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

Optimal. Leaf size=331 \[ \frac {\log \left (-\sqrt {2} \sqrt [4]{a} x \sqrt {\sqrt {a+b}-\sqrt {a}}+\sqrt {a+b}+\sqrt {a} x^2\right )}{4 \sqrt {2} a^{3/4} \sqrt {\sqrt {a+b}-\sqrt {a}}}-\frac {\log \left (\sqrt {2} \sqrt [4]{a} x \sqrt {\sqrt {a+b}-\sqrt {a}}+\sqrt {a+b}+\sqrt {a} x^2\right )}{4 \sqrt {2} a^{3/4} \sqrt {\sqrt {a+b}-\sqrt {a}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt {2} \sqrt [4]{a} x}{\sqrt {\sqrt {a+b}+\sqrt {a}}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {\sqrt {a+b}+\sqrt {a}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {a+b}-\sqrt {a}}+\sqrt {2} \sqrt [4]{a} x}{\sqrt {\sqrt {a+b}+\sqrt {a}}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {\sqrt {a+b}+\sqrt {a}}} \]

[Out]

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)

________________________________________________________________________________________

Rubi [A]  time = 0.26, antiderivative size = 331, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1129, 634, 618, 204, 628} \[ \frac {\log \left (-\sqrt {2} \sqrt [4]{a} x \sqrt {\sqrt {a+b}-\sqrt {a}}+\sqrt {a+b}+\sqrt {a} x^2\right )}{4 \sqrt {2} a^{3/4} \sqrt {\sqrt {a+b}-\sqrt {a}}}-\frac {\log \left (\sqrt {2} \sqrt [4]{a} x \sqrt {\sqrt {a+b}-\sqrt {a}}+\sqrt {a+b}+\sqrt {a} x^2\right )}{4 \sqrt {2} a^{3/4} \sqrt {\sqrt {a+b}-\sqrt {a}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {a+b}-\sqrt {a}}-\sqrt {2} \sqrt [4]{a} x}{\sqrt {\sqrt {a+b}+\sqrt {a}}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {\sqrt {a+b}+\sqrt {a}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {a+b}-\sqrt {a}}+\sqrt {2} \sqrt [4]{a} x}{\sqrt {\sqrt {a+b}+\sqrt {a}}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {\sqrt {a+b}+\sqrt {a}}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b + 2*a*x^2 + a*x^4),x]

[Out]

-ArcTan[(Sqrt[-Sqrt[a] + Sqrt[a + b]] - Sqrt[2]*a^(1/4)*x)/Sqrt[Sqrt[a] + Sqrt[a + b]]]/(2*Sqrt[2]*a^(3/4)*Sqr
t[Sqrt[a] + Sqrt[a + b]]) + ArcTan[(Sqrt[-Sqrt[a] + Sqrt[a + b]] + Sqrt[2]*a^(1/4)*x)/Sqrt[Sqrt[a] + Sqrt[a +
b]]]/(2*Sqrt[2]*a^(3/4)*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]/(4*Sqrt[2]*a^(3/4)*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]/(4*Sqrt[2]*a^(3/4)*Sqrt[-Sqrt[a] + Sqrt[a + b]])

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 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 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]

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 1129

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]}, Dist[1/(2*c*r), Int[x^(m - 1)/(q - r*x + x^2), x], x] - Dist[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]

