Integrand size = 20, antiderivative size = 331 \[ \int \frac {x^2}{a+b+2 a x^2+a x^4} \, dx=-\frac {\arctan \left (\frac {\sqrt {-\sqrt {a}+\sqrt {a+b}}-\sqrt {2} \sqrt [4]{a} x}{\sqrt {\sqrt {a}+\sqrt {a+b}}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {\sqrt {a}+\sqrt {a+b}}}+\frac {\arctan \left (\frac {\sqrt {-\sqrt {a}+\sqrt {a+b}}+\sqrt {2} \sqrt [4]{a} x}{\sqrt {\sqrt {a}+\sqrt {a+b}}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {\sqrt {a}+\sqrt {a+b}}}+\frac {\log \left (\sqrt {a+b}-\sqrt {2} \sqrt [4]{a} \sqrt {-\sqrt {a}+\sqrt {a+b}} x+\sqrt {a} x^2\right )}{4 \sqrt {2} a^{3/4} \sqrt {-\sqrt {a}+\sqrt {a+b}}}-\frac {\log \left (\sqrt {a+b}+\sqrt {2} \sqrt [4]{a} \sqrt {-\sqrt {a}+\sqrt {a+b}} x+\sqrt {a} x^2\right )}{4 \sqrt {2} a^{3/4} \sqrt {-\sqrt {a}+\sqrt {a+b}}} \]
1/8*ln(x^2*a^(1/2)+(a+b)^(1/2)-a^(1/4)*x*2^(1/2)*(-a^(1/2)+(a+b)^(1/2))^(1 /2))/a^(3/4)*2^(1/2)/(-a^(1/2)+(a+b)^(1/2))^(1/2)-1/8*ln(x^2*a^(1/2)+(a+b) ^(1/2)+a^(1/4)*x*2^(1/2)*(-a^(1/2)+(a+b)^(1/2))^(1/2))/a^(3/4)*2^(1/2)/(-a ^(1/2)+(a+b)^(1/2))^(1/2)-1/4*arctan((-a^(1/4)*x*2^(1/2)+(-a^(1/2)+(a+b)^( 1/2))^(1/2))/(a^(1/2)+(a+b)^(1/2))^(1/2))/a^(3/4)*2^(1/2)/(a^(1/2)+(a+b)^( 1/2))^(1/2)+1/4*arctan((a^(1/4)*x*2^(1/2)+(-a^(1/2)+(a+b)^(1/2))^(1/2))/(a ^(1/2)+(a+b)^(1/2))^(1/2))/a^(3/4)*2^(1/2)/(a^(1/2)+(a+b)^(1/2))^(1/2)
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.43 \[ \int \frac {x^2}{a+b+2 a x^2+a x^4} \, dx=\frac {\frac {\left (i \sqrt {a}+\sqrt {b}\right ) \arctan \left (\frac {\sqrt {a} x}{\sqrt {a-i \sqrt {a} \sqrt {b}}}\right )}{\sqrt {a-i \sqrt {a} \sqrt {b}}}+\frac {\left (-i \sqrt {a}+\sqrt {b}\right ) \arctan \left (\frac {\sqrt {a} x}{\sqrt {a+i \sqrt {a} \sqrt {b}}}\right )}{\sqrt {a+i \sqrt {a} \sqrt {b}}}}{2 \sqrt {a} \sqrt {b}} \]
(((I*Sqrt[a] + Sqrt[b])*ArcTan[(Sqrt[a]*x)/Sqrt[a - I*Sqrt[a]*Sqrt[b]]])/S qrt[a - I*Sqrt[a]*Sqrt[b]] + (((-I)*Sqrt[a] + Sqrt[b])*ArcTan[(Sqrt[a]*x)/ Sqrt[a + I*Sqrt[a]*Sqrt[b]]])/Sqrt[a + I*Sqrt[a]*Sqrt[b]])/(2*Sqrt[a]*Sqrt [b])
Time = 0.59 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.14, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1449, 1142, 25, 27, 1083, 217, 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.
