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

Optimal. Leaf size=234 \[ -\frac{\left (a-\sqrt{2} b\right ) \log \left (x^2-\sqrt{2 \sqrt{2}-1} x+\sqrt{2}\right )}{4 \sqrt{2 \left (2 \sqrt{2}-1\right )}}+\frac{\left (a-\sqrt{2} b\right ) \log \left (x^2+\sqrt{2 \sqrt{2}-1} x+\sqrt{2}\right )}{4 \sqrt{2 \left (2 \sqrt{2}-1\right )}}-\frac{1}{2} \sqrt{\frac{1}{14} \left (2 \sqrt{2}-1\right )} \left (a+\sqrt{2} b\right ) \tan ^{-1}\left (\frac{\sqrt{2 \sqrt{2}-1}-2 x}{\sqrt{1+2 \sqrt{2}}}\right )+\frac{1}{2} \sqrt{\frac{1}{14} \left (2 \sqrt{2}-1\right )} \left (a+\sqrt{2} b\right ) \tan ^{-1}\left (\frac{2 x+\sqrt{2 \sqrt{2}-1}}{\sqrt{1+2 \sqrt{2}}}\right ) \]

[Out]

-(Sqrt[(-1 + 2*Sqrt[2])/14]*(a + Sqrt[2]*b)*ArcTan[(Sqrt[-1 + 2*Sqrt[2]] - 2*x)/Sqrt[1 + 2*Sqrt[2]]])/2 + (Sqr
t[(-1 + 2*Sqrt[2])/14]*(a + Sqrt[2]*b)*ArcTan[(Sqrt[-1 + 2*Sqrt[2]] + 2*x)/Sqrt[1 + 2*Sqrt[2]]])/2 - ((a - Sqr
t[2]*b)*Log[Sqrt[2] - Sqrt[-1 + 2*Sqrt[2]]*x + x^2])/(4*Sqrt[2*(-1 + 2*Sqrt[2])]) + ((a - Sqrt[2]*b)*Log[Sqrt[
2] + Sqrt[-1 + 2*Sqrt[2]]*x + x^2])/(4*Sqrt[2*(-1 + 2*Sqrt[2])])

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Rubi [A]  time = 0.229521, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {1169, 634, 618, 204, 628} \[ -\frac{\left (a-\sqrt{2} b\right ) \log \left (x^2-\sqrt{2 \sqrt{2}-1} x+\sqrt{2}\right )}{4 \sqrt{2 \left (2 \sqrt{2}-1\right )}}+\frac{\left (a-\sqrt{2} b\right ) \log \left (x^2+\sqrt{2 \sqrt{2}-1} x+\sqrt{2}\right )}{4 \sqrt{2 \left (2 \sqrt{2}-1\right )}}-\frac{1}{2} \sqrt{\frac{1}{14} \left (2 \sqrt{2}-1\right )} \left (a+\sqrt{2} b\right ) \tan ^{-1}\left (\frac{\sqrt{2 \sqrt{2}-1}-2 x}{\sqrt{1+2 \sqrt{2}}}\right )+\frac{1}{2} \sqrt{\frac{1}{14} \left (2 \sqrt{2}-1\right )} \left (a+\sqrt{2} b\right ) \tan ^{-1}\left (\frac{2 x+\sqrt{2 \sqrt{2}-1}}{\sqrt{1+2 \sqrt{2}}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)/(2 + x^2 + x^4),x]

[Out]

-(Sqrt[(-1 + 2*Sqrt[2])/14]*(a + Sqrt[2]*b)*ArcTan[(Sqrt[-1 + 2*Sqrt[2]] - 2*x)/Sqrt[1 + 2*Sqrt[2]]])/2 + (Sqr
t[(-1 + 2*Sqrt[2])/14]*(a + Sqrt[2]*b)*ArcTan[(Sqrt[-1 + 2*Sqrt[2]] + 2*x)/Sqrt[1 + 2*Sqrt[2]]])/2 - ((a - Sqr
t[2]*b)*Log[Sqrt[2] - Sqrt[-1 + 2*Sqrt[2]]*x + x^2])/(4*Sqrt[2*(-1 + 2*Sqrt[2])]) + ((a - Sqrt[2]*b)*Log[Sqrt[
2] + Sqrt[-1 + 2*Sqrt[2]]*x + x^2])/(4*Sqrt[2*(-1 + 2*Sqrt[2])])

