\(\int \frac {a+b x^2}{(2+x^2+x^4)^2} \, dx\) [101]

Optimal result

Integrand size = 18, antiderivative size = 316 \[ \int \frac {a+b x^2}{\left (2+x^2+x^4\right )^2} \, dx=\frac {x \left (3 a+2 b-(a-4 b) x^2\right )}{28 \left (2+x^2+x^4\right )}-\frac {1}{56} \sqrt {\frac {1}{14} \left (-1+2 \sqrt {2}\right )} \left (\left (11-\sqrt {2}\right ) a-\left (2-4 \sqrt {2}\right ) b\right ) \arctan \left (\frac {\sqrt {-1+2 \sqrt {2}}-2 x}{\sqrt {1+2 \sqrt {2}}}\right )+\frac {1}{56} \sqrt {\frac {1}{14} \left (-1+2 \sqrt {2}\right )} \left (\left (11-\sqrt {2}\right ) a-\left (2-4 \sqrt {2}\right ) b\right ) \arctan \left (\frac {\sqrt {-1+2 \sqrt {2}}+2 x}{\sqrt {1+2 \sqrt {2}}}\right )-\frac {\left (11 a+\sqrt {2} (a-4 b)-2 b\right ) \log \left (\sqrt {2}-\sqrt {-1+2 \sqrt {2}} x+x^2\right )}{112 \sqrt {2 \left (-1+2 \sqrt {2}\right )}}+\frac {\left (\left (11+\sqrt {2}\right ) a-2 \left (b+2 \sqrt {2} b\right )\right ) \log \left (\sqrt {2}+\sqrt {-1+2 \sqrt {2}} x+x^2\right )}{112 \sqrt {2 \left (-1+2 \sqrt {2}\right )}} \]

[Out]

1/28*x*(3*a+2*b-(a-4*b)*x^2)/(x^4+x^2+2)-1/784*arctan((-2*x+(-1+2*2^(1/2))^(1/2))/(1+2*2^(1/2))^(1/2))*(-b*(2-
4*2^(1/2))+a*(11-2^(1/2)))*(-14+28*2^(1/2))^(1/2)+1/784*arctan((2*x+(-1+2*2^(1/2))^(1/2))/(1+2*2^(1/2))^(1/2))
*(-b*(2-4*2^(1/2))+a*(11-2^(1/2)))*(-14+28*2^(1/2))^(1/2)-1/112*ln(x^2+2^(1/2)-x*(-1+2*2^(1/2))^(1/2))*(11*a-2
*b+(a-4*b)*2^(1/2))/(-2+4*2^(1/2))^(1/2)+1/112*ln(x^2+2^(1/2)+x*(-1+2*2^(1/2))^(1/2))*(a*(11+2^(1/2))-2*b-4*b*
2^(1/2))/(-2+4*2^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1192, 1183, 648, 632, 210, 642} \[ \int \frac {a+b x^2}{\left (2+x^2+x^4\right )^2} \, dx=-\frac {1}{56} \sqrt {\frac {1}{14} \left (2 \sqrt {2}-1\right )} \left (\left (11-\sqrt {2}\right ) a-\left (2-4 \sqrt {2}\right ) b\right ) \arctan \left (\frac {\sqrt {2 \sqrt {2}-1}-2 x}{\sqrt {1+2 \sqrt {2}}}\right )+\frac {1}{56} \sqrt {\frac {1}{14} \left (2 \sqrt {2}-1\right )} \left (\left (11-\sqrt {2}\right ) a-\left (2-4 \sqrt {2}\right ) b\right ) \arctan \left (\frac {2 x+\sqrt {2 \sqrt {2}-1}}{\sqrt {1+2 \sqrt {2}}}\right )-\frac {\left (\sqrt {2} (a-4 b)+11 a-2 b\right ) \log \left (x^2-\sqrt {2 \sqrt {2}-1} x+\sqrt {2}\right )}{112 \sqrt {2 \left (2 \sqrt {2}-1\right )}}+\frac {\left (\left (11+\sqrt {2}\right ) a-2 \left (2 \sqrt {2} b+b\right )\right ) \log \left (x^2+\sqrt {2 \sqrt {2}-1} x+\sqrt {2}\right )}{112 \sqrt {2 \left (2 \sqrt {2}-1\right )}}+\frac {x \left (-\left (x^2 (a-4 b)\right )+3 a+2 b\right )}{28 \left (x^4+x^2+2\right )} \]

[In]

Int[(a + b*x^2)/(2 + x^2 + x^4)^2,x]

[Out]

(x*(3*a + 2*b - (a - 4*b)*x^2))/(28*(2 + x^2 + x^4)) - (Sqrt[(-1 + 2*Sqrt[2])/14]*((11 - Sqrt[2])*a - (2 - 4*S
qrt[2])*b)*ArcTan[(Sqrt[-1 + 2*Sqrt[2]] - 2*x)/Sqrt[1 + 2*Sqrt[2]]])/56 + (Sqrt[(-1 + 2*Sqrt[2])/14]*((11 - Sq
rt[2])*a - (2 - 4*Sqrt[2])*b)*ArcTan[(Sqrt[-1 + 2*Sqrt[2]] + 2*x)/Sqrt[1 + 2*Sqrt[2]]])/56 - ((11*a + Sqrt[2]*
(a - 4*b) - 2*b)*Log[Sqrt[2] - Sqrt[-1 + 2*Sqrt[2]]*x + x^2])/(112*Sqrt[2*(-1 + 2*Sqrt[2])]) + (((11 + Sqrt[2]
)*a - 2*(b + 2*Sqrt[2]*b))*Log[Sqrt[2] + Sqrt[-1 + 2*Sqrt[2]]*x + x^2])/(112*Sqrt[2*(-1 + 2*Sqrt[2])])

