3.10.12 \(\int \frac {x^2}{a+b+2 a x^2+a x^4} \, dx\) [912]

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

[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)

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Rubi [A]
time = 0.18, 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 = {1143, 648, 632, 210, 642} \begin {gather*} -\frac {\text {ArcTan}\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 {\text {ArcTan}\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 {\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}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b + 2*a*x^2 + a*x^4),x]

[Out]

-1/2*ArcTan[(Sqrt[-Sqrt[a] + Sqrt[a + b]] - Sqrt[2]*a^(1/4)*x)/Sqrt[Sqrt[a] + Sqrt[a + b]]]/(Sqrt[2]*a^(3/4)*S
qrt[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] + Sqr
t[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 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 1143

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 {\text {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 {\text {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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.08, size = 143, normalized size = 0.43 \begin {gather*} \frac {\frac {\left (i \sqrt {a}+\sqrt {b}\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 (-i \sqrt {a}+\sqrt {b}\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}} \end {gather*}

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])

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Maple [A]
time = 0.05, size = 339, normalized size = 1.02

method	result	size
risch	\(\frac {\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{4}+2 a \,\textit {\_Z}^{2}+a +b \right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}+\textit {\_R}}}{4 a}\)	\(41\)
default	\(\frac {\sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \left (\sqrt {a^{2}+a b}+a \right ) \left (\frac {\ln \left (-x^{2} \sqrt {a}+x \sqrt {2 \sqrt {a \left (a +b \right )}-2 a}-\sqrt {a +b}\right )}{2 \sqrt {a}}-\frac {\sqrt {2 \sqrt {a \left (a +b \right )}-2 a}\, \arctan \left (\frac {-2 x \sqrt {a}+\sqrt {2 \sqrt {a \left (a +b \right )}-2 a}}{\sqrt {4 \sqrt {a}\, \sqrt {a +b}-2 \sqrt {a \left (a +b \right )}+2 a}}\right )}{\sqrt {a}\, \sqrt {4 \sqrt {a}\, \sqrt {a +b}-2 \sqrt {a \left (a +b \right )}+2 a}}\right )}{4 a b}-\frac {\sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \left (\sqrt {a^{2}+a b}+a \right ) \left (\frac {\ln \left (x^{2} \sqrt {a}+x \sqrt {2 \sqrt {a \left (a +b \right )}-2 a}+\sqrt {a +b}\right )}{2 \sqrt {a}}-\frac {\sqrt {2 \sqrt {a \left (a +b \right )}-2 a}\, \arctan \left (\frac {2 x \sqrt {a}+\sqrt {2 \sqrt {a \left (a +b \right )}-2 a}}{\sqrt {4 \sqrt {a}\, \sqrt {a +b}-2 \sqrt {a \left (a +b \right )}+2 a}}\right )}{\sqrt {a}\, \sqrt {4 \sqrt {a}\, \sqrt {a +b}-2 \sqrt {a \left (a +b \right )}+2 a}}\right )}{4 a b}\)	\(339\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

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Fricas [A]
time = 0.35, size = 279, normalized size = 0.84 \begin {gather*} \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 ) \end {gather*}

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)

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Sympy [A]
time = 0.26, size = 44, normalized size = 0.13 \begin {gather*} \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 )} \end {gather*}

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)))

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Giac [A]
time = 3.44, size = 203, normalized size = 0.61 \begin {gather*} -\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 )}} \end {gather*}

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)

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Mupad [B]
time = 0.28, size = 222, normalized size = 0.67 \begin {gather*} -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}} \end {gather*}

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)

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