Rule 1169

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 634

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 618

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 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

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]

Rubi steps

\begin{align*} \int \frac{a+b x^2}{2+x^2+x^4} \, dx &=\frac{\int \frac{\sqrt{-1+2 \sqrt{2}} a-\left (a-\sqrt{2} b\right ) x}{\sqrt{2}-\sqrt{-1+2 \sqrt{2}} x+x^2} \, dx}{2 \sqrt{2 \left (-1+2 \sqrt{2}\right )}}+\frac{\int \frac{\sqrt{-1+2 \sqrt{2}} a+\left (a-\sqrt{2} b\right ) x}{\sqrt{2}+\sqrt{-1+2 \sqrt{2}} x+x^2} \, dx}{2 \sqrt{2 \left (-1+2 \sqrt{2}\right )}}\\ &=\frac{1}{8} \left (\sqrt{2} a+2 b\right ) \int \frac{1}{\sqrt{2}-\sqrt{-1+2 \sqrt{2}} x+x^2} \, dx+\frac{1}{8} \left (\sqrt{2} a+2 b\right ) \int \frac{1}{\sqrt{2}+\sqrt{-1+2 \sqrt{2}} x+x^2} \, dx-\frac{\left (a-\sqrt{2} b\right ) \int \frac{-\sqrt{-1+2 \sqrt{2}}+2 x}{\sqrt{2}-\sqrt{-1+2 \sqrt{2}} x+x^2} \, dx}{4 \sqrt{2 \left (-1+2 \sqrt{2}\right )}}+\frac{\left (a-\sqrt{2} b\right ) \int \frac{\sqrt{-1+2 \sqrt{2}}+2 x}{\sqrt{2}+\sqrt{-1+2 \sqrt{2}} x+x^2} \, dx}{4 \sqrt{2 \left (-1+2 \sqrt{2}\right )}}\\ &=-\frac{\left (a-\sqrt{2} b\right ) \log \left (\sqrt{2}-\sqrt{-1+2 \sqrt{2}} x+x^2\right )}{4 \sqrt{2 \left (-1+2 \sqrt{2}\right )}}+\frac{\left (a-\sqrt{2} b\right ) \log \left (\sqrt{2}+\sqrt{-1+2 \sqrt{2}} x+x^2\right )}{4 \sqrt{2 \left (-1+2 \sqrt{2}\right )}}-\frac{1}{4} \left (\sqrt{2} a+2 b\right ) \operatorname{Subst}\left (\int \frac{1}{-1-2 \sqrt{2}-x^2} \, dx,x,-\sqrt{-1+2 \sqrt{2}}+2 x\right )-\frac{1}{4} \left (\sqrt{2} a+2 b\right ) \operatorname{Subst}\left (\int \frac{1}{-1-2 \sqrt{2}-x^2} \, dx,x,\sqrt{-1+2 \sqrt{2}}+2 x\right )\\ &=-\frac{\left (a+\sqrt{2} b\right ) \tan ^{-1}\left (\frac{\sqrt{-1+2 \sqrt{2}}-2 x}{\sqrt{1+2 \sqrt{2}}}\right )}{2 \sqrt{2 \left (1+2 \sqrt{2}\right )}}+\frac{\left (a+\sqrt{2} b\right ) \tan ^{-1}\left (\frac{\sqrt{-1+2 \sqrt{2}}+2 x}{\sqrt{1+2 \sqrt{2}}}\right )}{2 \sqrt{2 \left (1+2 \sqrt{2}\right )}}-\frac{\left (a-\sqrt{2} b\right ) \log \left (\sqrt{2}-\sqrt{-1+2 \sqrt{2}} x+x^2\right )}{4 \sqrt{2 \left (-1+2 \sqrt{2}\right )}}+\frac{\left (a-\sqrt{2} b\right ) \log \left (\sqrt{2}+\sqrt{-1+2 \sqrt{2}} x+x^2\right )}{4 \sqrt{2 \left (-1+2 \sqrt{2}\right )}}\\ \end{align*}

