sqrt {2} a c^{3} e^{2}\right )}} + \frac {{\left (\left (a c^{3}\right )^{\frac {1}{4}} a c e - \left (a c^{3}\right )^{\frac {3}{4}} d\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{4 \, {\left (\sqrt {2} c^{4} d^{2} + \sqrt {2} a c^{3} e^{2}\right )}} + \frac {x}{c e} \]
-d^3*arctan(e*x/sqrt(d*e))/((c*d^2*e + a*e^3)*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)*c^4*d^2 + sqrt(2)*a*c^3*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)*c^4*d^2 + sqrt(2)*a*c^3*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) *c^4*d^2 + sqrt(2)*a*c^3*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)*c^4*d^2 + sqrt(2)* a*c^3*e^2) + x/(c*e)
Time = 8.44 (sec) , antiderivative size = 5908, normalized size of antiderivative = 17.12 \[ \int \frac {x^6}{\left (d+e x^2\right ) \left (a+c x^4\right )} \, dx=\text {Too large to display} \]
atan(((((((64*a^5*c^4*d*e^8 + 64*a^3*c^6*d^5*e^4 + 128*a^4*c^5*d^3*e^6)/(c *e) - (2*x*(-(a*e^2*(-a^3*c^5)^(1/2) - c*d^2*(-a^3*c^5)^(1/2) + 2*a^2*c^3* d*e)/(16*(c^7*d^4 + a^2*c^5*e^4 + 2*a*c^6*d^2*e^2)))^(1/2)*(256*a^5*c^5*e^ 10 - 256*a^2*c^8*d^6*e^4 - 256*a^3*c^7*d^4*e^6 + 256*a^4*c^6*d^2*e^8))/(c* e))*(-(a*e^2*(-a^3*c^5)^(1/2) - c*d^2*(-a^3*c^5)^(1/2) + 2*a^2*c^3*d*e)/(1 6*(c^7*d^4 + a^2*c^5*e^4 + 2*a*c^6*d^2*e^2)))^(1/2) + (2*x*(64*a^2*c^6*d^7 *e - 56*a^5*c^3*d*e^7 + 8*a^3*c^5*d^5*e^3 + 16*a^4*c^4*d^3*e^5))/(c*e))*(- (a*e^2*(-a^3*c^5)^(1/2) - c*d^2*(-a^3*c^5)^(1/2) + 2*a^2*c^3*d*e)/(16*(c^7 *d^4 + a^2*c^5*e^4 + 2*a*c^6*d^2*e^2)))^(1/2) - (48*a^3*c^4*d^6*e - 60*a^4 *c^3*d^4*e^3 + 4*a^5*c^2*d^2*e^5)/(c*e))*(-(a*e^2*(-a^3*c^5)^(1/2) - c*d^2 *(-a^3*c^5)^(1/2) + 2*a^2*c^3*d*e)/(16*(c^7*d^4 + a^2*c^5*e^4 + 2*a*c^6*d^ 2*e^2)))^(1/2) - (2*x*(a^6*e^6 - 2*a^3*c^3*d^6))/(c*e))*(-(a*e^2*(-a^3*c^5 )^(1/2) - c*d^2*(-a^3*c^5)^(1/2) + 2*a^2*c^3*d*e)/(16*(c^7*d^4 + a^2*c^5*e ^4 + 2*a*c^6*d^2*e^2)))^(1/2)*1i - (((((64*a^5*c^4*d*e^8 + 64*a^3*c^6*d^5* e^4 + 128*a^4*c^5*d^3*e^6)/(c*e) + (2*x*(-(a*e^2*(-a^3*c^5)^(1/2) - c*d^2* (-a^3*c^5)^(1/2) + 2*a^2*c^3*d*e)/(16*(c^7*d^4 + a^2*c^5*e^4 + 2*a*c^6*d^2 *e^2)))^(1/2)*(256*a^5*c^5*e^10 - 256*a^2*c^8*d^6*e^4 - 256*a^3*c^7*d^4*e^ 6 + 256*a^4*c^6*d^2*e^8))/(c*e))*(-(a*e^2*(-a^3*c^5)^(1/2) - c*d^2*(-a^3*c ^5)^(1/2) + 2*a^2*c^3*d*e)/(16*(c^7*d^4 + a^2*c^5*e^4 + 2*a*c^6*d^2*e^2))) ^(1/2) - (2*x*(64*a^2*c^6*d^7*e - 56*a^5*c^3*d*e^7 + 8*a^3*c^5*d^5*e^3 ...