Integrand size = 17, antiderivative size = 247 \[ \int \frac {d+e x^2}{a+c x^4} \, dx=-\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} c^{3/4}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} c^{3/4}}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} c^{3/4}}+\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} c^{3/4}} \]
-1/8*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^(3/4)/c^(3/4)*2^(1/2)+1/8*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^(3/4)/c^(3/4)*2^(1/2)+1/4*arctan(-1+c^(1 /4)*x*2^(1/2)/a^(1/4))*(e*a^(1/2)+d*c^(1/2))/a^(3/4)/c^(3/4)*2^(1/2)+1/4*a rctan(1+c^(1/4)*x*2^(1/2)/a^(1/4))*(e*a^(1/2)+d*c^(1/2))/a^(3/4)/c^(3/4)*2 ^(1/2)
Time = 0.03 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.74 \[ \int \frac {d+e x^2}{a+c x^4} \, dx=\frac {-2 \left (\sqrt {c} d+\sqrt {a} e\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+2 \left (\sqrt {c} d+\sqrt {a} e\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )-\left (\sqrt {c} d-\sqrt {a} e\right ) \left (\log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )-\log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )\right )}{4 \sqrt {2} a^{3/4} c^{3/4}} \]
(-2*(Sqrt[c]*d + Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + 2*(S qrt[c]*d + Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)] - (Sqrt[c]*d - Sqrt[a]*e)*(Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2] - Lo g[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2]))/(4*Sqrt[2]*a^(3/4)* c^(3/4))
Time = 0.43 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.93, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {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.
| \(\displaystyle \int \frac {d+e x^2}{a+c x^4} \, dx\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right ) \int \frac {\sqrt {c} \left (\sqrt {a}-\sqrt {c} x^2\right )}{c x^4+a}dx}{2 c}+\frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right ) \int \frac {\sqrt {c} \left (\sqrt {c} x^2+\sqrt {a}\right )}{c x^4+a}dx}{2 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {c}}+\frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{c x^4+a}dx}{2 \sqrt {c}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right ) \left (\frac {\int \frac {1}{x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {c}}\right )}{2 \sqrt {c}}+\frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {c}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {c}}+\frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right ) \left (\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )^2-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )^2-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{c x^4+a}dx}{2 \sqrt {c}}+\frac {\left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right ) \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right )}{2 \sqrt {c}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} x}{\sqrt [4]{c} \left (x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} x+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}+\frac {\left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right ) \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right )}{2 \sqrt {c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} x}{\sqrt [4]{c} \left (x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} x+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}+\frac {\left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right ) \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right )}{2 \sqrt {c}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} x}{x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{c} x+\sqrt [4]{a}}{x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt [4]{a} \sqrt {c}}\right )}{2 \sqrt {c}}+\frac {\left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right ) \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right )}{2 \sqrt {c}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right ) \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right )}{2 \sqrt {c}}+\frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right ) \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}\) |
(((Sqrt[c]*d)/Sqrt[a] + e)*(-(ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)]/(Sqr t[2]*a^(1/4)*c^(1/4))) + ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)]/(Sqrt[2]* a^(1/4)*c^(1/4))))/(2*Sqrt[c]) + (((Sqrt[c]*d)/Sqrt[a] - e)*(-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[c])
3.2.40.3.1 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[((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])
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]
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]
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]
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]
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.22 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.14
| method | result | size |
| risch | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+a \right )}{\sum }\frac {\left (\textit {\_R}^{2} e +d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{4 c}\) | \(34\) |
| default | \(\frac {d \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {e \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}\) | \(206\) |
Leaf count of result is larger than twice the leaf count of optimal. 767 vs. \(2 (166) = 332\).