Rule 210

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])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[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]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1183

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =
Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(
d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]

Rule 1192

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[x*(a*b*e - d*(b^2 - 2*a
*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Dist[1/(2*a*(p + 1)
*(b^2 - 4*a*c)), Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a +
b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e
^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rubi steps \begin{align*} \text {integral}& = \frac {x \left (3 a+2 b-(a-4 b) x^2\right )}{28 \left (2+x^2+x^4\right )}+\frac {1}{28} \int \frac {11 a-2 b+(-a+4 b) x^2}{2+x^2+x^4} \, dx \\ & = \frac {x \left (3 a+2 b-(a-4 b) x^2\right )}{28 \left (2+x^2+x^4\right )}+\frac {\int \frac {\sqrt {-1+2 \sqrt {2}} (11 a-2 b)-\left (11 a-2 b-\sqrt {2} (-a+4 b)\right ) x}{\sqrt {2}-\sqrt {-1+2 \sqrt {2}} x+x^2} \, dx}{56 \sqrt {2 \left (-1+2 \sqrt {2}\right )}}+\frac {\int \frac {\sqrt {-1+2 \sqrt {2}} (11 a-2 b)+\left (11 a-2 b-\sqrt {2} (-a+4 b)\right ) x}{\sqrt {2}+\sqrt {-1+2 \sqrt {2}} x+x^2} \, dx}{56 \sqrt {2 \left (-1+2 \sqrt {2}\right )}} \\ & = \frac {x \left (3 a+2 b-(a-4 b) x^2\right )}{28 \left (2+x^2+x^4\right )}-\frac {\left (11 a+\sqrt {2} (a-4 b)-2 b\right ) \int \frac {-\sqrt {-1+2 \sqrt {2}}+2 x}{\sqrt {2}-\sqrt {-1+2 \sqrt {2}} x+x^2} \, dx}{112 \sqrt {2 \left (-1+2 \sqrt {2}\right )}}+\frac {\left (\left (11-\sqrt {2}\right ) a-\left (2-4 \sqrt {2}\right ) b\right ) \int \frac {1}{\sqrt {2}-\sqrt {-1+2 \sqrt {2}} x+x^2} \, dx}{112 \sqrt {2}}+\frac {\left (\left (11-\sqrt {2}\right ) a-\left (2-4 \sqrt {2}\right ) b\right ) \int \frac {1}{\sqrt {2}+\sqrt {-1+2 \sqrt {2}} x+x^2} \, dx}{112 \sqrt {2}}+\frac {\left (\left (11+\sqrt {2}\right ) a-2 \left (b+2 \sqrt {2} b\right )\right ) \int \frac {\sqrt {-1+2 \sqrt {2}}+2 x}{\sqrt {2}+\sqrt {-1+2 \sqrt {2}} x+x^2} \, dx}{112 \sqrt {2 \left (-1+2 \sqrt {2}\right )}} \\ & = \frac {x \left (3 a+2 b-(a-4 b) x^2\right )}{28 \left (2+x^2+x^4\right )}-\frac {\left (11 a+\sqrt {2} (a-4 b)-2 b\right ) \log \left (\sqrt {2}-\sqrt {-1+2 \sqrt {2}} x+x^2\right )}{112 \sqrt {2 \left (-1+2 \sqrt {2}\right )}}+\frac {\left (\left (11+\sqrt {2}\right ) a-2 \left (b+2 \sqrt {2} b\right )\right ) \log \left (\sqrt {2}+\sqrt {-1+2 \sqrt {2}} x+x^2\right )}{112 \sqrt {2 \left (-1+2 \sqrt {2}\right )}}-\frac {\left (\left (11-\sqrt {2}\right ) a-\left (2-4 \sqrt {2}\right ) b\right ) \text {Subst}\left (\int \frac {1}{-1-2 \sqrt {2}-x^2} \, dx,x,-\sqrt {-1+2 \sqrt {2}}+2 x\right )}{56 \sqrt {2}}-\frac {\left (\left (11-\sqrt {2}\right ) a-\left (2-4 \sqrt {2}\right ) b\right ) \text {Subst}\left (\int \frac {1}{-1-2 \sqrt {2}-x^2} \, dx,x,\sqrt {-1+2 \sqrt {2}}+2 x\right )}{56 \sqrt {2}} \\ & = \frac {x \left (3 a+2 b-(a-4 b) x^2\right )}{28 \left (2+x^2+x^4\right )}-\frac {\left (\left (11-\sqrt {2}\right ) a-\left (2-4 \sqrt {2}\right ) b\right ) \tan ^{-1}\left (\frac {\sqrt {-1+2 \sqrt {2}}-2 x}{\sqrt {1+2 \sqrt {2}}}\right )}{56 \sqrt {2 \left (1+2 \sqrt {2}\right )}}+\frac {\left (\left (11-\sqrt {2}\right ) a-\left (2-4 \sqrt {2}\right ) b\right ) \tan ^{-1}\left (\frac {\sqrt {-1+2 \sqrt {2}}+2 x}{\sqrt {1+2 \sqrt {2}}}\right )}{56 \sqrt {2 \left (1+2 \sqrt {2}\right )}}-\frac {\left (11 a+\sqrt {2} (a-4 b)-2 b\right ) \log \left (\sqrt {2}-\sqrt {-1+2 \sqrt {2}} x+x^2\right )}{112 \sqrt {2 \left (-1+2 \sqrt {2}\right )}}+\frac {\left (\left (11+\sqrt {2}\right ) a-2 \left (b+2 \sqrt {2} b\right )\right ) \log \left (\sqrt {2}+\sqrt {-1+2 \sqrt {2}} x+x^2\right )}{112 \sqrt {2 \left (-1+2 \sqrt {2}\right )}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.14 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.53 \[ \int \frac {a+b x^2}{\left (2+x^2+x^4\right )^2} \, dx=\frac {3 a x+2 b x-a x^3+4 b x^3}{28 \left (2+x^2+x^4\right )}-\frac {\left (\left (23 i+\sqrt {7}\right ) a-4 \left (2 i+\sqrt {7}\right ) b\right ) \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \left (1-i \sqrt {7}\right )}}\right )}{28 \sqrt {14-14 i \sqrt {7}}}-\frac {\left (\left (-23 i+\sqrt {7}\right ) a-4 \left (-2 i+\sqrt {7}\right ) b\right ) \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \left (1+i \sqrt {7}\right )}}\right )}{28 \sqrt {14+14 i \sqrt {7}}} \]