Mathematica [C]  time = 0.115918, size = 111, normalized size = 0.47 \[ \frac{\left (\left (\sqrt{7}+i\right ) b-2 i a\right ) \tan ^{-1}\left (\frac{x}{\sqrt{\frac{1}{2} \left (1-i \sqrt{7}\right )}}\right )}{\sqrt{14-14 i \sqrt{7}}}+\frac{\left (2 i a+\left (\sqrt{7}-i\right ) b\right ) \tan ^{-1}\left (\frac{x}{\sqrt{\frac{1}{2} \left (1+i \sqrt{7}\right )}}\right )}{\sqrt{14+14 i \sqrt{7}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((-2*I)*a + (I + Sqrt[7])*b)*ArcTan[x/Sqrt[(1 - I*Sqrt[7])/2]])/Sqrt[14 - (14*I)*Sqrt[7]] + (((2*I)*a + (-I +
Sqrt[7])*b)*ArcTan[x/Sqrt[(1 + I*Sqrt[7])/2]])/Sqrt[14 + (14*I)*Sqrt[7]]

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Maple [B]  time = 0.08, size = 710, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/56*ln(x^2+2^(1/2)-x*(-1+2*2^(1/2))^(1/2))*2^(1/2)*(-1+2*2^(1/2))^(1/2)*a+1/14*ln(x^2+2^(1/2)-x*(-1+2*2^(1/2
))^(1/2))*2^(1/2)*(-1+2*2^(1/2))^(1/2)*b-1/14*ln(x^2+2^(1/2)-x*(-1+2*2^(1/2))^(1/2))*(-1+2*2^(1/2))^(1/2)*a+1/
28*ln(x^2+2^(1/2)-x*(-1+2*2^(1/2))^(1/2))*(-1+2*2^(1/2))^(1/2)*b-1/28/(1+2*2^(1/2))^(1/2)*arctan((2*x-(-1+2*2^
(1/2))^(1/2))/(1+2*2^(1/2))^(1/2))*2^(1/2)*(-1+2*2^(1/2))*a+1/7/(1+2*2^(1/2))^(1/2)*arctan((2*x-(-1+2*2^(1/2))
^(1/2))/(1+2*2^(1/2))^(1/2))*2^(1/2)*(-1+2*2^(1/2))*b-1/7/(1+2*2^(1/2))^(1/2)*arctan((2*x-(-1+2*2^(1/2))^(1/2)
)/(1+2*2^(1/2))^(1/2))*(-1+2*2^(1/2))*a+1/14/(1+2*2^(1/2))^(1/2)*arctan((2*x-(-1+2*2^(1/2))^(1/2))/(1+2*2^(1/2
))^(1/2))*(-1+2*2^(1/2))*b+1/2/(1+2*2^(1/2))^(1/2)*arctan((2*x-(-1+2*2^(1/2))^(1/2))/(1+2*2^(1/2))^(1/2))*2^(1
/2)*a+1/56*ln(x^2+2^(1/2)+x*(-1+2*2^(1/2))^(1/2))*2^(1/2)*(-1+2*2^(1/2))^(1/2)*a-1/14*ln(x^2+2^(1/2)+x*(-1+2*2
^(1/2))^(1/2))*2^(1/2)*(-1+2*2^(1/2))^(1/2)*b+1/14*ln(x^2+2^(1/2)+x*(-1+2*2^(1/2))^(1/2))*(-1+2*2^(1/2))^(1/2)
*a-1/28*ln(x^2+2^(1/2)+x*(-1+2*2^(1/2))^(1/2))*(-1+2*2^(1/2))^(1/2)*b-1/28/(1+2*2^(1/2))^(1/2)*arctan((2*x+(-1
+2*2^(1/2))^(1/2))/(1+2*2^(1/2))^(1/2))*2^(1/2)*(-1+2*2^(1/2))*a+1/7/(1+2*2^(1/2))^(1/2)*arctan((2*x+(-1+2*2^(
1/2))^(1/2))/(1+2*2^(1/2))^(1/2))*2^(1/2)*(-1+2*2^(1/2))*b-1/7/(1+2*2^(1/2))^(1/2)*arctan((2*x+(-1+2*2^(1/2))^
(1/2))/(1+2*2^(1/2))^(1/2))*(-1+2*2^(1/2))*a+1/14/(1+2*2^(1/2))^(1/2)*arctan((2*x+(-1+2*2^(1/2))^(1/2))/(1+2*2
^(1/2))^(1/2))*(-1+2*2^(1/2))*b+1/2/(1+2*2^(1/2))^(1/2)*arctan((2*x+(-1+2*2^(1/2))^(1/2))/(1+2*2^(1/2))^(1/2))
*2^(1/2)*a