Time = 0.30 (sec) , antiderivative size = 767, normalized size of antiderivative = 3.11 \[ \int \frac {d+e x^2}{a+c x^4} \, dx=-\frac {1}{4} \, \sqrt {-\frac {a c \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} + 2 \, d e}{a c}} \log \left (-{\left (c^{2} d^{4} - a^{2} e^{4}\right )} x + {\left (a^{3} c^{2} e \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} + a c^{2} d^{3} - a^{2} c d e^{2}\right )} \sqrt {-\frac {a c \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} + 2 \, d e}{a c}}\right ) + \frac {1}{4} \, \sqrt {-\frac {a c \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} + 2 \, d e}{a c}} \log \left (-{\left (c^{2} d^{4} - a^{2} e^{4}\right )} x - {\left (a^{3} c^{2} e \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} + a c^{2} d^{3} - a^{2} c d e^{2}\right )} \sqrt {-\frac {a c \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} + 2 \, d e}{a c}}\right ) + \frac {1}{4} \, \sqrt {\frac {a c \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} - 2 \, d e}{a c}} \log \left (-{\left (c^{2} d^{4} - a^{2} e^{4}\right )} x + {\left (a^{3} c^{2} e \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} - a c^{2} d^{3} + a^{2} c d e^{2}\right )} \sqrt {\frac {a c \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} - 2 \, d e}{a c}}\right ) - \frac {1}{4} \, \sqrt {\frac {a c \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} - 2 \, d e}{a c}} \log \left (-{\left (c^{2} d^{4} - a^{2} e^{4}\right )} x - {\left (a^{3} c^{2} e \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} - a c^{2} d^{3} + a^{2} c d e^{2}\right )} \sqrt {\frac {a c \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} - 2 \, d e}{a c}}\right ) \]
-1/4*sqrt(-(a*c*sqrt(-(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)/(a^3*c^3)) + 2*d *e)/(a*c))*log(-(c^2*d^4 - a^2*e^4)*x + (a^3*c^2*e*sqrt(-(c^2*d^4 - 2*a*c* d^2*e^2 + a^2*e^4)/(a^3*c^3)) + a*c^2*d^3 - a^2*c*d*e^2)*sqrt(-(a*c*sqrt(- (c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)/(a^3*c^3)) + 2*d*e)/(a*c))) + 1/4*sqrt (-(a*c*sqrt(-(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)/(a^3*c^3)) + 2*d*e)/(a*c) )*log(-(c^2*d^4 - a^2*e^4)*x - (a^3*c^2*e*sqrt(-(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)/(a^3*c^3)) + a*c^2*d^3 - a^2*c*d*e^2)*sqrt(-(a*c*sqrt(-(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)/(a^3*c^3)) + 2*d*e)/(a*c))) + 1/4*sqrt((a*c*sqr t(-(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)/(a^3*c^3)) - 2*d*e)/(a*c))*log(-(c^ 2*d^4 - a^2*e^4)*x + (a^3*c^2*e*sqrt(-(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)/ (a^3*c^3)) - a*c^2*d^3 + a^2*c*d*e^2)*sqrt((a*c*sqrt(-(c^2*d^4 - 2*a*c*d^2 *e^2 + a^2*e^4)/(a^3*c^3)) - 2*d*e)/(a*c))) - 1/4*sqrt((a*c*sqrt(-(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)/(a^3*c^3)) - 2*d*e)/(a*c))*log(-(c^2*d^4 - a^2 *e^4)*x - (a^3*c^2*e*sqrt(-(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)/(a^3*c^3)) - a*c^2*d^3 + a^2*c*d*e^2)*sqrt((a*c*sqrt(-(c^2*d^4 - 2*a*c*d^2*e^2 + a^2* e^4)/(a^3*c^3)) - 2*d*e)/(a*c)))