Rubi steps

\begin {align*} \int \frac {x^2}{a+b+2 a x^2+a x^4} \, dx &=\frac {\int \frac {x}{\frac {\sqrt {a+b}}{\sqrt {a}}-\frac {\sqrt {2} \sqrt {-\sqrt {a}+\sqrt {a+b}} x}{\sqrt [4]{a}}+x^2} \, dx}{2 \sqrt {2} a^{3/4} \sqrt {-\sqrt {a}+\sqrt {a+b}}}-\frac {\int \frac {x}{\frac {\sqrt {a+b}}{\sqrt {a}}+\frac {\sqrt {2} \sqrt {-\sqrt {a}+\sqrt {a+b}} x}{\sqrt [4]{a}}+x^2} \, dx}{2 \sqrt {2} a^{3/4} \sqrt {-\sqrt {a}+\sqrt {a+b}}}\\ &=\frac {\int \frac {1}{\frac {\sqrt {a+b}}{\sqrt {a}}-\frac {\sqrt {2} \sqrt {-\sqrt {a}+\sqrt {a+b}} x}{\sqrt [4]{a}}+x^2} \, dx}{4 a}+\frac {\int \frac {1}{\frac {\sqrt {a+b}}{\sqrt {a}}+\frac {\sqrt {2} \sqrt {-\sqrt {a}+\sqrt {a+b}} x}{\sqrt [4]{a}}+x^2} \, dx}{4 a}+\frac {\int \frac {-\frac {\sqrt {2} \sqrt {-\sqrt {a}+\sqrt {a+b}}}{\sqrt [4]{a}}+2 x}{\frac {\sqrt {a+b}}{\sqrt {a}}-\frac {\sqrt {2} \sqrt {-\sqrt {a}+\sqrt {a+b}} x}{\sqrt [4]{a}}+x^2} \, dx}{4 \sqrt {2} a^{3/4} \sqrt {-\sqrt {a}+\sqrt {a+b}}}-\frac {\int \frac {\frac {\sqrt {2} \sqrt {-\sqrt {a}+\sqrt {a+b}}}{\sqrt [4]{a}}+2 x}{\frac {\sqrt {a+b}}{\sqrt {a}}+\frac {\sqrt {2} \sqrt {-\sqrt {a}+\sqrt {a+b}} x}{\sqrt [4]{a}}+x^2} \, dx}{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}}}-\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 {\operatorname {Subst}\left (\int \frac {1}{-2 \left (1+\frac {\sqrt {a+b}}{\sqrt {a}}\right )-x^2} \, dx,x,-\frac {\sqrt {2} \sqrt {-\sqrt {a}+\sqrt {a+b}}}{\sqrt [4]{a}}+2 x\right )}{2 a}-\frac {\operatorname {Subst}\left (\int \frac {1}{-2 \left (1+\frac {\sqrt {a+b}}{\sqrt {a}}\right )-x^2} \, dx,x,\frac {\sqrt {2} \sqrt {-\sqrt {a}+\sqrt {a+b}}}{\sqrt [4]{a}}+2 x\right )}{2 a}\\ &=-\frac {\tan ^{-1}\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 {\tan ^{-1}\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}}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.12, size = 143, normalized size = 0.43 \[ \frac {\frac {\left (\sqrt {b}+i \sqrt {a}\right ) \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a-i \sqrt {a} \sqrt {b}}}\right )}{\sqrt {a-i \sqrt {a} \sqrt {b}}}+\frac {\left (\sqrt {b}-i \sqrt {a}\right ) \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a+i \sqrt {a} \sqrt {b}}}\right )}{\sqrt {a+i \sqrt {a} \sqrt {b}}}}{2 \sqrt {a} \sqrt {b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((I*Sqrt[a] + Sqrt[b])*ArcTan[(Sqrt[a]*x)/Sqrt[a - I*Sqrt[a]*Sqrt[b]]])/Sqrt[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]*Sq
rt[b])

________________________________________________________________________________________

fricas [A]  time = 0.70, size = 279, normalized size = 0.84 \[ \frac {1}{4} \, \sqrt {\frac {a b \sqrt {-\frac {1}{a^{3} b}} + 1}{a b}} \log \left (a^{2} b \sqrt {\frac {a b \sqrt {-\frac {1}{a^{3} b}} + 1}{a b}} \sqrt {-\frac {1}{a^{3} b}} + x\right ) - \frac {1}{4} \, \sqrt {\frac {a b \sqrt {-\frac {1}{a^{3} b}} + 1}{a b}} \log \left (-a^{2} b \sqrt {\frac {a b \sqrt {-\frac {1}{a^{3} b}} + 1}{a b}} \sqrt {-\frac {1}{a^{3} b}} + x\right ) - \frac {1}{4} \, \sqrt {-\frac {a b \sqrt {-\frac {1}{a^{3} b}} - 1}{a b}} \log \left (a^{2} b \sqrt {-\frac {a b \sqrt {-\frac {1}{a^{3} b}} - 1}{a b}} \sqrt {-\frac {1}{a^{3} b}} + x\right ) + \frac {1}{4} \, \sqrt {-\frac {a b \sqrt {-\frac {1}{a^{3} b}} - 1}{a b}} \log \left (-a^{2} b \sqrt {-\frac {a b \sqrt {-\frac {1}{a^{3} b}} - 1}{a b}} \sqrt {-\frac {1}{a^{3} b}} + x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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))*sqr
t(-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)

