Integrand size = 31, antiderivative size = 866 \[ \int \frac {1}{\sqrt {f x} \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx=\frac {c^{3/4} \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {f x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}} \sqrt {f}}\right )}{\sqrt [4]{2} \sqrt {b^2-4 a c} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4} \left (c d^2-b d e+a e^2\right ) \sqrt {f}}-\frac {c^{3/4} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {f x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}} \sqrt {f}}\right )}{\sqrt [4]{2} \sqrt {b^2-4 a c} \left (-b+\sqrt {b^2-4 a c}\right )^{3/4} \left (c d^2-b d e+a e^2\right ) \sqrt {f}}-\frac {e^{7/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {f x}}{\sqrt [4]{d} \sqrt {f}}\right )}{\sqrt {2} d^{3/4} \left (c d^2-b d e+a e^2\right ) \sqrt {f}}+\frac {e^{7/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {f x}}{\sqrt [4]{d} \sqrt {f}}\right )}{\sqrt {2} d^{3/4} \left (c d^2-b d e+a e^2\right ) \sqrt {f}}+\frac {c^{3/4} \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {f x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}} \sqrt {f}}\right )}{\sqrt [4]{2} \sqrt {b^2-4 a c} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4} \left (c d^2-b d e+a e^2\right ) \sqrt {f}}-\frac {c^{3/4} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {f x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}} \sqrt {f}}\right )}{\sqrt [4]{2} \sqrt {b^2-4 a c} \left (-b+\sqrt {b^2-4 a c}\right )^{3/4} \left (c d^2-b d e+a e^2\right ) \sqrt {f}}-\frac {e^{7/4} \log \left (\sqrt {d} \sqrt {f}+\sqrt {e} \sqrt {f} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {f x}\right )}{2 \sqrt {2} d^{3/4} \left (c d^2-b d e+a e^2\right ) \sqrt {f}}+\frac {e^{7/4} \log \left (\sqrt {d} \sqrt {f}+\sqrt {e} \sqrt {f} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {f x}\right )}{2 \sqrt {2} d^{3/4} \left (c d^2-b d e+a e^2\right ) \sqrt {f}} \]
-1/2*e^(7/4)*arctan(1-e^(1/4)*2^(1/2)*(f*x)^(1/2)/d^(1/4)/f^(1/2))/d^(3/4) /(a*e^2-b*d*e+c*d^2)*2^(1/2)/f^(1/2)+1/2*e^(7/4)*arctan(1+e^(1/4)*2^(1/2)* (f*x)^(1/2)/d^(1/4)/f^(1/2))/d^(3/4)/(a*e^2-b*d*e+c*d^2)*2^(1/2)/f^(1/2)-1 /4*e^(7/4)*ln(d^(1/2)*f^(1/2)+x*e^(1/2)*f^(1/2)-d^(1/4)*e^(1/4)*2^(1/2)*(f *x)^(1/2))/d^(3/4)/(a*e^2-b*d*e+c*d^2)*2^(1/2)/f^(1/2)+1/4*e^(7/4)*ln(d^(1 /2)*f^(1/2)+x*e^(1/2)*f^(1/2)+d^(1/4)*e^(1/4)*2^(1/2)*(f*x)^(1/2))/d^(3/4) /(a*e^2-b*d*e+c*d^2)*2^(1/2)/f^(1/2)+1/2*c^(3/4)*arctan(2^(1/4)*c^(1/4)*(f *x)^(1/2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4)/f^(1/2))*(2*c*d-e*(b-(-4*a*c+b^2)^ (1/2)))*2^(3/4)/(a*e^2-b*d*e+c*d^2)/(-b-(-4*a*c+b^2)^(1/2))^(3/4)/(-4*a*c+ b^2)^(1/2)/f^(1/2)+1/2*c^(3/4)*arctanh(2^(1/4)*c^(1/4)*(f*x)^(1/2)/(-b-(-4 *a*c+b^2)^(1/2))^(1/4)/f^(1/2))*(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))*2^(3/4)/( a*e^2-b*d*e+c*d^2)/(-b-(-4*a*c+b^2)^(1/2))^(3/4)/(-4*a*c+b^2)^(1/2)/f^(1/2 )-1/2*c^(3/4)*arctan(2^(1/4)*c^(1/4)*(f*x)^(1/2)/(-b+(-4*a*c+b^2)^(1/2))^( 1/4)/f^(1/2))*(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))*2^(3/4)/(a*e^2-b*d*e+c*d^2) /(-4*a*c+b^2)^(1/2)/(-b+(-4*a*c+b^2)^(1/2))^(3/4)/f^(1/2)-1/2*c^(3/4)*arct anh(2^(1/4)*c^(1/4)*(f*x)^(1/2)/(-b+(-4*a*c+b^2)^(1/2))^(1/4)/f^(1/2))*(2* c*d-e*(b+(-4*a*c+b^2)^(1/2)))*2^(3/4)/(a*e^2-b*d*e+c*d^2)/(-4*a*c+b^2)^(1/ 2)/(-b+(-4*a*c+b^2)^(1/2))^(3/4)/f^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.30 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.25 \[ \int \frac {1}{\sqrt {f x} \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx=-\frac {\sqrt {x} \left (\sqrt {2} e^{7/4} \left (\arctan \left (\frac {\sqrt {d}-\sqrt {e} x}{\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}}\right )-\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}}{\sqrt {d}+\sqrt {e} x}\right )\right )+d^{3/4} \text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {-c d \log \left (\sqrt {x}-\text {$\#$1}\right )+b e \log \left (\sqrt {x}-\text {$\#$1}\right )+c e \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4}{b \text {$\#$1}^3+2 c \text {$\#$1}^7}\&\right ]\right )}{2 d^{3/4} \left (c d^2+e (-b d+a e)\right ) \sqrt {f x}} \]
-1/2*(Sqrt[x]*(Sqrt[2]*e^(7/4)*(ArcTan[(Sqrt[d] - Sqrt[e]*x)/(Sqrt[2]*d^(1 /4)*e^(1/4)*Sqrt[x])] - ArcTanh[(Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[x])/(Sqrt[d] + Sqrt[e]*x)]) + d^(3/4)*RootSum[a + b*#1^4 + c*#1^8 & , (-(c*d*Log[Sqrt[ x] - #1]) + b*e*Log[Sqrt[x] - #1] + c*e*Log[Sqrt[x] - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ]))/(d^(3/4)*(c*d^2 + e*(-(b*d) + a*e))*Sqrt[f*x])
Time = 1.83 (sec) , antiderivative size = 878, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1592, 27, 1754, 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.
| \(\displaystyle \int \frac {1}{\sqrt {f x} \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx\) |
| \(\Big \downarrow \) 1592 |
| \(\displaystyle \frac {2 \int \frac {f^2}{\left (e x^2 f^2+d f^2\right ) \left (c x^4+b x^2+a\right )}d\sqrt {f x}}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 f \int \frac {1}{\left (e x^2 f^2+d f^2\right ) \left (c x^4+b x^2+a\right )}d\sqrt {f x}\) |
| \(\Big \downarrow \) 1754 |