[In]

Integrate[(a + b*x^2)/(2 + x^2 + x^4)^2,x]

[Out]

(3*a*x + 2*b*x - a*x^3 + 4*b*x^3)/(28*(2 + x^2 + x^4)) - (((23*I + Sqrt[7])*a - 4*(2*I + Sqrt[7])*b)*ArcTan[x/
Sqrt[(1 - I*Sqrt[7])/2]])/(28*Sqrt[14 - (14*I)*Sqrt[7]]) - (((-23*I + Sqrt[7])*a - 4*(-2*I + Sqrt[7])*b)*ArcTa
n[x/Sqrt[(1 + I*Sqrt[7])/2]])/(28*Sqrt[14 + (14*I)*Sqrt[7]])

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.40 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.26

method	result	size
risch	\(\frac {\left (\frac {b}{7}-\frac {a}{28}\right ) x^{3}+\left (\frac {b}{14}+\frac {3 a}{28}\right ) x}{x^{4}+x^{2}+2}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+\textit {\_Z}^{2}+2\right )}{\sum }\frac {\left (\left (-a +4 b \right ) \textit {\_R}^{2}-2 b +11 a \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3}+\textit {\_R}}\right )}{56}\)	\(82\)
default	\(\frac {\frac {\left (-14 a -28 \sqrt {2}\, a +112 b \sqrt {2}+56 b \right ) x}{1+2 \sqrt {2}}+\frac {\sqrt {-1+2 \sqrt {2}}\, \left (-70 a -42 \sqrt {2}\, a +56 b \sqrt {2}+28 b \right )}{1+2 \sqrt {2}}}{784 x \sqrt {-1+2 \sqrt {2}}+784 x^{2}+784 \sqrt {2}}+\frac {\frac {\left (107 \sqrt {-1+2 \sqrt {2}}\, \sqrt {2}\, a -50 \sqrt {-1+2 \sqrt {2}}\, \sqrt {2}\, b +106 \sqrt {-1+2 \sqrt {2}}\, a -88 \sqrt {-1+2 \sqrt {2}}\, b \right ) \ln \left (x^{2}+\sqrt {2}+x \sqrt {-1+2 \sqrt {2}}\right )}{2}+\frac {2 \left (308 a +77 \sqrt {2}\, a -56 b -14 b \sqrt {2}-\frac {\left (107 \sqrt {-1+2 \sqrt {2}}\, \sqrt {2}\, a -50 \sqrt {-1+2 \sqrt {2}}\, \sqrt {2}\, b +106 \sqrt {-1+2 \sqrt {2}}\, a -88 \sqrt {-1+2 \sqrt {2}}\, b \right ) \sqrt {-1+2 \sqrt {2}}}{2}\right ) \arctan \left (\frac {2 x +\sqrt {-1+2 \sqrt {2}}}{\sqrt {1+2 \sqrt {2}}}\right )}{\sqrt {1+2 \sqrt {2}}}}{784+1568 \sqrt {2}}-\frac {-\frac {\left (-14 a -28 \sqrt {2}\, a +112 b \sqrt {2}+56 b \right ) x}{1+2 \sqrt {2}}+\frac {\sqrt {-1+2 \sqrt {2}}\, \left (-70 a -42 \sqrt {2}\, a +56 b \sqrt {2}+28 b \right )}{1+2 \sqrt {2}}}{784 \left (x^{2}+\sqrt {2}-x \sqrt {-1+2 \sqrt {2}}\right )}-\frac {\frac {\left (107 \sqrt {-1+2 \sqrt {2}}\, \sqrt {2}\, a -50 \sqrt {-1+2 \sqrt {2}}\, \sqrt {2}\, b +106 \sqrt {-1+2 \sqrt {2}}\, a -88 \sqrt {-1+2 \sqrt {2}}\, b \right ) \ln \left (x^{2}+\sqrt {2}-x \sqrt {-1+2 \sqrt {2}}\right )}{2}+\frac {2 \left (-77 \sqrt {2}\, a +14 b \sqrt {2}-308 a +56 b +\frac {\left (107 \sqrt {-1+2 \sqrt {2}}\, \sqrt {2}\, a -50 \sqrt {-1+2 \sqrt {2}}\, \sqrt {2}\, b +106 \sqrt {-1+2 \sqrt {2}}\, a -88 \sqrt {-1+2 \sqrt {2}}\, b \right ) \sqrt {-1+2 \sqrt {2}}}{2}\right ) \arctan \left (\frac {2 x -\sqrt {-1+2 \sqrt {2}}}{\sqrt {1+2 \sqrt {2}}}\right )}{\sqrt {1+2 \sqrt {2}}}}{784 \left (1+2 \sqrt {2}\right )}\)	\(605\)

[In]

int((b*x^2+a)/(x^4+x^2+2)^2,x,method=_RETURNVERBOSE)

[Out]

((1/7*b-1/28*a)*x^3+(1/14*b+3/28*a)*x)/(x^4+x^2+2)+1/56*sum(((-a+4*b)*_R^2-2*b+11*a)/(2*_R^3+_R)*ln(x-_R),_R=R
ootOf(_Z^4+_Z^2+2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 953 vs. \(2 (235) = 470\).

