3.109 \(\int \frac{A+B x^2}{a^2-a x^2+x^4} \, dx\)

Optimal. Leaf size=136 \[ -\frac{(A-a B) \log \left (-\sqrt{3} \sqrt{a} x+a+x^2\right )}{4 \sqrt{3} a^{3/2}}+\frac{(A-a B) \log \left (\sqrt{3} \sqrt{a} x+a+x^2\right )}{4 \sqrt{3} a^{3/2}}-\frac{(a B+A) \tan ^{-1}\left (\sqrt{3}-\frac{2 x}{\sqrt{a}}\right )}{2 a^{3/2}}+\frac{(a B+A) \tan ^{-1}\left (\frac{2 x}{\sqrt{a}}+\sqrt{3}\right )}{2 a^{3/2}} \]

[Out]

-((A + a*B)*ArcTan[Sqrt[3] - (2*x)/Sqrt[a]])/(2*a^(3/2)) + ((A + a*B)*ArcTan[Sqrt[3] + (2*x)/Sqrt[a]])/(2*a^(3
/2)) - ((A - a*B)*Log[a - Sqrt[3]*Sqrt[a]*x + x^2])/(4*Sqrt[3]*a^(3/2)) + ((A - a*B)*Log[a + Sqrt[3]*Sqrt[a]*x
+ x^2])/(4*Sqrt[3]*a^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.103947, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {1169, 634, 617, 204, 628} \[ -\frac{(A-a B) \log \left (-\sqrt{3} \sqrt{a} x+a+x^2\right )}{4 \sqrt{3} a^{3/2}}+\frac{(A-a B) \log \left (\sqrt{3} \sqrt{a} x+a+x^2\right )}{4 \sqrt{3} a^{3/2}}-\frac{(a B+A) \tan ^{-1}\left (\sqrt{3}-\frac{2 x}{\sqrt{a}}\right )}{2 a^{3/2}}+\frac{(a B+A) \tan ^{-1}\left (\frac{2 x}{\sqrt{a}}+\sqrt{3}\right )}{2 a^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x^2)/(a^2 - a*x^2 + x^4),x]

[Out]

-((A + a*B)*ArcTan[Sqrt[3] - (2*x)/Sqrt[a]])/(2*a^(3/2)) + ((A + a*B)*ArcTan[Sqrt[3] + (2*x)/Sqrt[a]])/(2*a^(3
/2)) - ((A - a*B)*Log[a - Sqrt[3]*Sqrt[a]*x + x^2])/(4*Sqrt[3]*a^(3/2)) + ((A - a*B)*Log[a + Sqrt[3]*Sqrt[a]*x
+ x^2])/(4*Sqrt[3]*a^(3/2))

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 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[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])] /;
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}{a^2-a x^2+x^4} \, dx &=\frac{\int \frac{\sqrt{3} \sqrt{a} A-(A-a B) x}{a-\sqrt{3} \sqrt{a} x+x^2} \, dx}{2 \sqrt{3} a^{3/2}}+\frac{\int \frac{\sqrt{3} \sqrt{a} A+(A-a B) x}{a+\sqrt{3} \sqrt{a} x+x^2} \, dx}{2 \sqrt{3} a^{3/2}}\\ &=-\frac{(A-a B) \int \frac{-\sqrt{3} \sqrt{a}+2 x}{a-\sqrt{3} \sqrt{a} x+x^2} \, dx}{4 \sqrt{3} a^{3/2}}+\frac{(A-a B) \int \frac{\sqrt{3} \sqrt{a}+2 x}{a+\sqrt{3} \sqrt{a} x+x^2} \, dx}{4 \sqrt{3} a^{3/2}}+\frac{(A+a B) \int \frac{1}{a-\sqrt{3} \sqrt{a} x+x^2} \, dx}{4 a}+\frac{(A+a B) \int \frac{1}{a+\sqrt{3} \sqrt{a} x+x^2} \, dx}{4 a}\\ &=-\frac{(A-a B) \log \left (a-\sqrt{3} \sqrt{a} x+x^2\right )}{4 \sqrt{3} a^{3/2}}+\frac{(A-a B) \log \left (a+\sqrt{3} \sqrt{a} x+x^2\right )}{4 \sqrt{3} a^{3/2}}+\frac{(A+a B) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{3}-x^2} \, dx,x,1-\frac{2 x}{\sqrt{3} \sqrt{a}}\right )}{2 \sqrt{3} a^{3/2}}-\frac{(A+a B) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{3}-x^2} \, dx,x,1+\frac{2 x}{\sqrt{3} \sqrt{a}}\right )}{2 \sqrt{3} a^{3/2}}\\ &=-\frac{(A+a B) \tan ^{-1}\left (\sqrt{3}-\frac{2 x}{\sqrt{a}}\right )}{2 a^{3/2}}+\frac{(A+a B) \tan ^{-1}\left (\sqrt{3}+\frac{2 x}{\sqrt{a}}\right )}{2 a^{3/2}}-\frac{(A-a B) \log \left (a-\sqrt{3} \sqrt{a} x+x^2\right )}{4 \sqrt{3} a^{3/2}}+\frac{(A-a B) \log \left (a+\sqrt{3} \sqrt{a} x+x^2\right )}{4 \sqrt{3} a^{3/2}}\\ \end{align*}

