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

Optimal. Leaf size=316 \[ \frac{x \left (x^2 (-(a-4 b))+3 a+2 b\right )}{28 \left (x^4+x^2+2\right )}-\frac{\left (\sqrt{2} (a-4 b)+11 a-2 b\right ) \log \left (x^2-\sqrt{2 \sqrt{2}-1} x+\sqrt{2}\right )}{112 \sqrt{2 \left (2 \sqrt{2}-1\right )}}+\frac{\left (\left (11+\sqrt{2}\right ) a-2 \left (2 \sqrt{2} b+b\right )\right ) \log \left (x^2+\sqrt{2 \sqrt{2}-1} x+\sqrt{2}\right )}{112 \sqrt{2 \left (2 \sqrt{2}-1\right )}}-\frac{1}{56} \sqrt{\frac{1}{14} \left (2 \sqrt{2}-1\right )} \left (\left (11-\sqrt{2}\right ) a-\left (2-4 \sqrt{2}\right ) b\right ) \tan ^{-1}\left (\frac{\sqrt{2 \sqrt{2}-1}-2 x}{\sqrt{1+2 \sqrt{2}}}\right )+\frac{1}{56} \sqrt{\frac{1}{14} \left (2 \sqrt{2}-1\right )} \left (\left (11-\sqrt{2}\right ) a-\left (2-4 \sqrt{2}\right ) b\right ) \tan ^{-1}\left (\frac{2 x+\sqrt{2 \sqrt{2}-1}}{\sqrt{1+2 \sqrt{2}}}\right ) \]

[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])])

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Rubi [A]  time = 0.289108, antiderivative size = 316, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1178, 1169, 634, 618, 204, 628} \[ \frac{x \left (x^2 (-(a-4 b))+3 a+2 b\right )}{28 \left (x^4+x^2+2\right )}-\frac{\left (\sqrt{2} (a-4 b)+11 a-2 b\right ) \log \left (x^2-\sqrt{2 \sqrt{2}-1} x+\sqrt{2}\right )}{112 \sqrt{2 \left (2 \sqrt{2}-1\right )}}+\frac{\left (\left (11+\sqrt{2}\right ) a-2 \left (2 \sqrt{2} b+b\right )\right ) \log \left (x^2+\sqrt{2 \sqrt{2}-1} x+\sqrt{2}\right )}{112 \sqrt{2 \left (2 \sqrt{2}-1\right )}}-\frac{1}{56} \sqrt{\frac{1}{14} \left (2 \sqrt{2}-1\right )} \left (\left (11-\sqrt{2}\right ) a-\left (2-4 \sqrt{2}\right ) b\right ) \tan ^{-1}\left (\frac{\sqrt{2 \sqrt{2}-1}-2 x}{\sqrt{1+2 \sqrt{2}}}\right )+\frac{1}{56} \sqrt{\frac{1}{14} \left (2 \sqrt{2}-1\right )} \left (\left (11-\sqrt{2}\right ) a-\left (2-4 \sqrt{2}\right ) b\right ) \tan ^{-1}\left (\frac{2 x+\sqrt{2 \sqrt{2}-1}}{\sqrt{1+2 \sqrt{2}}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)/(2 + x^2 + x^4)^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 1178

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]

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}{\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}{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 ) \operatorname{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 ) \operatorname{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]  time = 0.20857, size = 165, normalized size = 0.52 \[ \frac{2 b \left (2 x^3+x\right )-a x \left (x^2-3\right )}{28 \left (x^4+x^2+2\right )}-\frac{\left (\left (\sqrt{7}+23 i\right ) a-4 \left (\sqrt{7}+2 i\right ) b\right ) \tan ^{-1}\left (\frac{x}{\sqrt{\frac{1}{2} \left (1-i \sqrt{7}\right )}}\right )}{28 \sqrt{14-14 i \sqrt{7}}}-\frac{\left (\left (\sqrt{7}-23 i\right ) a-4 \left (\sqrt{7}-2 i\right ) b\right ) \tan ^{-1}\left (\frac{x}{\sqrt{\frac{1}{2} \left (1+i \sqrt{7}\right )}}\right )}{28 \sqrt{14+14 i \sqrt{7}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.403, size = 756, normalized size = 2.4 \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)^2,x)

[Out]