Time = 0.29 (sec) , antiderivative size = 953, normalized size of antiderivative = 3.02 \[ \int \frac {a+b x^2}{\left (2+x^2+x^4\right )^2} \, dx=-\frac {28 \, {\left (a - 4 \, b\right )} x^{3} + \sqrt {7} {\left (x^{4} + x^{2} + 2\right )} \sqrt {211 \, a^{2} - 428 \, a b + 100 \, b^{2} + 7 \, \sqrt {7} \sqrt {-289 \, a^{4} + 136 \, a^{3} b + 120 \, a^{2} b^{2} - 32 \, a b^{3} - 16 \, b^{4}}} \log \left (-8 \, {\left (1139 \, a^{4} - 1169 \, a^{3} b + 318 \, a^{2} b^{2} + 124 \, a b^{3} - 88 \, b^{4}\right )} x + \sqrt {211 \, a^{2} - 428 \, a b + 100 \, b^{2} + 7 \, \sqrt {7} \sqrt {-289 \, a^{4} + 136 \, a^{3} b + 120 \, a^{2} b^{2} - 32 \, a b^{3} - 16 \, b^{4}}} {\left (\sqrt {7} {\left (187 \, a^{3} - 78 \, a^{2} b - 36 \, a b^{2} + 8 \, b^{3}\right )} + 3 \, \sqrt {-289 \, a^{4} + 136 \, a^{3} b + 120 \, a^{2} b^{2} - 32 \, a b^{3} - 16 \, b^{4}} {\left (5 \, a - 6 \, b\right )}\right )}\right ) - \sqrt {7} {\left (x^{4} + x^{2} + 2\right )} \sqrt {211 \, a^{2} - 428 \, a b + 100 \, b^{2} + 7 \, \sqrt {7} \sqrt {-289 \, a^{4} + 136 \, a^{3} b + 120 \, a^{2} b^{2} - 32 \, a b^{3} - 16 \, b^{4}}} \log \left (-8 \, {\left (1139 \, a^{4} - 1169 \, a^{3} b + 318 \, a^{2} b^{2} + 124 \, a b^{3} - 88 \, b^{4}\right )} x - \sqrt {211 \, a^{2} - 428 \, a b + 100 \, b^{2} + 7 \, \sqrt {7} \sqrt {-289 \, a^{4} + 136 \, a^{3} b + 120 \, a^{2} b^{2} - 32 \, a b^{3} - 16 \, b^{4}}} {\left (\sqrt {7} {\left (187 \, a^{3} - 78 \, a^{2} b - 36 \, a b^{2} + 8 \, b^{3}\right )} + 3 \, \sqrt {-289 \, a^{4} + 136 \, a^{3} b + 120 \, a^{2} b^{2} - 32 \, a b^{3} - 16 \, b^{4}} {\left (5 \, a - 6 \, b\right )}\right )}\right ) + \sqrt {7} {\left (x^{4} + x^{2} + 2\right )} \sqrt {211 \, a^{2} - 428 \, a b + 100 \, b^{2} - 7 \, \sqrt {7} \sqrt {-289 \, a^{4} + 136 \, a^{3} b + 120 \, a^{2} b^{2} - 32 \, a b^{3} - 16 \, b^{4}}} \log \left (-8 \, {\left (1139 \, a^{4} - 1169 \, a^{3} b + 318 \, a^{2} b^{2} + 124 \, a b^{3} - 88 \, b^{4}\right )} x + \sqrt {211 \, a^{2} - 428 \, a b + 100 \, b^{2} - 7 \, \sqrt {7} \sqrt {-289 \, a^{4} + 136 \, a^{3} b + 120 \, a^{2} b^{2} - 32 \, a b^{3} - 16 \, b^{4}}} {\left (\sqrt {7} {\left (187 \, a^{3} - 78 \, a^{2} b - 36 \, a b^{2} + 8 \, b^{3}\right )} - 3 \, \sqrt {-289 \, a^{4} + 136 \, a^{3} b + 120 \, a^{2} b^{2} - 32 \, a b^{3} - 16 \, b^{4}} {\left (5 \, a - 6 \, b\right )}\right )}\right ) - \sqrt {7} {\left (x^{4} + x^{2} + 2\right )} \sqrt {211 \, a^{2} - 428 \, a b + 100 \, b^{2} - 7 \, \sqrt {7} \sqrt {-289 \, a^{4} + 136 \, a^{3} b + 120 \, a^{2} b^{2} - 32 \, a b^{3} - 16 \, b^{4}}} \log \left (-8 \, {\left (1139 \, a^{4} - 1169 \, a^{3} b + 318 \, a^{2} b^{2} + 124 \, a b^{3} - 88 \, b^{4}\right )} x - \sqrt {211 \, a^{2} - 428 \, a b + 100 \, b^{2} - 7 \, \sqrt {7} \sqrt {-289 \, a^{4} + 136 \, a^{3} b + 120 \, a^{2} b^{2} - 32 \, a b^{3} - 16 \, b^{4}}} {\left (\sqrt {7} {\left (187 \, a^{3} - 78 \, a^{2} b - 36 \, a b^{2} + 8 \, b^{3}\right )} - 3 \, \sqrt {-289 \, a^{4} + 136 \, a^{3} b + 120 \, a^{2} b^{2} - 32 \, a b^{3} - 16 \, b^{4}} {\left (5 \, a - 6 \, b\right )}\right )}\right ) - 28 \, {\left (3 \, a + 2 \, b\right )} x}{784 \, {\left (x^{4} + x^{2} + 2\right )}} \]