| \(\displaystyle 2 f \int \left (\frac {e^2}{\left (c d^2-b e d+a e^2\right ) \left (e x^2 f^2+d f^2\right )}+\frac {-c e x^2 f^2+c d f^2-b e f^2}{\left (c d^2-b e d+a e^2\right ) \left (c x^4 f^4+b x^2 f^4+a f^4\right )}\right )d\sqrt {f x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 f \left (-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {f x}}{\sqrt [4]{d} \sqrt {f}}\right ) e^{7/4}}{2 \sqrt {2} d^{3/4} \left (c d^2-b e d+a e^2\right ) f^{3/2}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {f x}}{\sqrt [4]{d} \sqrt {f}}+1\right ) e^{7/4}}{2 \sqrt {2} d^{3/4} \left (c d^2-b e d+a e^2\right ) f^{3/2}}-\frac {\log \left (\sqrt {e} x f+\sqrt {d} f-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {f x} \sqrt {f}\right ) e^{7/4}}{4 \sqrt {2} d^{3/4} \left (c d^2-b e d+a e^2\right ) f^{3/2}}+\frac {\log \left (\sqrt {e} x f+\sqrt {d} f+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {f x} \sqrt {f}\right ) e^{7/4}}{4 \sqrt {2} d^{3/4} \left (c d^2-b e d+a e^2\right ) f^{3/2}}+\frac {c^{3/4} \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {f x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}} \sqrt {f}}\right )}{2 \sqrt [4]{2} \sqrt {b^2-4 a c} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4} \left (c d^2-b e d+a e^2\right ) f^{3/2}}-\frac {c^{3/4} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {f x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b} \sqrt {f}}\right )}{2 \sqrt [4]{2} \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}-b\right )^{3/4} \left (c d^2-b e d+a e^2\right ) f^{3/2}}+\frac {c^{3/4} \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {f x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}} \sqrt {f}}\right )}{2 \sqrt [4]{2} \sqrt {b^2-4 a c} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4} \left (c d^2-b e d+a e^2\right ) f^{3/2}}-\frac {c^{3/4} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {f x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b} \sqrt {f}}\right )}{2 \sqrt [4]{2} \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}-b\right )^{3/4} \left (c d^2-b e d+a e^2\right ) f^{3/2}}\right )\) |
2*f*((c^(3/4)*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)*ArcTan[(2^(1/4)*c^(1/4)* Sqrt[f*x])/((-b - Sqrt[b^2 - 4*a*c])^(1/4)*Sqrt[f])])/(2*2^(1/4)*Sqrt[b^2 - 4*a*c]*(-b - Sqrt[b^2 - 4*a*c])^(3/4)*(c*d^2 - b*d*e + a*e^2)*f^(3/2)) - (c^(3/4)*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*ArcTan[(2^(1/4)*c^(1/4)*Sqrt [f*x])/((-b + Sqrt[b^2 - 4*a*c])^(1/4)*Sqrt[f])])/(2*2^(1/4)*Sqrt[b^2 - 4* a*c]*(-b + Sqrt[b^2 - 4*a*c])^(3/4)*(c*d^2 - b*d*e + a*e^2)*f^(3/2)) - (e^ (7/4)*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[f*x])/(d^(1/4)*Sqrt[f])])/(2*Sqrt[2 ]*d^(3/4)*(c*d^2 - b*d*e + a*e^2)*f^(3/2)) + (e^(7/4)*ArcTan[1 + (Sqrt[2]* e^(1/4)*Sqrt[f*x])/(d^(1/4)*Sqrt[f])])/(2*Sqrt[2]*d^(3/4)*(c*d^2 - b*d*e + a*e^2)*f^(3/2)) + (c^(3/4)*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)*ArcTanh[(2 ^(1/4)*c^(1/4)*Sqrt[f*x])/((-b - Sqrt[b^2 - 4*a*c])^(1/4)*Sqrt[f])])/(2*2^ (1/4)*Sqrt[b^2 - 4*a*c]*(-b - Sqrt[b^2 - 4*a*c])^(3/4)*(c*d^2 - b*d*e + a* e^2)*f^(3/2)) - (c^(3/4)*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*ArcTanh[(2^(1 /4)*c^(1/4)*Sqrt[f*x])/((-b + Sqrt[b^2 - 4*a*c])^(1/4)*Sqrt[f])])/(2*2^(1/ 4)*Sqrt[b^2 - 4*a*c]*(-b + Sqrt[b^2 - 4*a*c])^(3/4)*(c*d^2 - b*d*e + a*e^2 )*f^(3/2)) - (e^(7/4)*Log[Sqrt[d]*f + Sqrt[e]*f*x - Sqrt[2]*d^(1/4)*e^(1/4 )*Sqrt[f]*Sqrt[f*x]])/(4*Sqrt[2]*d^(3/4)*(c*d^2 - b*d*e + a*e^2)*f^(3/2)) + (e^(7/4)*Log[Sqrt[d]*f + Sqrt[e]*f*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[f]*S qrt[f*x]])/(4*Sqrt[2]*d^(3/4)*(c*d^2 - b*d*e + a*e^2)*f^(3/2)))
3.4.10.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c _.)*(x_)^4)^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/f Subst[ Int[x^(k*(m + 1) - 1)*(d + e*(x^(2*k)/f^2))^q*(a + b*(x^(2*k)/f^k) + c*(x^( 4*k)/f^4))^p, x], x, (f*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f, p, q}, x ] && NeQ[b^2 - 4*a*c, 0] && FractionQ[m] && IntegerQ[p]
Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_ )), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^n)^q/(a + b*x^n + c*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[q]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.99 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.30
| method | result | size |
| derivativedivides | \(2 f^{5} \left (\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+b \,f^{2} \textit {\_Z}^{4}+a \,f^{4}\right )}{\sum }\frac {\left (-\textit {\_R}^{4} c e -b e \,f^{2}+c d \,f^{2}\right ) \ln \left (\sqrt {f x}-\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b \,f^{2}}}{4 f^{4} \left (a \,e^{2}-b d e +c \,d^{2}\right )}+\frac {e^{2} \left (\frac {d \,f^{2}}{e}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {f x +\left (\frac {d \,f^{2}}{e}\right )^{\frac {1}{4}} \sqrt {f x}\, \sqrt {2}+\sqrt {\frac {d \,f^{2}}{e}}}{f x -\left (\frac {d \,f^{2}}{e}\right )^{\frac {1}{4}} \sqrt {f x}\, \sqrt {2}+\sqrt {\frac {d \,f^{2}}{e}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {f x}}{\left (\frac {d \,f^{2}}{e}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {f x}}{\left (\frac {d \,f^{2}}{e}\right )^{\frac {1}{4}}}-1\right )\right )}{8 f^{6} \left (a \,e^{2}-b d e +c \,d^{2}\right ) d}\right )\) | \(264\) |