________________________________________________________________________________________

giac [A]  time = 0.34, size = 203, normalized size = 0.61 \[ -\frac {{\left (3 \, \sqrt {a^{2} + \sqrt {-a b} a} \sqrt {-a b} a + 4 \, \sqrt {a^{2} + \sqrt {-a b} a} \sqrt {-a b} b\right )} {\left | a \right |} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {\frac {2 \, a + \sqrt {-4 \, {\left (a + b\right )} a + 4 \, a^{2}}}{a}}}\right )}{2 \, {\left (3 \, a^{4} b + 4 \, a^{3} b^{2}\right )}} + \frac {{\left (3 \, \sqrt {a^{2} - \sqrt {-a b} a} \sqrt {-a b} a + 4 \, \sqrt {a^{2} - \sqrt {-a b} a} \sqrt {-a b} b\right )} {\left | a \right |} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {\frac {2 \, a - \sqrt {-4 \, {\left (a + b\right )} a + 4 \, a^{2}}}{a}}}\right )}{2 \, {\left (3 \, a^{4} b + 4 \, a^{3} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)*sqr
t(-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)

________________________________________________________________________________________

maple [B]  time = 0.06, size = 724, normalized size = 2.19 \[ -\frac {\sqrt {-2 a +2 \sqrt {\left (a +b \right ) a}}\, \sqrt {-2 a +2 \sqrt {a^{2}+a b}}\, \arctan \left (\frac {-2 \sqrt {a}\, x +\sqrt {-2 a +2 \sqrt {\left (a +b \right ) a}}}{\sqrt {2 a +4 \sqrt {a +b}\, \sqrt {a}-2 \sqrt {\left (a +b \right ) a}}}\right )}{4 \sqrt {2 a +4 \sqrt {a +b}\, \sqrt {a}-2 \sqrt {\left (a +b \right ) a}}\, \sqrt {a}\, b}+\frac {\sqrt {-2 a +2 \sqrt {\left (a +b \right ) a}}\, \sqrt {-2 a +2 \sqrt {a^{2}+a b}}\, \arctan \left (\frac {2 \sqrt {a}\, x +\sqrt {-2 a +2 \sqrt {\left (a +b \right ) a}}}{\sqrt {2 a +4 \sqrt {a +b}\, \sqrt {a}-2 \sqrt {\left (a +b \right ) a}}}\right )}{4 \sqrt {2 a +4 \sqrt {a +b}\, \sqrt {a}-2 \sqrt {\left (a +b \right ) a}}\, \sqrt {a}\, b}+\frac {\sqrt {-2 a +2 \sqrt {a^{2}+a b}}\, \ln \left (-\sqrt {a}\, x^{2}+\sqrt {-2 a +2 \sqrt {\left (a +b \right ) a}}\, x -\sqrt {a +b}\right )}{8 \sqrt {a}\, b}-\frac {\sqrt {-2 a +2 \sqrt {a^{2}+a b}}\, \ln \left (\sqrt {a}\, x^{2}+\sqrt {-2 a +2 \sqrt {\left (a +b \right ) a}}\, x +\sqrt {a +b}\right )}{8 \sqrt {a}\, b}-\frac {\sqrt {-2 a +2 \sqrt {\left (a +b \right ) a}}\, \sqrt {a^{2}+a b}\, \sqrt {-2 a +2 \sqrt {a^{2}+a b}}\, \arctan \left (\frac {-2 \sqrt {a}\, x +\sqrt {-2 a +2 \sqrt {\left (a +b \right ) a}}}{\sqrt {2 a +4 \sqrt {a +b}\, \sqrt {a}-2 \sqrt {\left (a +b \right ) a}}}\right )}{4 \sqrt {2 a +4 \sqrt {a +b}\, \sqrt {a}-2 \sqrt {\left (a +b \right ) a}}\, a^{\frac {3}{2}} b}+\frac {\sqrt {-2 a +2 \sqrt {\left (a +b \right ) a}}\, \sqrt {a^{2}+a b}\, \sqrt {-2 a +2 \sqrt {a^{2}+a b}}\, \arctan \left (\frac {2 \sqrt {a}\, x +\sqrt {-2 a +2 \sqrt {\left (a +b \right ) a}}}{\sqrt {2 a +4 \sqrt {a +b}\, \sqrt {a}-2 \sqrt {\left (a +b \right ) a}}}\right )}{4 \sqrt {2 a +4 \sqrt {a +b}\, \sqrt {a}-2 \sqrt {\left (a +b \right ) a}}\, a^{\frac {3}{2}} b}+\frac {\sqrt {a^{2}+a b}\, \sqrt {-2 a +2 \sqrt {a^{2}+a b}}\, \ln \left (-\sqrt {a}\, x^{2}+\sqrt {-2 a +2 \sqrt {\left (a +b \right ) a}}\, x -\sqrt {a +b}\right )}{8 a^{\frac {3}{2}} b}-\frac {\sqrt {a^{2}+a b}\, \sqrt {-2 a +2 \sqrt {a^{2}+a b}}\, \ln \left (\sqrt {a}\, x^{2}+\sqrt {-2 a +2 \sqrt {\left (a +b \right ) a}}\, x +\sqrt {a +b}\right )}{8 a^{\frac {3}{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(a^2+a*b)^(1/2)*(-2*a+2*(a^2+a*b)^(1/2))^(1/2)/a^(3/2)/b*ln(-a^(1/2)*x^2+(-2*a+2*((a+b)*a)^(1/2))^(1/2)*x-