[In]

integrate((b*x^2+a)/(x^4+x^2+2)^2,x, algorithm="fricas")

[Out]

-1/784*(28*(a - 4*b)*x^3 + sqrt(7)*(x^4 + x^2 + 2)*sqrt(211*a^2 - 428*a*b + 100*b^2 + 7*sqrt(7)*sqrt(-289*a^4
+ 136*a^3*b + 120*a^2*b^2 - 32*a*b^3 - 16*b^4))*log(-8*(1139*a^4 - 1169*a^3*b + 318*a^2*b^2 + 124*a*b^3 - 88*b
^4)*x + sqrt(211*a^2 - 428*a*b + 100*b^2 + 7*sqrt(7)*sqrt(-289*a^4 + 136*a^3*b + 120*a^2*b^2 - 32*a*b^3 - 16*b
^4))*(sqrt(7)*(187*a^3 - 78*a^2*b - 36*a*b^2 + 8*b^3) + 3*sqrt(-289*a^4 + 136*a^3*b + 120*a^2*b^2 - 32*a*b^3 -
16*b^4)*(5*a - 6*b))) - sqrt(7)*(x^4 + x^2 + 2)*sqrt(211*a^2 - 428*a*b + 100*b^2 + 7*sqrt(7)*sqrt(-289*a^4 +
136*a^3*b + 120*a^2*b^2 - 32*a*b^3 - 16*b^4))*log(-8*(1139*a^4 - 1169*a^3*b + 318*a^2*b^2 + 124*a*b^3 - 88*b^4
)*x - sqrt(211*a^2 - 428*a*b + 100*b^2 + 7*sqrt(7)*sqrt(-289*a^4 + 136*a^3*b + 120*a^2*b^2 - 32*a*b^3 - 16*b^4
))*(sqrt(7)*(187*a^3 - 78*a^2*b - 36*a*b^2 + 8*b^3) + 3*sqrt(-289*a^4 + 136*a^3*b + 120*a^2*b^2 - 32*a*b^3 - 1
6*b^4)*(5*a - 6*b))) + sqrt(7)*(x^4 + x^2 + 2)*sqrt(211*a^2 - 428*a*b + 100*b^2 - 7*sqrt(7)*sqrt(-289*a^4 + 13
6*a^3*b + 120*a^2*b^2 - 32*a*b^3 - 16*b^4))*log(-8*(1139*a^4 - 1169*a^3*b + 318*a^2*b^2 + 124*a*b^3 - 88*b^4)*
x + sqrt(211*a^2 - 428*a*b + 100*b^2 - 7*sqrt(7)*sqrt(-289*a^4 + 136*a^3*b + 120*a^2*b^2 - 32*a*b^3 - 16*b^4))
*(sqrt(7)*(187*a^3 - 78*a^2*b - 36*a*b^2 + 8*b^3) - 3*sqrt(-289*a^4 + 136*a^3*b + 120*a^2*b^2 - 32*a*b^3 - 16*
b^4)*(5*a - 6*b))) - sqrt(7)*(x^4 + x^2 + 2)*sqrt(211*a^2 - 428*a*b + 100*b^2 - 7*sqrt(7)*sqrt(-289*a^4 + 136*
a^3*b + 120*a^2*b^2 - 32*a*b^3 - 16*b^4))*log(-8*(1139*a^4 - 1169*a^3*b + 318*a^2*b^2 + 124*a*b^3 - 88*b^4)*x
- sqrt(211*a^2 - 428*a*b + 100*b^2 - 7*sqrt(7)*sqrt(-289*a^4 + 136*a^3*b + 120*a^2*b^2 - 32*a*b^3 - 16*b^4))*(
sqrt(7)*(187*a^3 - 78*a^2*b - 36*a*b^2 + 8*b^3) - 3*sqrt(-289*a^4 + 136*a^3*b + 120*a^2*b^2 - 32*a*b^3 - 16*b^
4)*(5*a - 6*b))) - 28*(3*a + 2*b)*x)/(x^4 + x^2 + 2)

Sympy [A] (verification not implemented)

Time = 0.92 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.52 \[ \int \frac {a+b x^2}{\left (2+x^2+x^4\right )^2} \, dx=\frac {x^{3} \left (- a + 4 b\right ) + x \left (3 a + 2 b\right )}{28 x^{4} + 28 x^{2} + 56} + \operatorname {RootSum} {\left (240945152 t^{4} + t^{2} \left (- 1157968 a^{2} + 2348864 a b - 548800 b^{2}\right ) + 4489 a^{4} - 7102 a^{3} b + 5757 a^{2} b^{2} - 2332 a b^{3} + 484 b^{4}, \left ( t \mapsto t \log {\left (x + \frac {2634240 t^{3} a - 3161088 t^{3} b + 11996 t a^{3} + 12792 t a^{2} b - 21936 t a b^{2} + 4384 t b^{3}}{1139 a^{4} - 1169 a^{3} b + 318 a^{2} b^{2} + 124 a b^{3} - 88 b^{4}} \right )} \right )\right )} \]

[In]

integrate((b*x**2+a)/(x**4+x**2+2)**2,x)

[Out]