-1/784*(-(-14*a-28*2^(1/2)*a+112*b*2^(1/2)+56*b)/(1+2*2^(1/2))*x+1/(1+2*2^(1/2))*(-1+2*2^(1/2))^(1/2)*(-70*a-4
2*2^(1/2)*a+56*b*2^(1/2)+28*b))/(x^2+2^(1/2)-x*(-1+2*2^(1/2))^(1/2))-107/1568/(1+2*2^(1/2))*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+25/784/(1+2*2^(1/2))*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-53/784/(1+2*2^(1/2))*ln(x^2+2^(1/2)-x*(-1+2*2^(1/2))^(1/2))*(-1+2*2^(1/2))^(1
/2)*a+11/196/(1+2*2^(1/2))*ln(x^2+2^(1/2)-x*(-1+2*2^(1/2))^(1/2))*(-1+2*2^(1/2))^(1/2)*b+1/16/(1+2*2^(1/2))^(3
/2)*arctan((2*x-(-1+2*2^(1/2))^(1/2))/(1+2*2^(1/2))^(1/2))*2^(1/2)*a+3/8/(1+2*2^(1/2))^(3/2)*arctan((2*x-(-1+2
*2^(1/2))^(1/2))/(1+2*2^(1/2))^(1/2))*a+1/8/(1+2*2^(1/2))^(3/2)*arctan((2*x-(-1+2*2^(1/2))^(1/2))/(1+2*2^(1/2)
)^(1/2))*2^(1/2)*b+1/784*((-14*a-28*2^(1/2)*a+112*b*2^(1/2)+56*b)/(1+2*2^(1/2))*x+1/(1+2*2^(1/2))*(-1+2*2^(1/2
))^(1/2)*(-70*a-42*2^(1/2)*a+56*b*2^(1/2)+28*b))/(x^2+2^(1/2)+x*(-1+2*2^(1/2))^(1/2))+107/1568/(1+2*2^(1/2))*l
n(x^2+2^(1/2)+x*(-1+2*2^(1/2))^(1/2))*2^(1/2)*(-1+2*2^(1/2))^(1/2)*a-25/784/(1+2*2^(1/2))*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+53/784/(1+2*2^(1/2))*ln(x^2+2^(1/2)+x*(-1+2*2^(1/2))^(1/2))*
(-1+2*2^(1/2))^(1/2)*a-11/196/(1+2*2^(1/2))*ln(x^2+2^(1/2)+x*(-1+2*2^(1/2))^(1/2))*(-1+2*2^(1/2))^(1/2)*b+1/16
/(1+2*2^(1/2))^(3/2)*arctan((2*x+(-1+2*2^(1/2))^(1/2))/(1+2*2^(1/2))^(1/2))*2^(1/2)*a+3/8/(1+2*2^(1/2))^(3/2)*
arctan((2*x+(-1+2*2^(1/2))^(1/2))/(1+2*2^(1/2))^(1/2))*a+1/8/(1+2*2^(1/2))^(3/2)*arctan((2*x+(-1+2*2^(1/2))^(1
/2))/(1+2*2^(1/2))^(1/2))*2^(1/2)*b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (a - 4 \, b\right )} x^{3} -{\left (3 \, a + 2 \, b\right )} x}{28 \,{\left (x^{4} + x^{2} + 2\right )}} + \frac{1}{28} \, \int -\frac{{\left (a - 4 \, b\right )} x^{2} - 11 \, a + 2 \, b}{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)^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)

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Fricas [B]  time = 2.60127, size = 11750, normalized size = 37.18 \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)^2,x, algorithm="fricas")

[Out]