Mathematica [C]  time = 0.147658, size = 130, normalized size = 0.96 \[ \frac{\sqrt [4]{-1} \left (\frac{\left (\left (\sqrt{3}-i\right ) a B-2 i A\right ) \tan ^{-1}\left (\frac{(1+i) x}{\sqrt{\sqrt{3}-i} \sqrt{a}}\right )}{\sqrt{\sqrt{3}-i}}-\frac{\left (\left (\sqrt{3}+i\right ) a B+2 i A\right ) \tanh ^{-1}\left (\frac{(1+i) x}{\sqrt{\sqrt{3}+i} \sqrt{a}}\right )}{\sqrt{\sqrt{3}+i}}\right )}{\sqrt{6} a^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^2)/(a^2 - a*x^2 + x^4),x]

[Out]

((-1)^(1/4)*((((-2*I)*A + (-I + Sqrt[3])*a*B)*ArcTan[((1 + I)*x)/(Sqrt[-I + Sqrt[3]]*Sqrt[a])])/Sqrt[-I + Sqrt
[3]] - (((2*I)*A + (I + Sqrt[3])*a*B)*ArcTanh[((1 + I)*x)/(Sqrt[I + Sqrt[3]]*Sqrt[a])])/Sqrt[I + Sqrt[3]]))/(S
qrt[6]*a^(3/2))

________________________________________________________________________________________

Maple [A]  time = 0.061, size = 190, normalized size = 1.4 \begin{align*} -{\frac{B\sqrt{3}}{12}\ln \left ( a+{x}^{2}+x\sqrt{3}\sqrt{a} \right ){\frac{1}{\sqrt{a}}}}+{\frac{A\sqrt{3}}{12}\ln \left ( a+{x}^{2}+x\sqrt{3}\sqrt{a} \right ){a}^{-{\frac{3}{2}}}}+{\frac{B}{2}\arctan \left ({ \left ( 2\,x+\sqrt{3}\sqrt{a} \right ){\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{a}}}}+{\frac{A}{2}\arctan \left ({ \left ( 2\,x+\sqrt{3}\sqrt{a} \right ){\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{3}{2}}}}+{\frac{B\sqrt{3}}{12}\ln \left ( x\sqrt{3}\sqrt{a}-{x}^{2}-a \right ){\frac{1}{\sqrt{a}}}}-{\frac{A\sqrt{3}}{12}\ln \left ( x\sqrt{3}\sqrt{a}-{x}^{2}-a \right ){a}^{-{\frac{3}{2}}}}-{\frac{B}{2}\arctan \left ({ \left ( \sqrt{3}\sqrt{a}-2\,x \right ){\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{a}}}}-{\frac{A}{2}\arctan \left ({ \left ( \sqrt{3}\sqrt{a}-2\,x \right ){\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^2+A)/(x^4-a*x^2+a^2),x)

[Out]

-1/12/a^(1/2)*ln(a+x^2+x*3^(1/2)*a^(1/2))*B*3^(1/2)+1/12/a^(3/2)*ln(a+x^2+x*3^(1/2)*a^(1/2))*A*3^(1/2)+1/2/a^(
1/2)*arctan((2*x+3^(1/2)*a^(1/2))/a^(1/2))*B+1/2/a^(3/2)*arctan((2*x+3^(1/2)*a^(1/2))/a^(1/2))*A+1/12/a^(1/2)*
ln(x*3^(1/2)*a^(1/2)-x^2-a)*B*3^(1/2)-1/12/a^(3/2)*ln(x*3^(1/2)*a^(1/2)-x^2-a)*A*3^(1/2)-1/2/a^(1/2)*arctan((3
^(1/2)*a^(1/2)-2*x)/a^(1/2))*B-1/2/a^(3/2)*arctan((3^(1/2)*a^(1/2)-2*x)/a^(1/2))*A