(x**3*(-a + 4*b) + x*(3*a + 2*b))/(28*x**4 + 28*x**2 + 56) + RootSum(240945152*_t**4 + _t**2*(-1157968*a**2 +
2348864*a*b - 548800*b**2) + 4489*a**4 - 7102*a**3*b + 5757*a**2*b**2 - 2332*a*b**3 + 484*b**4, Lambda(_t, _t*
log(x + (2634240*_t**3*a - 3161088*_t**3*b + 11996*_t*a**3 + 12792*_t*a**2*b - 21936*_t*a*b**2 + 4384*_t*b**3)
/(1139*a**4 - 1169*a**3*b + 318*a**2*b**2 + 124*a*b**3 - 88*b**4))))

Maxima [F]

\[ \int \frac {a+b x^2}{\left (2+x^2+x^4\right )^2} \, dx=\int { \frac {b x^{2} + a}{{\left (x^{4} + x^{2} + 2\right )}^{2}} \,d x } \]

[In]

integrate((b*x^2+a)/(x^4+x^2+2)^2,x, algorithm="maxima")

[Out]

-1/28*((a - 4*b)*x^3 - (3*a + 2*b)*x)/(x^4 + x^2 + 2) + 1/28*integrate(-((a - 4*b)*x^2 - 11*a + 2*b)/(x^4 + x^
2 + 2), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1112 vs. \(2 (235) = 470\).

Time = 0.57 (sec) , antiderivative size = 1112, normalized size of antiderivative = 3.52 \[ \int \frac {a+b x^2}{\left (2+x^2+x^4\right )^2} \, dx=\text {Too large to display} \]

[In]

integrate((b*x^2+a)/(x^4+x^2+2)^2,x, algorithm="giac")

[Out]

1/25088*sqrt(7)*(sqrt(7)*2^(3/4)*a*sqrt(2*sqrt(2) + 8)*(sqrt(2) + 4) - 4*sqrt(7)*2^(3/4)*b*sqrt(2*sqrt(2) + 8)
*(sqrt(2) + 4) + 3*sqrt(7)*2^(3/4)*a*sqrt(2*sqrt(2) + 8)*(sqrt(2) - 4) - 12*sqrt(7)*2^(3/4)*b*sqrt(2*sqrt(2) +
8)*(sqrt(2) - 4) - 3*2^(3/4)*a*(sqrt(2) + 4)*sqrt(-2*sqrt(2) + 8) + 12*2^(3/4)*b*(sqrt(2) + 4)*sqrt(-2*sqrt(2
) + 8) - 2^(3/4)*a*(sqrt(2) - 4)*sqrt(-2*sqrt(2) + 8) + 4*2^(3/4)*b*(sqrt(2) - 4)*sqrt(-2*sqrt(2) + 8) + 88*sq
rt(7)*2^(1/4)*a*sqrt(2*sqrt(2) + 8) - 16*sqrt(7)*2^(1/4)*b*sqrt(2*sqrt(2) + 8) - 88*2^(1/4)*a*sqrt(-2*sqrt(2)
+ 8) + 16*2^(1/4)*b*sqrt(-2*sqrt(2) + 8))*arctan(2*2^(3/4)*sqrt(1/2)*(x + 2^(1/4)*sqrt(-1/8*sqrt(2) + 1/2))/sq
rt(sqrt(2) + 4)) + 1/25088*sqrt(7)*(sqrt(7)*2^(3/4)*a*sqrt(2*sqrt(2) + 8)*(sqrt(2) + 4) - 4*sqrt(7)*2^(3/4)*b*
sqrt(2*sqrt(2) + 8)*(sqrt(2) + 4) + 3*sqrt(7)*2^(3/4)*a*sqrt(2*sqrt(2) + 8)*(sqrt(2) - 4) - 12*sqrt(7)*2^(3/4)
*b*sqrt(2*sqrt(2) + 8)*(sqrt(2) - 4) - 3*2^(3/4)*a*(sqrt(2) + 4)*sqrt(-2*sqrt(2) + 8) + 12*2^(3/4)*b*(sqrt(2)
+ 4)*sqrt(-2*sqrt(2) + 8) - 2^(3/4)*a*(sqrt(2) - 4)*sqrt(-2*sqrt(2) + 8) + 4*2^(3/4)*b*(sqrt(2) - 4)*sqrt(-2*s
qrt(2) + 8) + 88*sqrt(7)*2^(1/4)*a*sqrt(2*sqrt(2) + 8) - 16*sqrt(7)*2^(1/4)*b*sqrt(2*sqrt(2) + 8) - 88*2^(1/4)
*a*sqrt(-2*sqrt(2) + 8) + 16*2^(1/4)*b*sqrt(-2*sqrt(2) + 8))*arctan(2*2^(3/4)*sqrt(1/2)*(x - 2^(1/4)*sqrt(-1/8
*sqrt(2) + 1/2))/sqrt(sqrt(2) + 4)) + 1/50176*sqrt(7)*(3*sqrt(7)*2^(3/4)*a*(sqrt(2) + 4)*sqrt(-2*sqrt(2) + 8)
- 12*sqrt(7)*2^(3/4)*b*(sqrt(2) + 4)*sqrt(-2*sqrt(2) + 8) + sqrt(7)*2^(3/4)*a*(sqrt(2) - 4)*sqrt(-2*sqrt(2) +
8) - 4*sqrt(7)*2^(3/4)*b*(sqrt(2) - 4)*sqrt(-2*sqrt(2) + 8) + 2^(3/4)*a*sqrt(2*sqrt(2) + 8)*(sqrt(2) + 4) - 4*
2^(3/4)*b*sqrt(2*sqrt(2) + 8)*(sqrt(2) + 4) + 3*2^(3/4)*a*sqrt(2*sqrt(2) + 8)*(sqrt(2) - 4) - 12*2^(3/4)*b*sqr
t(2*sqrt(2) + 8)*(sqrt(2) - 4) + 88*sqrt(7)*2^(1/4)*a*sqrt(-2*sqrt(2) + 8) - 16*sqrt(7)*2^(1/4)*b*sqrt(-2*sqrt
(2) + 8) + 88*2^(1/4)*a*sqrt(2*sqrt(2) + 8) - 16*2^(1/4)*b*sqrt(2*sqrt(2) + 8))*log(x^2 + 2*2^(1/4)*x*sqrt(-1/
8*sqrt(2) + 1/2) + sqrt(2)) - 1/50176*sqrt(7)*(3*sqrt(7)*2^(3/4)*a*(sqrt(2) + 4)*sqrt(-2*sqrt(2) + 8) - 12*sqr
t(7)*2^(3/4)*b*(sqrt(2) + 4)*sqrt(-2*sqrt(2) + 8) + sqrt(7)*2^(3/4)*a*(sqrt(2) - 4)*sqrt(-2*sqrt(2) + 8) - 4*s
qrt(7)*2^(3/4)*b*(sqrt(2) - 4)*sqrt(-2*sqrt(2) + 8) + 2^(3/4)*a*sqrt(2*sqrt(2) + 8)*(sqrt(2) + 4) - 4*2^(3/4)*
b*sqrt(2*sqrt(2) + 8)*(sqrt(2) + 4) + 3*2^(3/4)*a*sqrt(2*sqrt(2) + 8)*(sqrt(2) - 4) - 12*2^(3/4)*b*sqrt(2*sqrt
(2) + 8)*(sqrt(2) - 4) + 88*sqrt(7)*2^(1/4)*a*sqrt(-2*sqrt(2) + 8) - 16*sqrt(7)*2^(1/4)*b*sqrt(-2*sqrt(2) + 8)
+ 88*2^(1/4)*a*sqrt(2*sqrt(2) + 8) - 16*2^(1/4)*b*sqrt(2*sqrt(2) + 8))*log(x^2 - 2*2^(1/4)*x*sqrt(-1/8*sqrt(2
) + 1/2) + sqrt(2)) - 1/28*(a*x^3 - 4*b*x^3 - 3*a*x - 2*b*x)/(x^4 + x^2 + 2)