-1/21952*(196*2^(3/4)*sqrt(2/7)*(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4)^(3/4)*sqrt(289*a
^4 - 136*a^3*b - 120*a^2*b^2 + 32*a*b^3 + 16*b^4)*(x^4 + x^2 + 2)*sqrt((35912*a^4 - 56816*a^3*b + 46056*a^2*b^
2 - 18656*a*b^3 + 3872*b^4 - sqrt(2)*sqrt(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4)*(211*a^
2 - 428*a*b + 100*b^2))/(289*a^4 - 136*a^3*b - 120*a^2*b^2 + 32*a*b^3 + 16*b^4))*arctan(1/14*(2^(3/4)*sqrt(2/7
)*sqrt(1/14)*(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4)^(3/4)*(sqrt(2)*sqrt(4489*a^4 - 7102
*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4)*sqrt(289*a^4 - 136*a^3*b - 120*a^2*b^2 + 32*a*b^3 + 16*b^4)*(11*
a - 2*b) + 2*sqrt(289*a^4 - 136*a^3*b - 120*a^2*b^2 + 32*a*b^3 + 16*b^4)*(67*a^3 - 321*a^2*b + 234*a*b^2 - 88*
b^3))*sqrt((35912*a^4 - 56816*a^3*b + 46056*a^2*b^2 - 18656*a*b^3 + 3872*b^4 - sqrt(2)*sqrt(4489*a^4 - 7102*a^
3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4)*(211*a^2 - 428*a*b + 100*b^2))/(289*a^4 - 136*a^3*b - 120*a^2*b^2 +
32*a*b^3 + 16*b^4))*sqrt((14*(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4)*x^2 + 2^(1/4)*sqrt
(2/7)*(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4)^(1/4)*(sqrt(7)*sqrt(2)*sqrt(4489*a^4 - 710
2*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4)*(a - 4*b)*x + sqrt(7)*(737*a^3 - 717*a^2*b + 348*a*b^2 - 44*b^3
)*x)*sqrt((35912*a^4 - 56816*a^3*b + 46056*a^2*b^2 - 18656*a*b^3 + 3872*b^4 - sqrt(2)*sqrt(4489*a^4 - 7102*a^3
*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4)*(211*a^2 - 428*a*b + 100*b^2))/(289*a^4 - 136*a^3*b - 120*a^2*b^2 +
32*a*b^3 + 16*b^4)) + 14*sqrt(2)*sqrt(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4)*(67*a^2 - 5
3*a*b + 22*b^2))/(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4)) - 2^(3/4)*sqrt(2/7)*(4489*a^4
- 7102*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4)^(3/4)*(sqrt(2)*sqrt(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 -
2332*a*b^3 + 484*b^4)*sqrt(289*a^4 - 136*a^3*b - 120*a^2*b^2 + 32*a*b^3 + 16*b^4)*(11*a - 2*b)*x + 2*sqrt(289
*a^4 - 136*a^3*b - 120*a^2*b^2 + 32*a*b^3 + 16*b^4)*(67*a^3 - 321*a^2*b + 234*a*b^2 - 88*b^3)*x)*sqrt((35912*a
^4 - 56816*a^3*b + 46056*a^2*b^2 - 18656*a*b^3 + 3872*b^4 - sqrt(2)*sqrt(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2
- 2332*a*b^3 + 484*b^4)*(211*a^2 - 428*a*b + 100*b^2))/(289*a^4 - 136*a^3*b - 120*a^2*b^2 + 32*a*b^3 + 16*b^4)
) - 4*sqrt(7)*sqrt(2)*(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4)^(3/2)*sqrt(289*a^4 - 136*a
^3*b - 120*a^2*b^2 + 32*a*b^3 + 16*b^4) + 2*sqrt(7)*(300763*a^6 - 713751*a^5*b + 860883*a^4*b^2 - 617609*a^3*b
^3 + 282678*a^2*b^4 - 76956*a*b^5 + 10648*b^6)*sqrt(289*a^4 - 136*a^3*b - 120*a^2*b^2 + 32*a*b^3 + 16*b^4))/(5
112971*a^8 - 13336819*a^7*b + 16286963*a^6*b^2 - 11087881*a^5*b^3 + 3832430*a^4*b^4 + 31472*a^3*b^5 - 641872*a
^2*b^6 + 265232*a*b^7 - 42592*b^8)) + 196*2^(3/4)*sqrt(2/7)*(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332*a*b^3
+ 484*b^4)^(3/4)*sqrt(289*a^4 - 136*a^3*b - 120*a^2*b^2 + 32*a*b^3 + 16*b^4)*(x^4 + x^2 + 2)*sqrt((35912*a^4