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((B*x^2 + A)/(x^4 - a*x^2 + a^2), x)

________________________________________________________________________________________

Fricas [B]  time = 6.15726, size = 9567, normalized size = 70.35 \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-a*x^2+a^2),x, algorithm="fricas")

[Out]

1/4*(4*(1/9)^(1/4)*a^6*sqrt((2*B^4*a^4 + 4*A*B^3*a^3 + 6*A^2*B^2*a^2 + 4*A^3*B*a + 2*A^4 + (B^2*a^5 + 4*A*B*a^
4 + A^2*a^3)*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6))/(B^4*a^4 - 2*A^2*B^2*a^2 + A
^4))*((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6)^(3/4)*sqrt((B^4*a^4 - 2*A^2*B^2*a^2 + A^4
)/a^6)*arctan((18*sqrt(1/3)*(1/9)^(3/4)*(sqrt(1/3)*A*a^10*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*
B*a + A^4)/a^6)*sqrt((B^4*a^4 - 2*A^2*B^2*a^2 + A^4)/a^6) - sqrt(1/3)*(B^3*a^10 + A*B^2*a^9 + A^2*B*a^8)*sqrt(
(B^4*a^4 - 2*A^2*B^2*a^2 + A^4)/a^6))*sqrt((2*B^4*a^4 + 4*A*B^3*a^3 + 6*A^2*B^2*a^2 + 4*A^3*B*a + 2*A^4 + (B^2
*a^5 + 4*A*B*a^4 + A^2*a^3)*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6))/(B^4*a^4 - 2*
A^2*B^2*a^2 + A^4))*sqrt(((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)*x^2 + 3*sqrt(1/3)*(1/9)^(1
/4)*(B*a^6*x*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6) - (A*B^2*a^4 + A^2*B*a^3 + A^
3*a^2)*x)*sqrt((2*B^4*a^4 + 4*A*B^3*a^3 + 6*A^2*B^2*a^2 + 4*A^3*B*a + 2*A^4 + (B^2*a^5 + 4*A*B*a^4 + A^2*a^3)*
sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6))/(B^4*a^4 - 2*A^2*B^2*a^2 + A^4))*((B^4*a^
4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6)^(1/4) + (B^2*a^6 + A*B*a^5 + A^2*a^4)*sqrt((B^4*a^4 +
2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6))/(B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)
)*((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6)^(3/4) - 18*sqrt(1/3)*(1/9)^(3/4)*(sqrt(1/3)*
A*a^10*x*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6)*sqrt((B^4*a^4 - 2*A^2*B^2*a^2 + A
^4)/a^6) - sqrt(1/3)*(B^3*a^10 + A*B^2*a^9 + A^2*B*a^8)*x*sqrt((B^4*a^4 - 2*A^2*B^2*a^2 + A^4)/a^6))*sqrt((2*B
^4*a^4 + 4*A*B^3*a^3 + 6*A^2*B^2*a^2 + 4*A^3*B*a + 2*A^4 + (B^2*a^5 + 4*A*B*a^4 + A^2*a^3)*sqrt((B^4*a^4 + 2*A
*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6))/(B^4*a^4 - 2*A^2*B^2*a^2 + A^4))*((B^4*a^4 + 2*A*B^3*a^3 + 3
*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6)^(3/4) + 2*sqrt(1/3)*(B^4*a^10 + 2*A*B^3*a^9 + 3*A^2*B^2*a^8 + 2*A^3*B*a^7
+ A^4*a^6)*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6)*sqrt((B^4*a^4 - 2*A^2*B^2*a^2
+ A^4)/a^6) + sqrt(1/3)*(B^6*a^9 + 3*A*B^5*a^8 + 6*A^2*B^4*a^7 + 7*A^3*B^3*a^6 + 6*A^4*B^2*a^5 + 3*A^5*B*a^4 +
A^6*a^3)*sqrt((B^4*a^4 - 2*A^2*B^2*a^2 + A^4)/a^6))/(B^8*a^8 + 3*A*B^7*a^7 + 5*A^2*B^6*a^6 + 4*A^3*B^5*a^5 -
4*A^5*B^3*a^3 - 5*A^6*B^2*a^2 - 3*A^7*B*a - A^8)) + 4*(1/9)^(1/4)*a^6*sqrt((2*B^4*a^4 + 4*A*B^3*a^3 + 6*A^2*B^
2*a^2 + 4*A^3*B*a + 2*A^4 + (B^2*a^5 + 4*A*B*a^4 + A^2*a^3)*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^
3*B*a + A^4)/a^6))/(B^4*a^4 - 2*A^2*B^2*a^2 + A^4))*((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)
/a^6)^(3/4)*sqrt((B^4*a^4 - 2*A^2*B^2*a^2 + A^4)/a^6)*arctan((18*sqrt(1/3)*(1/9)^(3/4)*(sqrt(1/3)*A*a^10*sqrt(
(B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6)*sqrt((B^4*a^4 - 2*A^2*B^2*a^2 + A^4)/a^6) - sqr
t(1/3)*(B^3*a^10 + A*B^2*a^9 + A^2*B*a^8)*sqrt((B^4*a^4 - 2*A^2*B^2*a^2 + A^4)/a^6))*sqrt((2*B^4*a^4 + 4*A*B^3
*a^3 + 6*A^2*B^2*a^2 + 4*A^3*B*a + 2*A^4 + (B^2*a^5 + 4*A*B*a^4 + A^2*a^3)*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2
*B^2*a^2 + 2*A^3*B*a + A^4)/a^6))/(B^4*a^4 - 2*A^2*B^2*a^2 + A^4))*sqrt(((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^
2 + 2*A^3*B*a + A^4)*x^2 - 3*sqrt(1/3)*(1/9)^(1/4)*(B*a^6*x*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^