| \(\displaystyle \int \frac {x^2}{a x^4+2 a x^2+a+b} \, dx\) |
| \(\Big \downarrow \) 1449 |
| \(\displaystyle \frac {\int \frac {x}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{2 \sqrt {2} a^{3/4} \sqrt {\sqrt {a+b}-\sqrt {a}}}-\frac {\int \frac {x}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{2 \sqrt {2} a^{3/4} \sqrt {\sqrt {a+b}-\sqrt {a}}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \int \frac {1}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2} \sqrt [4]{a}}+\frac {1}{2} \int -\frac {\sqrt {2} \left (\sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt {2} \sqrt [4]{a} x\right )}{\sqrt [4]{a} \left (x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}\right )}dx}{2 \sqrt {2} a^{3/4} \sqrt {\sqrt {a+b}-\sqrt {a}}}-\frac {\frac {1}{2} \int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{a} x+\sqrt {\sqrt {a+b}-\sqrt {a}}\right )}{\sqrt [4]{a} \left (x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}\right )}dx-\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \int \frac {1}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2} \sqrt [4]{a}}}{2 \sqrt {2} a^{3/4} \sqrt {\sqrt {a+b}-\sqrt {a}}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \int \frac {1}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2} \sqrt [4]{a}}-\frac {1}{2} \int \frac {\sqrt {2} \left (\sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt {2} \sqrt [4]{a} x\right )}{\sqrt [4]{a} \left (x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}\right )}dx}{2 \sqrt {2} a^{3/4} \sqrt {\sqrt {a+b}-\sqrt {a}}}-\frac {\frac {1}{2} \int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{a} x+\sqrt {\sqrt {a+b}-\sqrt {a}}\right )}{\sqrt [4]{a} \left (x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}\right )}dx-\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \int \frac {1}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2} \sqrt [4]{a}}}{2 \sqrt {2} a^{3/4} \sqrt {\sqrt {a+b}-\sqrt {a}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \int \frac {1}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2} \sqrt [4]{a}}-\frac {\int \frac {\sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt {2} \sqrt [4]{a} x}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2} \sqrt [4]{a}}}{2 \sqrt {2} a^{3/4} \sqrt {\sqrt {a+b}-\sqrt {a}}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a} x+\sqrt {\sqrt {a+b}-\sqrt {a}}}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2} \sqrt [4]{a}}-\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \int \frac {1}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2} \sqrt [4]{a}}}{2 \sqrt {2} a^{3/4} \sqrt {\sqrt {a+b}-\sqrt {a}}}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {-\frac {\int \frac {\sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt {2} \sqrt [4]{a} x}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2} \sqrt [4]{a}}-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} \int \frac {1}{-\left (2 x-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}\right )^2-2 \left (\frac {\sqrt {a+b}}{\sqrt {a}}+1\right )}d\left (2 x-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a}}}{2 \sqrt {2} a^{3/4} \sqrt {\sqrt {a+b}-\sqrt {a}}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a} x+\sqrt {\sqrt {a+b}-\sqrt {a}}}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2} \sqrt [4]{a}}+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} \int \frac {1}{-\left (2 x+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}\right )^2-2 \left (\frac {\sqrt {a+b}}{\sqrt {a}}+1\right )}d\left (2 x+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a}}}{2 \sqrt {2} a^{3/4} \sqrt {\sqrt {a+b}-\sqrt {a}}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \arctan \left (\frac {\sqrt [4]{a} \left (2 x-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt {\sqrt {a+b}+\sqrt {a}}}\right )}{\sqrt {\sqrt {a+b}+\sqrt {a}}}-\frac {\int \frac {\sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt {2} \sqrt [4]{a} x}{x^2-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2} \sqrt [4]{a}}}{2 \sqrt {2} a^{3/4} \sqrt {\sqrt {a+b}-\sqrt {a}}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a} x+\sqrt {\sqrt {a+b}-\sqrt {a}}}{x^2+\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}} x}{\sqrt [4]{a}}+\frac {\sqrt {a+b}}{\sqrt {a}}}dx}{\sqrt {2} \sqrt [4]{a}}-\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \arctan \left (\frac {\sqrt [4]{a} \left (\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}+2 x\right )}{\sqrt {2} \sqrt {\sqrt {a+b}+\sqrt {a}}}\right )}{\sqrt {\sqrt {a+b}+\sqrt {a}}}}{2 \sqrt {2} a^{3/4} \sqrt {\sqrt {a+b}-\sqrt {a}}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \arctan \left (\frac {\sqrt [4]{a} \left (2 x-\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt {\sqrt {a+b}+\sqrt {a}}}\right )}{\sqrt {\sqrt {a+b}+\sqrt {a}}}+\frac {1}{2} \log \left (-\sqrt {2} \sqrt [4]{a} x \sqrt {\sqrt {a+b}-\sqrt {a}}+\sqrt {a+b}+\sqrt {a} x^2\right )}{2 \sqrt {2} a^{3/4} \sqrt {\sqrt {a+b}-\sqrt {a}}}-\frac {\frac {1}{2} \log \left (\sqrt {2} \sqrt [4]{a} x \sqrt {\sqrt {a+b}-\sqrt {a}}+\sqrt {a+b}+\sqrt {a} x^2\right )-\frac {\sqrt {\sqrt {a+b}-\sqrt {a}} \arctan \left (\frac {\sqrt [4]{a} \left (\frac {\sqrt {2} \sqrt {\sqrt {a+b}-\sqrt {a}}}{\sqrt [4]{a}}+2 x\right )}{\sqrt {2} \sqrt {\sqrt {a+b}+\sqrt {a}}}\right )}{\sqrt {\sqrt {a+b}+\sqrt {a}}}}{2 \sqrt {2} a^{3/4} \sqrt {\sqrt {a+b}-\sqrt {a}}}\) |
((Sqrt[-Sqrt[a] + Sqrt[a + b]]*ArcTan[(a^(1/4)*(-((Sqrt[2]*Sqrt[-Sqrt[a] + Sqrt[a + b]])/a^(1/4)) + 2*x))/(Sqrt[2]*Sqrt[Sqrt[a] + Sqrt[a + b]])])/Sq rt[Sqrt[a] + Sqrt[a + b]] + Log[Sqrt[a + b] - Sqrt[2]*a^(1/4)*Sqrt[-Sqrt[a ] + Sqrt[a + b]]*x + Sqrt[a]*x^2]/2)/(2*Sqrt[2]*a^(3/4)*Sqrt[-Sqrt[a] + Sq rt[a + b]]) - (-((Sqrt[-Sqrt[a] + Sqrt[a + b]]*ArcTan[(a^(1/4)*((Sqrt[2]*S qrt[-Sqrt[a] + Sqrt[a + b]])/a^(1/4) + 2*x))/(Sqrt[2]*Sqrt[Sqrt[a] + Sqrt[ a + b]])])/Sqrt[Sqrt[a] + Sqrt[a + b]]) + Log[Sqrt[a + b] + Sqrt[2]*a^(1/4 )*Sqrt[-Sqrt[a] + Sqrt[a + b]]*x + Sqrt[a]*x^2]/2)/(2*Sqrt[2]*a^(3/4)*Sqrt [-Sqrt[a] + Sqrt[a + b]])