- 56816*a^3*b + 46056*a^2*b^2 - 18656*a*b^3 + 3872*b^4 - sqrt(2)*sqrt(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2
332*a*b^3 + 484*b^4)*(211*a^2 - 428*a*b + 100*b^2))/(289*a^4 - 136*a^3*b - 120*a^2*b^2 + 32*a*b^3 + 16*b^4))*a
rctan(1/14*(2^(3/4)*sqrt(2/7)*sqrt(1/14)*(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4)^(3/4)*(
sqrt(2)*sqrt(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4)*sqrt(289*a^4 - 136*a^3*b - 120*a^2*b
^2 + 32*a*b^3 + 16*b^4)*(11*a - 2*b) + 2*sqrt(289*a^4 - 136*a^3*b - 120*a^2*b^2 + 32*a*b^3 + 16*b^4)*(67*a^3 -
321*a^2*b + 234*a*b^2 - 88*b^3))*sqrt((35912*a^4 - 56816*a^3*b + 46056*a^2*b^2 - 18656*a*b^3 + 3872*b^4 - sqr
t(2)*sqrt(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4)*(211*a^2 - 428*a*b + 100*b^2))/(289*a^4
- 136*a^3*b - 120*a^2*b^2 + 32*a*b^3 + 16*b^4))*sqrt((14*(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 +
484*b^4)*x^2 - 2^(1/4)*sqrt(2/7)*(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4)^(1/4)*(sqrt(7)
*sqrt(2)*sqrt(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4)*(a - 4*b)*x + sqrt(7)*(737*a^3 - 71
7*a^2*b + 348*a*b^2 - 44*b^3)*x)*sqrt((35912*a^4 - 56816*a^3*b + 46056*a^2*b^2 - 18656*a*b^3 + 3872*b^4 - sqrt
(2)*sqrt(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4)*(211*a^2 - 428*a*b + 100*b^2))/(289*a^4
- 136*a^3*b - 120*a^2*b^2 + 32*a*b^3 + 16*b^4)) + 14*sqrt(2)*sqrt(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332*
a*b^3 + 484*b^4)*(67*a^2 - 53*a*b + 22*b^2))/(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4)) -
2^(3/4)*sqrt(2/7)*(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4)^(3/4)*(sqrt(2)*sqrt(4489*a^4 -
7102*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4)*sqrt(289*a^4 - 136*a^3*b - 120*a^2*b^2 + 32*a*b^3 + 16*b^4)
*(11*a - 2*b)*x + 2*sqrt(289*a^4 - 136*a^3*b - 120*a^2*b^2 + 32*a*b^3 + 16*b^4)*(67*a^3 - 321*a^2*b + 234*a*b^
2 - 88*b^3)*x)*sqrt((35912*a^4 - 56816*a^3*b + 46056*a^2*b^2 - 18656*a*b^3 + 3872*b^4 - sqrt(2)*sqrt(4489*a^4
- 7102*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4)*(211*a^2 - 428*a*b + 100*b^2))/(289*a^4 - 136*a^3*b - 120*
a^2*b^2 + 32*a*b^3 + 16*b^4)) + 4*sqrt(7)*sqrt(2)*(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4
)^(3/2)*sqrt(289*a^4 - 136*a^3*b - 120*a^2*b^2 + 32*a*b^3 + 16*b^4) - 2*sqrt(7)*(300763*a^6 - 713751*a^5*b + 8
60883*a^4*b^2 - 617609*a^3*b^3 + 282678*a^2*b^4 - 76956*a*b^5 + 10648*b^6)*sqrt(289*a^4 - 136*a^3*b - 120*a^2*
b^2 + 32*a*b^3 + 16*b^4))/(5112971*a^8 - 13336819*a^7*b + 16286963*a^6*b^2 - 11087881*a^5*b^3 + 3832430*a^4*b^
4 + 31472*a^3*b^5 - 641872*a^2*b^6 + 265232*a*b^7 - 42592*b^8)) + 784*(4489*a^5 - 25058*a^4*b + 34165*a^3*b^2
- 25360*a^2*b^3 + 9812*a*b^4 - 1936*b^5)*x^3 - 2^(1/4)*sqrt(2/7)*(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332*
a*b^3 + 484*b^4)^(1/4)*(sqrt(7)*sqrt(2)*((211*a^2 - 428*a*b + 100*b^2)*x^4 + (211*a^2 - 428*a*b + 100*b^2)*x^2
+ 422*a^2 - 856*a*b + 200*b^2)*sqrt(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4) + 8*sqrt(7)*
((4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4)*x^4 + 8978*a^4 - 14204*a^3*b + 11514*a^2*b^2 -