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b x^{2} + a}{x^{4} + x^{2} + 2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.27976, size = 7710, normalized size = 32.95 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/112*(28*sqrt(2)*sqrt(1/7)*(8*a^4 - 16*a^3*b + 40*a^2*b^2 - 32*a*b^3 + 32*b^4)^(1/4)*sqrt(a^4 - 2*a^3*b + 5*a
^2*b^2 - 4*a*b^3 + 4*b^4)*sqrt(a^4 - 4*a^2*b^2 + 4*b^4)*sqrt((4*a^4 - 8*a^3*b + 20*a^2*b^2 - 16*a*b^3 + 16*b^4
- sqrt(2)*sqrt(a^4 - 2*a^3*b + 5*a^2*b^2 - 4*a*b^3 + 4*b^4)*(a^2 - 8*a*b + 2*b^2))/(a^4 - 4*a^2*b^2 + 4*b^4))
*arctan(-1/28*(7*sqrt(1/2)*sqrt(1/7)*(8*a^4 - 16*a^3*b + 40*a^2*b^2 - 32*a*b^3 + 32*b^4)^(3/4)*(sqrt(2)*sqrt(a
^4 - 2*a^3*b + 5*a^2*b^2 - 4*a*b^3 + 4*b^4)*sqrt(a^4 - 4*a^2*b^2 + 4*b^4)*a - 2*sqrt(a^4 - 4*a^2*b^2 + 4*b^4)*
(a^2*b - a*b^2 + 2*b^3))*sqrt((4*a^4 - 8*a^3*b + 20*a^2*b^2 - 16*a*b^3 + 16*b^4 - sqrt(2)*sqrt(a^4 - 2*a^3*b +
5*a^2*b^2 - 4*a*b^3 + 4*b^4)*(a^2 - 8*a*b + 2*b^2))/(a^4 - 4*a^2*b^2 + 4*b^4))*sqrt((2*(a^4 - 2*a^3*b + 5*a^2
*b^2 - 4*a*b^3 + 4*b^4)*x^2 + sqrt(1/7)*(8*a^4 - 16*a^3*b + 40*a^2*b^2 - 32*a*b^3 + 32*b^4)^(1/4)*(sqrt(7)*sqr
t(2)*sqrt(a^4 - 2*a^3*b + 5*a^2*b^2 - 4*a*b^3 + 4*b^4)*b*x - sqrt(7)*(a^3 - a^2*b + 2*a*b^2)*x)*sqrt((4*a^4 -
8*a^3*b + 20*a^2*b^2 - 16*a*b^3 + 16*b^4 - sqrt(2)*sqrt(a^4 - 2*a^3*b + 5*a^2*b^2 - 4*a*b^3 + 4*b^4)*(a^2 - 8*
a*b + 2*b^2))/(a^4 - 4*a^2*b^2 + 4*b^4)) + 2*sqrt(2)*sqrt(a^4 - 2*a^3*b + 5*a^2*b^2 - 4*a*b^3 + 4*b^4)*(a^2 -
a*b + 2*b^2))/(a^4 - 2*a^3*b + 5*a^2*b^2 - 4*a*b^3 + 4*b^4)) + 8*sqrt(7)*sqrt(2)*(a^4 - 2*a^3*b + 5*a^2*b^2 -
4*a*b^3 + 4*b^4)^(3/2)*sqrt(a^4 - 4*a^2*b^2 + 4*b^4) - 7*sqrt(1/7)*(8*a^4 - 16*a^3*b + 40*a^2*b^2 - 32*a*b^3 +
32*b^4)^(3/4)*(sqrt(2)*sqrt(a^4 - 2*a^3*b + 5*a^2*b^2 - 4*a*b^3 + 4*b^4)*sqrt(a^4 - 4*a^2*b^2 + 4*b^4)*a*x -