| default | \(\frac {\left (\frac {d \,f^{2}}{e}\right )^{\frac {1}{4}} e^{2} \left (2 \arctan \left (\frac {\sqrt {2}\, \sqrt {f x}+\left (\frac {d \,f^{2}}{e}\right )^{\frac {1}{4}}}{\left (\frac {d \,f^{2}}{e}\right )^{\frac {1}{4}}}\right )+\ln \left (\frac {f x +\left (\frac {d \,f^{2}}{e}\right )^{\frac {1}{4}} \sqrt {f x}\, \sqrt {2}+\sqrt {\frac {d \,f^{2}}{e}}}{f x -\left (\frac {d \,f^{2}}{e}\right )^{\frac {1}{4}} \sqrt {f x}\, \sqrt {2}+\sqrt {\frac {d \,f^{2}}{e}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {f x}-\left (\frac {d \,f^{2}}{e}\right )^{\frac {1}{4}}}{\left (\frac {d \,f^{2}}{e}\right )^{\frac {1}{4}}}\right )\right ) \sqrt {2}-2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+b \,f^{2} \textit {\_Z}^{4}+a \,f^{4}\right )}{\sum }\frac {\left (\textit {\_R}^{4} c e +b e \,f^{2}-c d \,f^{2}\right ) \ln \left (\sqrt {f x}-\textit {\_R} \right )}{\textit {\_R}^{3} \left (2 \textit {\_R}^{4} c +b \,f^{2}\right )}\right ) f^{2} d}{4 f \left (a \,e^{2}-b d e +c \,d^{2}\right ) d}\) | \(264\) |
| pseudoelliptic | \(\frac {\left (\frac {d \,f^{2}}{e}\right )^{\frac {1}{4}} e^{2} \left (2 \arctan \left (\frac {\sqrt {2}\, \sqrt {f x}+\left (\frac {d \,f^{2}}{e}\right )^{\frac {1}{4}}}{\left (\frac {d \,f^{2}}{e}\right )^{\frac {1}{4}}}\right )+\ln \left (\frac {f x +\left (\frac {d \,f^{2}}{e}\right )^{\frac {1}{4}} \sqrt {f x}\, \sqrt {2}+\sqrt {\frac {d \,f^{2}}{e}}}{f x -\left (\frac {d \,f^{2}}{e}\right )^{\frac {1}{4}} \sqrt {f x}\, \sqrt {2}+\sqrt {\frac {d \,f^{2}}{e}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {f x}-\left (\frac {d \,f^{2}}{e}\right )^{\frac {1}{4}}}{\left (\frac {d \,f^{2}}{e}\right )^{\frac {1}{4}}}\right )\right ) \sqrt {2}-2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+b \,f^{2} \textit {\_Z}^{4}+a \,f^{4}\right )}{\sum }\frac {\left (\textit {\_R}^{4} c e +b e \,f^{2}-c d \,f^{2}\right ) \ln \left (\sqrt {f x}-\textit {\_R} \right )}{\textit {\_R}^{3} \left (2 \textit {\_R}^{4} c +b \,f^{2}\right )}\right ) f^{2} d}{4 f \left (a \,e^{2}-b d e +c \,d^{2}\right ) d}\) | \(264\) |
2*f^5*(1/4/f^4/(a*e^2-b*d*e+c*d^2)*sum((-_R^4*c*e-b*e*f^2+c*d*f^2)/(2*_R^7 *c+_R^3*b*f^2)*ln((f*x)^(1/2)-_R),_R=RootOf(_Z^8*c+_Z^4*b*f^2+a*f^4))+1/8* e^2/f^6/(a*e^2-b*d*e+c*d^2)*(d*f^2/e)^(1/4)/d*2^(1/2)*(ln((f*x+(d*f^2/e)^( 1/4)*(f*x)^(1/2)*2^(1/2)+(d*f^2/e)^(1/2))/(f*x-(d*f^2/e)^(1/4)*(f*x)^(1/2) *2^(1/2)+(d*f^2/e)^(1/2)))+2*arctan(2^(1/2)/(d*f^2/e)^(1/4)*(f*x)^(1/2)+1) +2*arctan(2^(1/2)/(d*f^2/e)^(1/4)*(f*x)^(1/2)-1)))
Timed out. \[ \int \frac {1}{\sqrt {f x} \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\sqrt {f x} \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx=\int \frac {1}{\sqrt {f x} \left (d + e x^{2}\right ) \left (a + b x^{2} + c x^{4}\right )}\, dx \]
\[ \int \frac {1}{\sqrt {f x} \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx=\int { \frac {1}{{\left (c x^{4} + b x^{2} + a\right )} {\left (e x^{2} + d\right )} \sqrt {f x}} \,d x } \]