4664*a*b^3 + 968*b^4 + (4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4)*x^2))*sqrt((35912*a^4 - 5
6816*a^3*b + 46056*a^2*b^2 - 18656*a*b^3 + 3872*b^4 - sqrt(2)*sqrt(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332
*a*b^3 + 484*b^4)*(211*a^2 - 428*a*b + 100*b^2))/(289*a^4 - 136*a^3*b - 120*a^2*b^2 + 32*a*b^3 + 16*b^4))*log(
32*(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4)*x^2 + 16/7*2^(1/4)*sqrt(2/7)*(4489*a^4 - 7102
*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4)^(1/4)*(sqrt(7)*sqrt(2)*sqrt(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2
- 2332*a*b^3 + 484*b^4)*(a - 4*b)*x + sqrt(7)*(737*a^3 - 717*a^2*b + 348*a*b^2 - 44*b^3)*x)*sqrt((35912*a^4 -
56816*a^3*b + 46056*a^2*b^2 - 18656*a*b^3 + 3872*b^4 - sqrt(2)*sqrt(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 23
32*a*b^3 + 484*b^4)*(211*a^2 - 428*a*b + 100*b^2))/(289*a^4 - 136*a^3*b - 120*a^2*b^2 + 32*a*b^3 + 16*b^4)) +
32*sqrt(2)*sqrt(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4)*(67*a^2 - 53*a*b + 22*b^2)) + 2^(
1/4)*sqrt(2/7)*(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4)^(1/4)*(sqrt(7)*sqrt(2)*((211*a^2
- 428*a*b + 100*b^2)*x^4 + (211*a^2 - 428*a*b + 100*b^2)*x^2 + 422*a^2 - 856*a*b + 200*b^2)*sqrt(4489*a^4 - 71
02*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4) + 8*sqrt(7)*((4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332*a*b^
3 + 484*b^4)*x^4 + 8978*a^4 - 14204*a^3*b + 11514*a^2*b^2 - 4664*a*b^3 + 968*b^4 + (4489*a^4 - 7102*a^3*b + 57
57*a^2*b^2 - 2332*a*b^3 + 484*b^4)*x^2))*sqrt((35912*a^4 - 56816*a^3*b + 46056*a^2*b^2 - 18656*a*b^3 + 3872*b^
4 - sqrt(2)*sqrt(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4)*(211*a^2 - 428*a*b + 100*b^2))/(
289*a^4 - 136*a^3*b - 120*a^2*b^2 + 32*a*b^3 + 16*b^4))*log(32*(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332*a*
b^3 + 484*b^4)*x^2 - 16/7*2^(1/4)*sqrt(2/7)*(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4)^(1/4
)*(sqrt(7)*sqrt(2)*sqrt(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4)*(a - 4*b)*x + sqrt(7)*(73
7*a^3 - 717*a^2*b + 348*a*b^2 - 44*b^3)*x)*sqrt((35912*a^4 - 56816*a^3*b + 46056*a^2*b^2 - 18656*a*b^3 + 3872*
b^4 - sqrt(2)*sqrt(4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4)*(211*a^2 - 428*a*b + 100*b^2))
/(289*a^4 - 136*a^3*b - 120*a^2*b^2 + 32*a*b^3 + 16*b^4)) + 32*sqrt(2)*sqrt(4489*a^4 - 7102*a^3*b + 5757*a^2*b
^2 - 2332*a*b^3 + 484*b^4)*(67*a^2 - 53*a*b + 22*b^2)) - 784*(13467*a^5 - 12328*a^4*b + 3067*a^3*b^2 + 4518*a^
2*b^3 - 3212*a*b^4 + 968*b^5)*x)/((4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 484*b^4)*x^4 + 8978*a^4
- 14204*a^3*b + 11514*a^2*b^2 - 4664*a*b^3 + 968*b^4 + (4489*a^4 - 7102*a^3*b + 5757*a^2*b^2 - 2332*a*b^3 + 4
84*b^4)*x^2)

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Sympy [A]  time = 1.38854, size = 167, normalized size = 0.53 \begin{align*} - \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b x^{2} + a}{{\left (x^{4} + x^{2} + 2\right )}^{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)^2,x, algorithm="giac")

[Out]

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