Mupad [B] (verification not implemented)

Time = 13.52 (sec) , antiderivative size = 1491, normalized size of antiderivative = 4.72 \[ \int \frac {a+b x^2}{\left (2+x^2+x^4\right )^2} \, dx=\text {Too large to display} \]

[In]

int((a + b*x^2)/(x^2 + x^4 + 2)^2,x)

[Out]

atan((b^2*x*((7^(1/2)*a^2*17i)/12544 - (107*a*b)/21952 - (7^(1/2)*b^2*1i)/3136 + (211*a^2)/87808 + (25*b^2)/21
952 - (7^(1/2)*a*b*1i)/3136)^(1/2)*1i)/(4*((7^(1/2)*a^3*187i)/6272 + (7^(1/2)*b^3*1i)/784 + (3*a*b^2)/1568 - (
183*a^2*b)/3136 + (255*a^3)/6272 + (9*b^3)/784 - (7^(1/2)*a*b^2*9i)/1568 - (7^(1/2)*a^2*b*39i)/3136)) - (a^2*x
*((7^(1/2)*a^2*17i)/12544 - (107*a*b)/21952 - (7^(1/2)*b^2*1i)/3136 + (211*a^2)/87808 + (25*b^2)/21952 - (7^(1
/2)*a*b*1i)/3136)^(1/2)*17i)/(16*((7^(1/2)*a^3*187i)/6272 + (7^(1/2)*b^3*1i)/784 + (3*a*b^2)/1568 - (183*a^2*b
)/3136 + (255*a^3)/6272 + (9*b^3)/784 - (7^(1/2)*a*b^2*9i)/1568 - (7^(1/2)*a^2*b*39i)/3136)) + (a*b*x*((7^(1/2
)*a^2*17i)/12544 - (107*a*b)/21952 - (7^(1/2)*b^2*1i)/3136 + (211*a^2)/87808 + (25*b^2)/21952 - (7^(1/2)*a*b*1
i)/3136)^(1/2)*1i)/(4*((7^(1/2)*a^3*187i)/6272 + (7^(1/2)*b^3*1i)/784 + (3*a*b^2)/1568 - (183*a^2*b)/3136 + (2
55*a^3)/6272 + (9*b^3)/784 - (7^(1/2)*a*b^2*9i)/1568 - (7^(1/2)*a^2*b*39i)/3136)) - (17*7^(1/2)*a^2*x*((7^(1/2
)*a^2*17i)/12544 - (107*a*b)/21952 - (7^(1/2)*b^2*1i)/3136 + (211*a^2)/87808 + (25*b^2)/21952 - (7^(1/2)*a*b*1
i)/3136)^(1/2))/(112*((7^(1/2)*a^3*187i)/6272 + (7^(1/2)*b^3*1i)/784 + (3*a*b^2)/1568 - (183*a^2*b)/3136 + (25
5*a^3)/6272 + (9*b^3)/784 - (7^(1/2)*a*b^2*9i)/1568 - (7^(1/2)*a^2*b*39i)/3136)) + (7^(1/2)*b^2*x*((7^(1/2)*a^
2*17i)/12544 - (107*a*b)/21952 - (7^(1/2)*b^2*1i)/3136 + (211*a^2)/87808 + (25*b^2)/21952 - (7^(1/2)*a*b*1i)/3
136)^(1/2))/(28*((7^(1/2)*a^3*187i)/6272 + (7^(1/2)*b^3*1i)/784 + (3*a*b^2)/1568 - (183*a^2*b)/3136 + (255*a^3
)/6272 + (9*b^3)/784 - (7^(1/2)*a*b^2*9i)/1568 - (7^(1/2)*a^2*b*39i)/3136)) + (7^(1/2)*a*b*x*((7^(1/2)*a^2*17i
)/12544 - (107*a*b)/21952 - (7^(1/2)*b^2*1i)/3136 + (211*a^2)/87808 + (25*b^2)/21952 - (7^(1/2)*a*b*1i)/3136)^
(1/2))/(28*((7^(1/2)*a^3*187i)/6272 + (7^(1/2)*b^3*1i)/784 + (3*a*b^2)/1568 - (183*a^2*b)/3136 + (255*a^3)/627
2 + (9*b^3)/784 - (7^(1/2)*a*b^2*9i)/1568 - (7^(1/2)*a^2*b*39i)/3136)))*((7^(1/2)*a^2*17i)/12544 - (107*a*b)/2