3.10.12.3.1 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] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{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_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*r) Int[x^(m - 1)/(q - r*x + x^2), x], x] - Simp[1/(2*c*r) Int[x^(m - 1)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 1] && LtQ[m, 3] && NegQ[b^2 - 4*a*c]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.10 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.12
| method | result | size |
| risch | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{4}+2 a \,\textit {\_Z}^{2}+a +b \right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}+\textit {\_R}}}{4 a}\) | \(41\) |
| default | \(\frac {\sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \left (\sqrt {a^{2}+a b}+a \right ) \left (\frac {\ln \left (-x^{2} \sqrt {a}+x \sqrt {2 \sqrt {a \left (a +b \right )}-2 a}-\sqrt {a +b}\right )}{2 \sqrt {a}}-\frac {\sqrt {2 \sqrt {a \left (a +b \right )}-2 a}\, \arctan \left (\frac {-2 x \sqrt {a}+\sqrt {2 \sqrt {a \left (a +b \right )}-2 a}}{\sqrt {4 \sqrt {a}\, \sqrt {a +b}-2 \sqrt {a \left (a +b \right )}+2 a}}\right )}{\sqrt {a}\, \sqrt {4 \sqrt {a}\, \sqrt {a +b}-2 \sqrt {a \left (a +b \right )}+2 a}}\right )}{4 a b}-\frac {\sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \left (\sqrt {a^{2}+a b}+a \right ) \left (\frac {\ln \left (x^{2} \sqrt {a}+x \sqrt {2 \sqrt {a \left (a +b \right )}-2 a}+\sqrt {a +b}\right )}{2 \sqrt {a}}-\frac {\sqrt {2 \sqrt {a \left (a +b \right )}-2 a}\, \arctan \left (\frac {2 x \sqrt {a}+\sqrt {2 \sqrt {a \left (a +b \right )}-2 a}}{\sqrt {4 \sqrt {a}\, \sqrt {a +b}-2 \sqrt {a \left (a +b \right )}+2 a}}\right )}{\sqrt {a}\, \sqrt {4 \sqrt {a}\, \sqrt {a +b}-2 \sqrt {a \left (a +b \right )}+2 a}}\right )}{4 a b}\) | \(339\) |
Time = 0.25 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.84 \[ \int \frac {x^2}{a+b+2 a x^2+a x^4} \, dx=\frac {1}{4} \, \sqrt {\frac {a b \sqrt {-\frac {1}{a^{3} b}} + 1}{a b}} \log \left (a^{2} b \sqrt {\frac {a b \sqrt {-\frac {1}{a^{3} b}} + 1}{a b}} \sqrt {-\frac {1}{a^{3} b}} + x\right ) - \frac {1}{4} \, \sqrt {\frac {a b \sqrt {-\frac {1}{a^{3} b}} + 1}{a b}} \log \left (-a^{2} b \sqrt {\frac {a b \sqrt {-\frac {1}{a^{3} b}} + 1}{a b}} \sqrt {-\frac {1}{a^{3} b}} + x\right ) - \frac {1}{4} \, \sqrt {-\frac {a b \sqrt {-\frac {1}{a^{3} b}} - 1}{a b}} \log \left (a^{2} b \sqrt {-\frac {a b \sqrt {-\frac {1}{a^{3} b}} - 1}{a b}} \sqrt {-\frac {1}{a^{3} b}} + x\right ) + \frac {1}{4} \, \sqrt {-\frac {a b \sqrt {-\frac {1}{a^{3} b}} - 1}{a b}} \log \left (-a^{2} b \sqrt {-\frac {a b \sqrt {-\frac {1}{a^{3} b}} - 1}{a b}} \sqrt {-\frac {1}{a^{3} b}} + x\right ) \]