2*sqrt(a^4 - 4*a^2*b^2 + 4*b^4)*(a^2*b - a*b^2 + 2*b^3)*x)*sqrt((4*a^4 - 8*a^3*b + 20*a^2*b^2 - 16*a*b^3 + 16*
b^4 - sqrt(2)*sqrt(a^4 - 2*a^3*b + 5*a^2*b^2 - 4*a*b^3 + 4*b^4)*(a^2 - 8*a*b + 2*b^2))/(a^4 - 4*a^2*b^2 + 4*b^
4)) - 4*sqrt(7)*(a^6 - 3*a^5*b + 9*a^4*b^2 - 13*a^3*b^3 + 18*a^2*b^4 - 12*a*b^5 + 8*b^6)*sqrt(a^4 - 4*a^2*b^2
+ 4*b^4))/(a^8 - 3*a^7*b + 7*a^6*b^2 - 7*a^5*b^3 + 14*a^3*b^5 - 28*a^2*b^6 + 24*a*b^7 - 16*b^8)) + 28*sqrt(2)*
sqrt(1/7)*(8*a^4 - 16*a^3*b + 40*a^2*b^2 - 32*a*b^3 + 32*b^4)^(1/4)*sqrt(a^4 - 2*a^3*b + 5*a^2*b^2 - 4*a*b^3 +
4*b^4)*sqrt(a^4 - 4*a^2*b^2 + 4*b^4)*sqrt((4*a^4 - 8*a^3*b + 20*a^2*b^2 - 16*a*b^3 + 16*b^4 - sqrt(2)*sqrt(a^
4 - 2*a^3*b + 5*a^2*b^2 - 4*a*b^3 + 4*b^4)*(a^2 - 8*a*b + 2*b^2))/(a^4 - 4*a^2*b^2 + 4*b^4))*arctan(-1/28*(7*s
qrt(1/2)*sqrt(1/7)*(8*a^4 - 16*a^3*b + 40*a^2*b^2 - 32*a*b^3 + 32*b^4)^(3/4)*(sqrt(2)*sqrt(a^4 - 2*a^3*b + 5*a
^2*b^2 - 4*a*b^3 + 4*b^4)*sqrt(a^4 - 4*a^2*b^2 + 4*b^4)*a - 2*sqrt(a^4 - 4*a^2*b^2 + 4*b^4)*(a^2*b - a*b^2 + 2
*b^3))*sqrt((4*a^4 - 8*a^3*b + 20*a^2*b^2 - 16*a*b^3 + 16*b^4 - sqrt(2)*sqrt(a^4 - 2*a^3*b + 5*a^2*b^2 - 4*a*b
^3 + 4*b^4)*(a^2 - 8*a*b + 2*b^2))/(a^4 - 4*a^2*b^2 + 4*b^4))*sqrt((2*(a^4 - 2*a^3*b + 5*a^2*b^2 - 4*a*b^3 + 4
*b^4)*x^2 - sqrt(1/7)*(8*a^4 - 16*a^3*b + 40*a^2*b^2 - 32*a*b^3 + 32*b^4)^(1/4)*(sqrt(7)*sqrt(2)*sqrt(a^4 - 2*
a^3*b + 5*a^2*b^2 - 4*a*b^3 + 4*b^4)*b*x - sqrt(7)*(a^3 - a^2*b + 2*a*b^2)*x)*sqrt((4*a^4 - 8*a^3*b + 20*a^2*b
^2 - 16*a*b^3 + 16*b^4 - sqrt(2)*sqrt(a^4 - 2*a^3*b + 5*a^2*b^2 - 4*a*b^3 + 4*b^4)*(a^2 - 8*a*b + 2*b^2))/(a^4
- 4*a^2*b^2 + 4*b^4)) + 2*sqrt(2)*sqrt(a^4 - 2*a^3*b + 5*a^2*b^2 - 4*a*b^3 + 4*b^4)*(a^2 - a*b + 2*b^2))/(a^4