Time = 0.34 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.44 \[ \int \frac {d+e x^2}{a+c x^4} \, dx=\operatorname {RootSum} {\left (256 t^{4} a^{3} c^{3} + 64 t^{2} a^{2} c^{2} d e + a^{2} e^{4} + 2 a c d^{2} e^{2} + c^{2} d^{4}, \left ( t \mapsto t \log {\left (x + \frac {64 t^{3} a^{3} c^{2} e + 12 t a^{2} c d e^{2} - 4 t a c^{2} d^{3}}{a^{2} e^{4} - c^{2} d^{4}} \right )} \right )\right )} \]
RootSum(256*_t**4*a**3*c**3 + 64*_t**2*a**2*c**2*d*e + a**2*e**4 + 2*a*c*d **2*e**2 + c**2*d**4, Lambda(_t, _t*log(x + (64*_t**3*a**3*c**2*e + 12*_t* a**2*c*d*e**2 - 4*_t*a*c**2*d**3)/(a**2*e**4 - c**2*d**4))))
Time = 0.29 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.89 \[ \int \frac {d+e x^2}{a+c x^4} \, dx=\frac {\sqrt {2} {\left (\sqrt {c} d + \sqrt {a} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{4 \, \sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (\sqrt {c} d + \sqrt {a} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{4 \, \sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (\sqrt {c} d - \sqrt {a} e\right )} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{8 \, a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (\sqrt {c} d - \sqrt {a} e\right )} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{8 \, a^{\frac {3}{4}} c^{\frac {3}{4}}} \]
1/4*sqrt(2)*(sqrt(c)*d + sqrt(a)*e)*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)) + 1/4*sqrt(2)*(sqrt(c)*d + sqrt(a)*e)*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)) + 1/8*sqrt(2)*(sqrt(c)*d - sqrt(a)*e)*log(sqrt(c)*x^ 2 + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(3/4)) - 1/8*sqrt(2)*( sqrt(c)*d - sqrt(a)*e)*log(sqrt(c)*x^2 - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt( a))/(a^(3/4)*c^(3/4))
Time = 0.28 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.98 \[ \int \frac {d+e x^2}{a+c x^4} \, dx=\frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d + \left (a c^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{4 \, a c^{3}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d + \left (a c^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{4 \, a c^{3}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d - \left (a c^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{8 \, a c^{3}} - \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d - \left (a c^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{8 \, a c^{3}} \]