-2*e^2*sqrt(x)/(c*d^3*sqrt(f) - b*d^2*e*sqrt(f) + a*d*e^2*sqrt(f)) + 1/4*( 2*sqrt(2)*e^2*arctan(1/2*sqrt(2)*(sqrt(2)*d^(1/4)*e^(1/4) + 2*sqrt(e)*sqrt (x))/sqrt(sqrt(d)*sqrt(e)))/(sqrt(d)*sqrt(sqrt(d)*sqrt(e))) + 2*sqrt(2)*e^ 2*arctan(-1/2*sqrt(2)*(sqrt(2)*d^(1/4)*e^(1/4) - 2*sqrt(e)*sqrt(x))/sqrt(s qrt(d)*sqrt(e)))/(sqrt(d)*sqrt(sqrt(d)*sqrt(e))) + sqrt(2)*e^(7/4)*log(sqr t(2)*d^(1/4)*e^(1/4)*sqrt(x) + sqrt(e)*x + sqrt(d))/d^(3/4) - sqrt(2)*e^(7 /4)*log(-sqrt(2)*d^(1/4)*e^(1/4)*sqrt(x) + sqrt(e)*x + sqrt(d))/d^(3/4))/( c*d^2*sqrt(f) - b*d*e*sqrt(f) + a*e^2*sqrt(f)) + 2*sqrt(x)/(a*d*sqrt(f)) + integrate(-((c^2*d - b*c*e)*x^(7/2) + (b*c*d - b^2*e + a*c*e)*x^(3/2))/(a ^3*e^2*sqrt(f) + (a^2*c*e^2*sqrt(f) + (c^2*d^2*sqrt(f) - b*c*d*e*sqrt(f))* a)*x^4 + (c*d^2*sqrt(f) - b*d*e*sqrt(f))*a^2 + (a^2*b*e^2*sqrt(f) + (b*c*d ^2*sqrt(f) - b^2*d*e*sqrt(f))*a)*x^2), x)
Timed out. \[ \int \frac {1}{\sqrt {f x} \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx=\text {Timed out} \]
Time = 12.01 (sec) , antiderivative size = 43112, normalized size of antiderivative = 49.78 \[ \int \frac {1}{\sqrt {f x} \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx=\text {Too large to display} \]
symsum(log(-root(8388608*a^7*b*c^11*d^18*e*f^6*h^12 - 513802240*a^10*b^2*c ^7*d^11*e^8*f^6*h^12 - 381681664*a^11*b^2*c^6*d^9*e^10*f^6*h^12 - 38168166 4*a^9*b^2*c^8*d^13*e^6*f^6*h^12 - 300941312*a^9*b^5*c^5*d^10*e^9*f^6*h^12 - 300941312*a^8*b^5*c^6*d^12*e^7*f^6*h^12 + 293601280*a^10*b^3*c^6*d^10*e^ 9*f^6*h^12 + 293601280*a^9*b^3*c^7*d^12*e^7*f^6*h^12 - 168820736*a^10*b^5* c^4*d^8*e^11*f^6*h^12 - 168820736*a^7*b^5*c^7*d^14*e^5*f^6*h^12 + 16606822 4*a^8*b^6*c^5*d^11*e^8*f^6*h^12 - 146800640*a^12*b^2*c^5*d^7*e^12*f^6*h^12 - 146800640*a^8*b^2*c^9*d^15*e^4*f^6*h^12 + 124780544*a^10*b^4*c^5*d^9*e^ 10*f^6*h^12 + 124780544*a^8*b^4*c^7*d^13*e^6*f^6*h^12 + 119275520*a^9*b^4* c^6*d^11*e^8*f^6*h^12 + 117440512*a^11*b^3*c^5*d^8*e^11*f^6*h^12 + 1174405 12*a^8*b^3*c^8*d^14*e^5*f^6*h^12 + 102760448*a^9*b^6*c^4*d^9*e^10*f^6*h^12 + 102760448*a^7*b^6*c^6*d^13*e^6*f^6*h^12 + 91750400*a^11*b^4*c^4*d^7*e^1 2*f^6*h^12 + 91750400*a^7*b^4*c^8*d^15*e^4*f^6*h^12 - 71065600*a^7*b^8*c^4 *d^11*e^8*f^6*h^12 - 53444608*a^8*b^8*c^3*d^9*e^10*f^6*h^12 - 53444608*a^6 *b^8*c^5*d^13*e^6*f^6*h^12 + 40370176*a^9*b^7*c^3*d^8*e^11*f^6*h^12 + 4037 0176*a^6*b^7*c^6*d^14*e^5*f^6*h^12 - 36700160*a^11*b^5*c^3*d^6*e^13*f^6*h^ 12 - 36700160*a^6*b^5*c^8*d^16*e^3*f^6*h^12 + 34078720*a^8*b^7*c^4*d^10*e^ 9*f^6*h^12 + 34078720*a^7*b^7*c^5*d^12*e^7*f^6*h^12 + 26214400*a^12*b^4*c^ 3*d^5*e^14*f^6*h^12 + 26214400*a^6*b^4*c^9*d^17*e^2*f^6*h^12 + 22118400*a^ 7*b^9*c^3*d^10*e^9*f^6*h^12 + 22118400*a^6*b^9*c^4*d^12*e^7*f^6*h^12 - ...