1952 - (7^(1/2)*b^2*1i)/3136 + (211*a^2)/87808 + (25*b^2)/21952 - (7^(1/2)*a*b*1i)/3136)^(1/2)*2i - atan((a^2*
x*((7^(1/2)*b^2*1i)/3136 - (7^(1/2)*a^2*17i)/12544 - (107*a*b)/21952 + (211*a^2)/87808 + (25*b^2)/21952 + (7^(
1/2)*a*b*1i)/3136)^(1/2)*17i)/(16*((3*a*b^2)/1568 - (7^(1/2)*b^3*1i)/784 - (7^(1/2)*a^3*187i)/6272 - (183*a^2*
b)/3136 + (255*a^3)/6272 + (9*b^3)/784 + (7^(1/2)*a*b^2*9i)/1568 + (7^(1/2)*a^2*b*39i)/3136)) - (b^2*x*((7^(1/
2)*b^2*1i)/3136 - (7^(1/2)*a^2*17i)/12544 - (107*a*b)/21952 + (211*a^2)/87808 + (25*b^2)/21952 + (7^(1/2)*a*b*
1i)/3136)^(1/2)*1i)/(4*((3*a*b^2)/1568 - (7^(1/2)*b^3*1i)/784 - (7^(1/2)*a^3*187i)/6272 - (183*a^2*b)/3136 + (
255*a^3)/6272 + (9*b^3)/784 + (7^(1/2)*a*b^2*9i)/1568 + (7^(1/2)*a^2*b*39i)/3136)) - (a*b*x*((7^(1/2)*b^2*1i)/
3136 - (7^(1/2)*a^2*17i)/12544 - (107*a*b)/21952 + (211*a^2)/87808 + (25*b^2)/21952 + (7^(1/2)*a*b*1i)/3136)^(
1/2)*1i)/(4*((3*a*b^2)/1568 - (7^(1/2)*b^3*1i)/784 - (7^(1/2)*a^3*187i)/6272 - (183*a^2*b)/3136 + (255*a^3)/62
72 + (9*b^3)/784 + (7^(1/2)*a*b^2*9i)/1568 + (7^(1/2)*a^2*b*39i)/3136)) - (17*7^(1/2)*a^2*x*((7^(1/2)*b^2*1i)/
3136 - (7^(1/2)*a^2*17i)/12544 - (107*a*b)/21952 + (211*a^2)/87808 + (25*b^2)/21952 + (7^(1/2)*a*b*1i)/3136)^(
1/2))/(112*((3*a*b^2)/1568 - (7^(1/2)*b^3*1i)/784 - (7^(1/2)*a^3*187i)/6272 - (183*a^2*b)/3136 + (255*a^3)/627
2 + (9*b^3)/784 + (7^(1/2)*a*b^2*9i)/1568 + (7^(1/2)*a^2*b*39i)/3136)) + (7^(1/2)*b^2*x*((7^(1/2)*b^2*1i)/3136
- (7^(1/2)*a^2*17i)/12544 - (107*a*b)/21952 + (211*a^2)/87808 + (25*b^2)/21952 + (7^(1/2)*a*b*1i)/3136)^(1/2)
)/(28*((3*a*b^2)/1568 - (7^(1/2)*b^3*1i)/784 - (7^(1/2)*a^3*187i)/6272 - (183*a^2*b)/3136 + (255*a^3)/6272 + (
9*b^3)/784 + (7^(1/2)*a*b^2*9i)/1568 + (7^(1/2)*a^2*b*39i)/3136)) + (7^(1/2)*a*b*x*((7^(1/2)*b^2*1i)/3136 - (7
^(1/2)*a^2*17i)/12544 - (107*a*b)/21952 + (211*a^2)/87808 + (25*b^2)/21952 + (7^(1/2)*a*b*1i)/3136)^(1/2))/(28
*((3*a*b^2)/1568 - (7^(1/2)*b^3*1i)/784 - (7^(1/2)*a^3*187i)/6272 - (183*a^2*b)/3136 + (255*a^3)/6272 + (9*b^3
)/784 + (7^(1/2)*a*b^2*9i)/1568 + (7^(1/2)*a^2*b*39i)/3136)))*((7^(1/2)*b^2*1i)/3136 - (7^(1/2)*a^2*17i)/12544
- (107*a*b)/21952 + (211*a^2)/87808 + (25*b^2)/21952 + (7^(1/2)*a*b*1i)/3136)^(1/2)*2i - (x^3*(a/28 - b/7) -
x*((3*a)/28 + b/14))/(x^2 + x^4 + 2)