- 2*a^3*b + 5*a^2*b^2 - 4*a*b^3 + 4*b^4)) - 8*sqrt(7)*sqrt(2)*(a^4 - 2*a^3*b + 5*a^2*b^2 - 4*a*b^3 + 4*b^4)^(
3/2)*sqrt(a^4 - 4*a^2*b^2 + 4*b^4) - 7*sqrt(1/7)*(8*a^4 - 16*a^3*b + 40*a^2*b^2 - 32*a*b^3 + 32*b^4)^(3/4)*(sq
rt(2)*sqrt(a^4 - 2*a^3*b + 5*a^2*b^2 - 4*a*b^3 + 4*b^4)*sqrt(a^4 - 4*a^2*b^2 + 4*b^4)*a*x - 2*sqrt(a^4 - 4*a^2
*b^2 + 4*b^4)*(a^2*b - a*b^2 + 2*b^3)*x)*sqrt((4*a^4 - 8*a^3*b + 20*a^2*b^2 - 16*a*b^3 + 16*b^4 - sqrt(2)*sqrt
(a^4 - 2*a^3*b + 5*a^2*b^2 - 4*a*b^3 + 4*b^4)*(a^2 - 8*a*b + 2*b^2))/(a^4 - 4*a^2*b^2 + 4*b^4)) + 4*sqrt(7)*(a
^6 - 3*a^5*b + 9*a^4*b^2 - 13*a^3*b^3 + 18*a^2*b^4 - 12*a*b^5 + 8*b^6)*sqrt(a^4 - 4*a^2*b^2 + 4*b^4))/(a^8 - 3
*a^7*b + 7*a^6*b^2 - 7*a^5*b^3 + 14*a^3*b^5 - 28*a^2*b^6 + 24*a*b^7 - 16*b^8)) - sqrt(1/7)*(8*a^4 - 16*a^3*b +
40*a^2*b^2 - 32*a*b^3 + 32*b^4)^(1/4)*(sqrt(7)*sqrt(2)*sqrt(a^4 - 2*a^3*b + 5*a^2*b^2 - 4*a*b^3 + 4*b^4)*(a^2
- 8*a*b + 2*b^2) + 4*sqrt(7)*(a^4 - 2*a^3*b + 5*a^2*b^2 - 4*a*b^3 + 4*b^4))*sqrt((4*a^4 - 8*a^3*b + 20*a^2*b^
2 - 16*a*b^3 + 16*b^4 - sqrt(2)*sqrt(a^4 - 2*a^3*b + 5*a^2*b^2 - 4*a*b^3 + 4*b^4)*(a^2 - 8*a*b + 2*b^2))/(a^4
- 4*a^2*b^2 + 4*b^4))*log(8*(a^4 - 2*a^3*b + 5*a^2*b^2 - 4*a*b^3 + 4*b^4)*x^2 + 4*sqrt(1/7)*(8*a^4 - 16*a^3*b
+ 40*a^2*b^2 - 32*a*b^3 + 32*b^4)^(1/4)*(sqrt(7)*sqrt(2)*sqrt(a^4 - 2*a^3*b + 5*a^2*b^2 - 4*a*b^3 + 4*b^4)*b*x
- sqrt(7)*(a^3 - a^2*b + 2*a*b^2)*x)*sqrt((4*a^4 - 8*a^3*b + 20*a^2*b^2 - 16*a*b^3 + 16*b^4 - sqrt(2)*sqrt(a^
4 - 2*a^3*b + 5*a^2*b^2 - 4*a*b^3 + 4*b^4)*(a^2 - 8*a*b + 2*b^2))/(a^4 - 4*a^2*b^2 + 4*b^4)) + 8*sqrt(2)*sqrt(
a^4 - 2*a^3*b + 5*a^2*b^2 - 4*a*b^3 + 4*b^4)*(a^2 - a*b + 2*b^2)) + sqrt(1/7)*(8*a^4 - 16*a^3*b + 40*a^2*b^2 -
32*a*b^3 + 32*b^4)^(1/4)*(sqrt(7)*sqrt(2)*sqrt(a^4 - 2*a^3*b + 5*a^2*b^2 - 4*a*b^3 + 4*b^4)*(a^2 - 8*a*b + 2*
