\(\int \frac {A+B x^2}{a-\sqrt {a} x^2+x^4} \, dx\) [110]

Optimal result

Integrand size = 25, antiderivative size = 160 \[ \int \frac {A+B x^2}{a-\sqrt {a} x^2+x^4} \, dx=-\frac {\left (A+\sqrt {a} B\right ) \arctan \left (\sqrt {3}-\frac {2 x}{\sqrt [4]{a}}\right )}{2 a^{3/4}}+\frac {\left (A+\sqrt {a} B\right ) \arctan \left (\sqrt {3}+\frac {2 x}{\sqrt [4]{a}}\right )}{2 a^{3/4}}-\frac {\left (A-\sqrt {a} B\right ) \log \left (\sqrt {a}-\sqrt {3} \sqrt [4]{a} x+x^2\right )}{4 \sqrt {3} a^{3/4}}+\frac {\left (A-\sqrt {a} B\right ) \log \left (\sqrt {a}+\sqrt {3} \sqrt [4]{a} x+x^2\right )}{4 \sqrt {3} a^{3/4}} \]

[Out]

-1/12*ln(x^2-a^(1/4)*x*3^(1/2)+a^(1/2))*(A-B*a^(1/2))/a^(3/4)*3^(1/2)+1/12*ln(x^2+a^(1/4)*x*3^(1/2)+a^(1/2))*(
A-B*a^(1/2))/a^(3/4)*3^(1/2)+1/2*arctan(2*x/a^(1/4)-3^(1/2))*(A+B*a^(1/2))/a^(3/4)+1/2*arctan(2*x/a^(1/4)+3^(1
/2))*(A+B*a^(1/2))/a^(3/4)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1183, 648, 631, 210, 642} \[ \int \frac {A+B x^2}{a-\sqrt {a} x^2+x^4} \, dx=-\frac {\left (\sqrt {a} B+A\right ) \arctan \left (\sqrt {3}-\frac {2 x}{\sqrt [4]{a}}\right )}{2 a^{3/4}}+\frac {\left (\sqrt {a} B+A\right ) \arctan \left (\frac {2 x}{\sqrt [4]{a}}+\sqrt {3}\right )}{2 a^{3/4}}-\frac {\left (A-\sqrt {a} B\right ) \log \left (-\sqrt {3} \sqrt [4]{a} x+\sqrt {a}+x^2\right )}{4 \sqrt {3} a^{3/4}}+\frac {\left (A-\sqrt {a} B\right ) \log \left (\sqrt {3} \sqrt [4]{a} x+\sqrt {a}+x^2\right )}{4 \sqrt {3} a^{3/4}} \]

[In]

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

[Out]

-1/2*((A + Sqrt[a]*B)*ArcTan[Sqrt[3] - (2*x)/a^(1/4)])/a^(3/4) + ((A + Sqrt[a]*B)*ArcTan[Sqrt[3] + (2*x)/a^(1/
4)])/(2*a^(3/4)) - ((A - Sqrt[a]*B)*Log[Sqrt[a] - Sqrt[3]*a^(1/4)*x + x^2])/(4*Sqrt[3]*a^(3/4)) + ((A - Sqrt[a
]*B)*Log[Sqrt[a] + Sqrt[3]*a^(1/4)*x + x^2])/(4*Sqrt[3]*a^(3/4))

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 631

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 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 1183