1/4*sqrt(2)*((a*c^3)^(1/4)*c^2*d + (a*c^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2* x + sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a*c^3) + 1/4*sqrt(2)*((a*c^3)^(1/4) *c^2*d + (a*c^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4))/( a/c)^(1/4))/(a*c^3) + 1/8*sqrt(2)*((a*c^3)^(1/4)*c^2*d - (a*c^3)^(3/4)*e)* log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a*c^3) - 1/8*sqrt(2)*((a*c^3 )^(1/4)*c^2*d - (a*c^3)^(3/4)*e)*log(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/ c))/(a*c^3)
Time = 0.35 (sec) , antiderivative size = 599, normalized size of antiderivative = 2.43 \[ \int \frac {d+e x^2}{a+c x^4} \, dx=-2\,\mathrm {atanh}\left (\frac {8\,c^3\,d^2\,x\,\sqrt {\frac {e^2\,\sqrt {-a^3\,c^3}}{16\,a^2\,c^3}-\frac {d^2\,\sqrt {-a^3\,c^3}}{16\,a^3\,c^2}-\frac {d\,e}{8\,a\,c}}}{2\,c^2\,d^2\,e-2\,a\,c\,e^3+\frac {2\,c\,d^3\,\sqrt {-a^3\,c^3}}{a^2}-\frac {2\,d\,e^2\,\sqrt {-a^3\,c^3}}{a}}-\frac {8\,a\,c^2\,e^2\,x\,\sqrt {\frac {e^2\,\sqrt {-a^3\,c^3}}{16\,a^2\,c^3}-\frac {d^2\,\sqrt {-a^3\,c^3}}{16\,a^3\,c^2}-\frac {d\,e}{8\,a\,c}}}{2\,c^2\,d^2\,e-2\,a\,c\,e^3+\frac {2\,c\,d^3\,\sqrt {-a^3\,c^3}}{a^2}-\frac {2\,d\,e^2\,\sqrt {-a^3\,c^3}}{a}}\right )\,\sqrt {-\frac {c\,d^2\,\sqrt {-a^3\,c^3}-a\,e^2\,\sqrt {-a^3\,c^3}+2\,a^2\,c^2\,d\,e}{16\,a^3\,c^3}}-2\,\mathrm {atanh}\left (\frac {8\,c^3\,d^2\,x\,\sqrt {\frac {d^2\,\sqrt {-a^3\,c^3}}{16\,a^3\,c^2}-\frac {d\,e}{8\,a\,c}-\frac {e^2\,\sqrt {-a^3\,c^3}}{16\,a^2\,c^3}}}{2\,c^2\,d^2\,e-2\,a\,c\,e^3-\frac {2\,c\,d^3\,\sqrt {-a^3\,c^3}}{a^2}+\frac {2\,d\,e^2\,\sqrt {-a^3\,c^3}}{a}}-\frac {8\,a\,c^2\,e^2\,x\,\sqrt {\frac {d^2\,\sqrt {-a^3\,c^3}}{16\,a^3\,c^2}-\frac {d\,e}{8\,a\,c}-\frac {e^2\,\sqrt {-a^3\,c^3}}{16\,a^2\,c^3}}}{2\,c^2\,d^2\,e-2\,a\,c\,e^3-\frac {2\,c\,d^3\,\sqrt {-a^3\,c^3}}{a^2}+\frac {2\,d\,e^2\,\sqrt {-a^3\,c^3}}{a}}\right )\,\sqrt {-\frac {a\,e^2\,\sqrt {-a^3\,c^3}-c\,d^2\,\sqrt {-a^3\,c^3}+2\,a^2\,c^2\,d\,e}{16\,a^3\,c^3}} \]
- 2*atanh((8*c^3*d^2*x*((e^2*(-a^3*c^3)^(1/2))/(16*a^2*c^3) - (d^2*(-a^3*c ^3)^(1/2))/(16*a^3*c^2) - (d*e)/(8*a*c))^(1/2))/(2*c^2*d^2*e - 2*a*c*e^3 + (2*c*d^3*(-a^3*c^3)^(1/2))/a^2 - (2*d*e^2*(-a^3*c^3)^(1/2))/a) - (8*a*c^2 *e^2*x*((e^2*(-a^3*c^3)^(1/2))/(16*a^2*c^3) - (d^2*(-a^3*c^3)^(1/2))/(16*a ^3*c^2) - (d*e)/(8*a*c))^(1/2))/(2*c^2*d^2*e - 2*a*c*e^3 + (2*c*d^3*(-a^3* c^3)^(1/2))/a^2 - (2*d*e^2*(-a^3*c^3)^(1/2))/a))*(-(c*d^2*(-a^3*c^3)^(1/2) - a*e^2*(-a^3*c^3)^(1/2) + 2*a^2*c^2*d*e)/(16*a^3*c^3))^(1/2) - 2*atanh(( 8*c^3*d^2*x*((d^2*(-a^3*c^3)^(1/2))/(16*a^3*c^2) - (d*e)/(8*a*c) - (e^2*(- a^3*c^3)^(1/2))/(16*a^2*c^3))^(1/2))/(2*c^2*d^2*e - 2*a*c*e^3 - (2*c*d^3*( -a^3*c^3)^(1/2))/a^2 + (2*d*e^2*(-a^3*c^3)^(1/2))/a) - (8*a*c^2*e^2*x*((d^ 2*(-a^3*c^3)^(1/2))/(16*a^3*c^2) - (d*e)/(8*a*c) - (e^2*(-a^3*c^3)^(1/2))/ (16*a^2*c^3))^(1/2))/(2*c^2*d^2*e - 2*a*c*e^3 - (2*c*d^3*(-a^3*c^3)^(1/2)) /a^2 + (2*d*e^2*(-a^3*c^3)^(1/2))/a))*(-(a*e^2*(-a^3*c^3)^(1/2) - c*d^2*(- a^3*c^3)^(1/2) + 2*a^2*c^2*d*e)/(16*a^3*c^3))^(1/2)