b^2) + 4*sqrt(7)*(a^4 - 2*a^3*b + 5*a^2*b^2 - 4*a*b^3 + 4*b^4))*sqrt((4*a^4 - 8*a^3*b + 20*a^2*b^2 - 16*a*b^3
+ 16*b^4 - sqrt(2)*sqrt(a^4 - 2*a^3*b + 5*a^2*b^2 - 4*a*b^3 + 4*b^4)*(a^2 - 8*a*b + 2*b^2))/(a^4 - 4*a^2*b^2 +
4*b^4))*log(8*(a^4 - 2*a^3*b + 5*a^2*b^2 - 4*a*b^3 + 4*b^4)*x^2 - 4*sqrt(1/7)*(8*a^4 - 16*a^3*b + 40*a^2*b^2
- 32*a*b^3 + 32*b^4)^(1/4)*(sqrt(7)*sqrt(2)*sqrt(a^4 - 2*a^3*b + 5*a^2*b^2 - 4*a*b^3 + 4*b^4)*b*x - sqrt(7)*(a
^3 - a^2*b + 2*a*b^2)*x)*sqrt((4*a^4 - 8*a^3*b + 20*a^2*b^2 - 16*a*b^3 + 16*b^4 - sqrt(2)*sqrt(a^4 - 2*a^3*b +
5*a^2*b^2 - 4*a*b^3 + 4*b^4)*(a^2 - 8*a*b + 2*b^2))/(a^4 - 4*a^2*b^2 + 4*b^4)) + 8*sqrt(2)*sqrt(a^4 - 2*a^3*b
+ 5*a^2*b^2 - 4*a*b^3 + 4*b^4)*(a^2 - a*b + 2*b^2)))/(a^4 - 2*a^3*b + 5*a^2*b^2 - 4*a*b^3 + 4*b^4)

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Sympy [A]  time = 0.899314, size = 122, normalized size = 0.52 \begin{align*} \operatorname{RootSum}{\left (1568 t^{4} + t^{2} \left (- 28 a^{2} + 224 a b - 56 b^{2}\right ) + a^{4} - 2 a^{3} b + 5 a^{2} b^{2} - 4 a b^{3} + 4 b^{4}, \left ( t \mapsto t \log{\left (x + \frac{112 t^{3} a - 448 t^{3} b + 6 t a^{3} + 12 t a^{2} b - 48 t a b^{2} + 8 t b^{3}}{a^{4} - a^{3} b + 2 a b^{3} - 4 b^{4}} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(1568*_t**4 + _t**2*(-28*a**2 + 224*a*b - 56*b**2) + a**4 - 2*a**3*b + 5*a**2*b**2 - 4*a*b**3 + 4*b**4,
Lambda(_t, _t*log(x + (112*_t**3*a - 448*_t**3*b + 6*_t*a**3 + 12*_t*a**2*b - 48*_t*a*b**2 + 8*_t*b**3)/(a**4
- a**3*b + 2*a*b**3 - 4*b**4))))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b x^{2} + a}{x^{4} + x^{2} + 2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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