3*B*a + A^4)/a^6) - (A*B^2*a^4 + A^2*B*a^3 + A^3*a^2)*x)*sqrt((2*B^4*a^4 + 4*A*B^3*a^3 + 6*A^2*B^2*a^2 + 4*A^3
*B*a + 2*A^4 + (B^2*a^5 + 4*A*B*a^4 + A^2*a^3)*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/
a^6))/(B^4*a^4 - 2*A^2*B^2*a^2 + A^4))*((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6)^(1/4) +
(B^2*a^6 + A*B*a^5 + A^2*a^4)*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6))/(B^4*a^4 +
2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4))*((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^
6)^(3/4) - 18*sqrt(1/3)*(1/9)^(3/4)*(sqrt(1/3)*A*a^10*x*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*
a + A^4)/a^6)*sqrt((B^4*a^4 - 2*A^2*B^2*a^2 + A^4)/a^6) - sqrt(1/3)*(B^3*a^10 + A*B^2*a^9 + A^2*B*a^8)*x*sqrt(
(B^4*a^4 - 2*A^2*B^2*a^2 + A^4)/a^6))*sqrt((2*B^4*a^4 + 4*A*B^3*a^3 + 6*A^2*B^2*a^2 + 4*A^3*B*a + 2*A^4 + (B^2
*a^5 + 4*A*B*a^4 + A^2*a^3)*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6))/(B^4*a^4 - 2*
A^2*B^2*a^2 + A^4))*((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6)^(3/4) - 2*sqrt(1/3)*(B^4*a
^10 + 2*A*B^3*a^9 + 3*A^2*B^2*a^8 + 2*A^3*B*a^7 + A^4*a^6)*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3
*B*a + A^4)/a^6)*sqrt((B^4*a^4 - 2*A^2*B^2*a^2 + A^4)/a^6) - sqrt(1/3)*(B^6*a^9 + 3*A*B^5*a^8 + 6*A^2*B^4*a^7
+ 7*A^3*B^3*a^6 + 6*A^4*B^2*a^5 + 3*A^5*B*a^4 + A^6*a^3)*sqrt((B^4*a^4 - 2*A^2*B^2*a^2 + A^4)/a^6))/(B^8*a^8 +
3*A*B^7*a^7 + 5*A^2*B^6*a^6 + 4*A^3*B^5*a^5 - 4*A^5*B^3*a^3 - 5*A^6*B^2*a^2 - 3*A^7*B*a - A^8)) - sqrt(1/3)*(
1/9)^(1/4)*(2*B^4*a^4 + 4*A*B^3*a^3 + 6*A^2*B^2*a^2 + 4*A^3*B*a + 2*A^4 - (B^2*a^5 + 4*A*B*a^4 + A^2*a^3)*sqrt
((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6))*sqrt((2*B^4*a^4 + 4*A*B^3*a^3 + 6*A^2*B^2*a^2
+ 4*A^3*B*a + 2*A^4 + (B^2*a^5 + 4*A*B*a^4 + A^2*a^3)*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a
+ A^4)/a^6))/(B^4*a^4 - 2*A^2*B^2*a^2 + A^4))*((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6)
^(1/4)*log(2*(B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)*x^2 + 6*sqrt(1/3)*(1/9)^(1/4)*(B*a^6*x*
sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6) - (A*B^2*a^4 + A^2*B*a^3 + A^3*a^2)*x)*sqr
t((2*B^4*a^4 + 4*A*B^3*a^3 + 6*A^2*B^2*a^2 + 4*A^3*B*a + 2*A^4 + (B^2*a^5 + 4*A*B*a^4 + A^2*a^3)*sqrt((B^4*a^4
+ 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6))/(B^4*a^4 - 2*A^2*B^2*a^2 + A^4))*((B^4*a^4 + 2*A*B^3*a
^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6)^(1/4) + 2*(B^2*a^6 + A*B*a^5 + A^2*a^4)*sqrt((B^4*a^4 + 2*A*B^3*a^3
+ 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6)) + sqrt(1/3)*(1/9)^(1/4)*(2*B^4*a^4 + 4*A*B^3*a^3 + 6*A^2*B^2*a^2 + 4
*A^3*B*a + 2*A^4 - (B^2*a^5 + 4*A*B*a^4 + A^2*a^3)*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A
^4)/a^6))*sqrt((2*B^4*a^4 + 4*A*B^3*a^3 + 6*A^2*B^2*a^2 + 4*A^3*B*a + 2*A^4 + (B^2*a^5 + 4*A*B*a^4 + A^2*a^3)*
sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6))/(B^4*a^4 - 2*A^2*B^2*a^2 + A^4))*((B^4*a^
4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6)^(1/4)*log(2*(B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2
*A^3*B*a + A^4)*x^2 - 6*sqrt(1/3)*(1/9)^(1/4)*(B*a^6*x*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a
+ A^4)/a^6) - (A*B^2*a^4 + A^2*B*a^3 + A^3*a^2)*x)*sqrt((2*B^4*a^4 + 4*A*B^3*a^3 + 6*A^2*B^2*a^2 + 4*A^3*B*a
+ 2*A^4 + (B^2*a^5 + 4*A*B*a^4 + A^2*a^3)*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6))
/(B^4*a^4 - 2*A^2*B^2*a^2 + A^4))*((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6)^(1/4) + 2*(B
^2*a^6 + A*B*a^5 + A^2*a^4)*sqrt((B^4*a^4 + 2*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)/a^6)))/(B^4*a^4 + 2
*A*B^3*a^3 + 3*A^2*B^2*a^2 + 2*A^3*B*a + A^4)