1/4*sqrt((a*b*sqrt(-1/(a^3*b)) + 1)/(a*b))*log(a^2*b*sqrt((a*b*sqrt(-1/(a^ 3*b)) + 1)/(a*b))*sqrt(-1/(a^3*b)) + x) - 1/4*sqrt((a*b*sqrt(-1/(a^3*b)) + 1)/(a*b))*log(-a^2*b*sqrt((a*b*sqrt(-1/(a^3*b)) + 1)/(a*b))*sqrt(-1/(a^3* b)) + x) - 1/4*sqrt(-(a*b*sqrt(-1/(a^3*b)) - 1)/(a*b))*log(a^2*b*sqrt(-(a* b*sqrt(-1/(a^3*b)) - 1)/(a*b))*sqrt(-1/(a^3*b)) + x) + 1/4*sqrt(-(a*b*sqrt (-1/(a^3*b)) - 1)/(a*b))*log(-a^2*b*sqrt(-(a*b*sqrt(-1/(a^3*b)) - 1)/(a*b) )*sqrt(-1/(a^3*b)) + x)
Time = 0.32 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.13 \[ \int \frac {x^2}{a+b+2 a x^2+a x^4} \, dx=\operatorname {RootSum} {\left (256 t^{4} a^{3} b^{2} - 32 t^{2} a^{2} b + a + b, \left ( t \mapsto t \log {\left (64 t^{3} a^{2} b - 4 t a + x \right )} \right )\right )} \]
RootSum(256*_t**4*a**3*b**2 - 32*_t**2*a**2*b + a + b, Lambda(_t, _t*log(6 4*_t**3*a**2*b - 4*_t*a + x)))
\[ \int \frac {x^2}{a+b+2 a x^2+a x^4} \, dx=\int { \frac {x^{2}}{a x^{4} + 2 \, a x^{2} + a + b} \,d x } \]
Time = 0.37 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.61 \[ \int \frac {x^2}{a+b+2 a x^2+a x^4} \, dx=-\frac {{\left (3 \, \sqrt {a^{2} + \sqrt {-a b} a} \sqrt {-a b} a + 4 \, \sqrt {a^{2} + \sqrt {-a b} a} \sqrt {-a b} b\right )} {\left | a \right |} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {\frac {2 \, a + \sqrt {-4 \, {\left (a + b\right )} a + 4 \, a^{2}}}{a}}}\right )}{2 \, {\left (3 \, a^{4} b + 4 \, a^{3} b^{2}\right )}} + \frac {{\left (3 \, \sqrt {a^{2} - \sqrt {-a b} a} \sqrt {-a b} a + 4 \, \sqrt {a^{2} - \sqrt {-a b} a} \sqrt {-a b} b\right )} {\left | a \right |} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {\frac {2 \, a - \sqrt {-4 \, {\left (a + b\right )} a + 4 \, a^{2}}}{a}}}\right )}{2 \, {\left (3 \, a^{4} b + 4 \, a^{3} b^{2}\right )}} \]
-1/2*(3*sqrt(a^2 + sqrt(-a*b)*a)*sqrt(-a*b)*a + 4*sqrt(a^2 + sqrt(-a*b)*a) *sqrt(-a*b)*b)*abs(a)*arctan(2*sqrt(1/2)*x/sqrt((2*a + sqrt(-4*(a + b)*a + 4*a^2))/a))/(3*a^4*b + 4*a^3*b^2) + 1/2*(3*sqrt(a^2 - sqrt(-a*b)*a)*sqrt( -a*b)*a + 4*sqrt(a^2 - sqrt(-a*b)*a)*sqrt(-a*b)*b)*abs(a)*arctan(2*sqrt(1/ 2)*x/sqrt((2*a - sqrt(-4*(a + b)*a + 4*a^2))/a))/(3*a^4*b + 4*a^3*b^2)
Time = 0.30 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.67 \[ \int \frac {x^2}{a+b+2 a x^2+a x^4} \, dx=-2\,\mathrm {atanh}\left (\frac {2\,\left (x\,\left (4\,a^2\,b-4\,a^3\right )+\frac {4\,a\,x\,\left (\sqrt {-a^3\,b^3}+a^2\,b\right )}{b}\right )\,\sqrt {\frac {\sqrt {-a^3\,b^3}+a^2\,b}{16\,a^3\,b^2}}}{2\,a^2+2\,b\,a}\right )\,\sqrt {\frac {\sqrt {-a^3\,b^3}+a^2\,b}{16\,a^3\,b^2}}-2\,\mathrm {atanh}\left (\frac {2\,\left (x\,\left (4\,a^2\,b-4\,a^3\right )-\frac {4\,a\,x\,\left (\sqrt {-a^3\,b^3}-a^2\,b\right )}{b}\right )\,\sqrt {-\frac {\sqrt {-a^3\,b^3}-a^2\,b}{16\,a^3\,b^2}}}{2\,a^2+2\,b\,a}\right )\,\sqrt {-\frac {\sqrt {-a^3\,b^3}-a^2\,b}{16\,a^3\,b^2}} \]
- 2*atanh((2*(x*(4*a^2*b - 4*a^3) + (4*a*x*((-a^3*b^3)^(1/2) + a^2*b))/b)* (((-a^3*b^3)^(1/2) + a^2*b)/(16*a^3*b^2))^(1/2))/(2*a*b + 2*a^2))*(((-a^3* b^3)^(1/2) + a^2*b)/(16*a^3*b^2))^(1/2) - 2*atanh((2*(x*(4*a^2*b - 4*a^3) - (4*a*x*((-a^3*b^3)^(1/2) - a^2*b))/b)*(-((-a^3*b^3)^(1/2) - a^2*b)/(16*a ^3*b^2))^(1/2))/(2*a*b + 2*a^2))*(-((-a^3*b^3)^(1/2) - a^2*b)/(16*a^3*b^2) )^(1/2)