(a+b)^(1/2))-1/4/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2)*(-2*a+2*((a+b)*a)^(1/2))^(1/2)*(a^2+a*b)^
(1/2)*(-2*a+2*(a^2+a*b)^(1/2))^(1/2)/a^(3/2)/b*arctan((-2*a^(1/2)*x+(-2*a+2*((a+b)*a)^(1/2))^(1/2))/(2*a+4*(a+
b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2))+1/8*(-2*a+2*(a^2+a*b)^(1/2))^(1/2)/a^(1/2)/b*ln(-a^(1/2)*x^2+(-2*a+
2*((a+b)*a)^(1/2))^(1/2)*x-(a+b)^(1/2))-1/4/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2)*(-2*a+2*((a+b)
*a)^(1/2))^(1/2)*(-2*a+2*(a^2+a*b)^(1/2))^(1/2)/a^(1/2)/b*arctan((-2*a^(1/2)*x+(-2*a+2*((a+b)*a)^(1/2))^(1/2))
/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2))-1/8*(a^2+a*b)^(1/2)*(-2*a+2*(a^2+a*b)^(1/2))^(1/2)/a^(3/
2)/b*ln(a^(1/2)*x^2+(-2*a+2*((a+b)*a)^(1/2))^(1/2)*x+(a+b)^(1/2))+1/4/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(
1/2))^(1/2)*(-2*a+2*((a+b)*a)^(1/2))^(1/2)*(a^2+a*b)^(1/2)*(-2*a+2*(a^2+a*b)^(1/2))^(1/2)/a^(3/2)/b*arctan((2*
a^(1/2)*x+(-2*a+2*((a+b)*a)^(1/2))^(1/2))/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2))-1/8*(-2*a+2*(a^
2+a*b)^(1/2))^(1/2)/a^(1/2)/b*ln(a^(1/2)*x^2+(-2*a+2*((a+b)*a)^(1/2))^(1/2)*x+(a+b)^(1/2))+1/4/(2*a+4*(a+b)^(1
/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2)*(-2*a+2*((a+b)*a)^(1/2))^(1/2)*(-2*a+2*(a^2+a*b)^(1/2))^(1/2)/a^(1/2)/b*a
rctan((2*a^(1/2)*x+(-2*a+2*((a+b)*a)^(1/2))^(1/2))/(2*a+4*(a+b)^(1/2)*a^(1/2)-2*((a+b)*a)^(1/2))^(1/2))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{a x^{4} + 2 \, a x^{2} + a + b}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.28, size = 222, normalized size = 0.67 \[ -2\,\mathrm {atanh}\left (\frac {2\,\left (x\,\left (4\,a^2\,b-4\,a^3\right )+\frac {4\,a\,x\,\left (\sqrt {-a^3\,b^3}+a^2\,b\right )}{b}\right )\,\sqrt {\frac {\sqrt {-a^3\,b^3}+a^2\,b}{16\,a^3\,b^2}}}{2\,a^2+2\,b\,a}\right )\,\sqrt {\frac {\sqrt {-a^3\,b^3}+a^2\,b}{16\,a^3\,b^2}}-2\,\mathrm {atanh}\left (\frac {2\,\left (x\,\left (4\,a^2\,b-4\,a^3\right )-\frac {4\,a\,x\,\left (\sqrt {-a^3\,b^3}-a^2\,b\right )}{b}\right )\,\sqrt {-\frac {\sqrt {-a^3\,b^3}-a^2\,b}{16\,a^3\,b^2}}}{2\,a^2+2\,b\,a}\right )\,\sqrt {-\frac {\sqrt {-a^3\,b^3}-a^2\,b}{16\,a^3\,b^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a + b + 2*a*x^2 + a*x^4),x)

[Out]

- 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)

________________________________________________________________________________________

sympy [A]  time = 0.83, size = 44, normalized size = 0.13 \[ \operatorname {RootSum} {\left (256 t^{4} a^{3} b^{2} - 32 t^{2} a^{2} b + a + b, \left (t \mapsto t \log {\left (64 t^{3} a^{2} b - 4 t a + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(256*_t**4*a**3*b**2 - 32*_t**2*a**2*b + a + b, Lambda(_t, _t*log(64*_t**3*a**2*b - 4*_t*a + x)))

________________________________________________________________________________________