________________________________________________________________________________________

Sympy [A]  time = 1.20255, size = 172, normalized size = 1.26 \begin{align*} \operatorname{RootSum}{\left (144 t^{4} a^{6} + t^{2} \left (12 A^{2} a^{3} + 48 A B a^{4} + 12 B^{2} a^{5}\right ) + A^{4} + 2 A^{3} B a + 3 A^{2} B^{2} a^{2} + 2 A B^{3} a^{3} + B^{4} a^{4}, \left ( t \mapsto t \log{\left (x + \frac{24 t^{3} A a^{5} + 48 t^{3} B a^{6} - 2 t A^{3} a^{2} + 6 t A^{2} B a^{3} + 12 t A B^{2} a^{4} + 2 t B^{3} a^{5}}{- A^{4} - A^{3} B a + A B^{3} a^{3} + B^{4} a^{4}} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)/(x**4-a*x**2+a**2),x)

[Out]

RootSum(144*_t**4*a**6 + _t**2*(12*A**2*a**3 + 48*A*B*a**4 + 12*B**2*a**5) + A**4 + 2*A**3*B*a + 3*A**2*B**2*a
**2 + 2*A*B**3*a**3 + B**4*a**4, Lambda(_t, _t*log(x + (24*_t**3*A*a**5 + 48*_t**3*B*a**6 - 2*_t*A**3*a**2 + 6
*_t*A**2*B*a**3 + 12*_t*A*B**2*a**4 + 2*_t*B**3*a**5)/(-A**4 - A**3*B*a + A*B**3*a**3 + B**4*a**4))))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((B*x^2 + A)/(x^4 - a*x^2 + a^2), x)