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\sqrt {3} \sqrt [4]{a} A-\left (A-\sqrt {a} B\right ) x}{\sqrt {a}-\sqrt {3} \sqrt [4]{a} x+x^2} \, dx}{2 \sqrt {3} a^{3/4}}+\frac {\int \frac {\sqrt {3} \sqrt [4]{a} A+\left (A-\sqrt {a} B\right ) x}{\sqrt {a}+\sqrt {3} \sqrt [4]{a} x+x^2} \, dx}{2 \sqrt {3} a^{3/4}} \\ & = \frac {1}{4} \left (\frac {A}{\sqrt {a}}+B\right ) \int \frac {1}{\sqrt {a}-\sqrt {3} \sqrt [4]{a} x+x^2} \, dx+\frac {1}{4} \left (\frac {A}{\sqrt {a}}+B\right ) \int \frac {1}{\sqrt {a}+\sqrt {3} \sqrt [4]{a} x+x^2} \, dx-\frac {\left (A-\sqrt {a} B\right ) \int \frac {-\sqrt {3} \sqrt [4]{a}+2 x}{\sqrt {a}-\sqrt {3} \sqrt [4]{a} x+x^2} \, dx}{4 \sqrt {3} a^{3/4}}+\frac {\left (A-\sqrt {a} B\right ) \int \frac {\sqrt {3} \sqrt [4]{a}+2 x}{\sqrt {a}+\sqrt {3} \sqrt [4]{a} x+x^2} \, dx}{4 \sqrt {3} a^{3/4}} \\ & = -\frac {\left (A-\sqrt {a} B\right ) \log \left (\sqrt {a}-\sqrt {3} \sqrt [4]{a} x+x^2\right )}{4 \sqrt {3} a^{3/4}}+\frac {\left (A-\sqrt {a} B\right ) \log \left (\sqrt {a}+\sqrt {3} \sqrt [4]{a} x+x^2\right )}{4 \sqrt {3} a^{3/4}}+\frac {\left (A+\sqrt {a} B\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 x}{\sqrt {3} \sqrt [4]{a}}\right )}{2 \sqrt {3} a^{3/4}}-\frac {\left (A+\sqrt {a} B\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 x}{\sqrt {3} \sqrt [4]{a}}\right )}{2 \sqrt {3} a^{3/4}} \\ & = -\frac {\left (A+\sqrt {a} B\right ) \tan ^{-1}\left (\sqrt {3}-\frac {2 x}{\sqrt [4]{a}}\right )}{2 a^{3/4}}+\frac {\left (A+\sqrt {a} B\right ) \tan ^{-1}\left (\sqrt {3}+\frac {2 x}{\sqrt [4]{a}}\right )}{2 a^{3/4}}-\frac {\left (A-\sqrt {a} B\right ) \log \left (\sqrt {a}-\sqrt {3} \sqrt [4]{a} x+x^2\right )}{4 \sqrt {3} a^{3/4}}+\frac {\left (A-\sqrt {a} B\right ) \log \left (\sqrt {a}+\sqrt {3} \sqrt [4]{a} x+x^2\right )}{4 \sqrt {3} a^{3/4}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.86 \[ \int \frac {A+B x^2}{a-\sqrt {a} x^2+x^4} \, dx=\frac {\sqrt [4]{-1} \left (\frac {\left (-2 i A+\left (-i+\sqrt {3}\right ) \sqrt {a} B\right ) \arctan \left (\frac {(1+i) x}{\sqrt {-i+\sqrt {3}} \sqrt [4]{a}}\right )}{\sqrt {-i+\sqrt {3}}}-\frac {\left (2 i A+\left (i+\sqrt {3}\right ) \sqrt {a} B\right ) \text {arctanh}\left (\frac {(1+i) x}{\sqrt {i+\sqrt {3}} \sqrt [4]{a}}\right )}{\sqrt {i+\sqrt {3}}}\right )}{\sqrt {6} a^{3/4}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.18

method	result	size
default	\(\frac {\frac {\left (A \sqrt {a}\, \sqrt {3}-B \sqrt {3}\, a \right ) \ln \left (x^{2}+a^{\frac {1}{4}} x \sqrt {3}+\sqrt {a}\right )}{2}+\frac {2 \left (3 A \,a^{\frac {3}{4}}-\frac {\left (A \sqrt {a}\, \sqrt {3}-B \sqrt {3}\, a \right ) a^{\frac {1}{4}} \sqrt {3}}{2}\right ) \arctan \left (\frac {2 x +a^{\frac {1}{4}} \sqrt {3}}{a^{\frac {1}{4}}}\right )}{a^{\frac {1}{4}}}}{6 a^{\frac {5}{4}}}+\frac {\frac {\left (-A \sqrt {a}\, \sqrt {3}+B \sqrt {3}\, a \right ) \ln \left (x^{2}-a^{\frac {1}{4}} x \sqrt {3}+\sqrt {a}\right )}{2}+\frac {2 \left (3 A \,a^{\frac {3}{4}}+\frac {\left (-A \sqrt {a}\, \sqrt {3}+B \sqrt {3}\, a \right ) a^{\frac {1}{4}} \sqrt {3}}{2}\right ) \arctan \left (\frac {2 x -a^{\frac {1}{4}} \sqrt {3}}{a^{\frac {1}{4}}}\right )}{a^{\frac {1}{4}}}}{6 a^{\frac {5}{4}}}\)	\(188\)

[In]

int((B*x^2+A)/(a+x^4-x^2*a^(1/2)),x,method=_RETURNVERBOSE)

[Out]

1/6/a^(5/4)*(1/2*(A*a^(1/2)*3^(1/2)-B*3^(1/2)*a)*ln(x^2+a^(1/4)*x*3^(1/2)+a^(1/2))+2*(3*A*a^(3/4)-1/2*(A*a^(1/
2)*3^(1/2)-B*3^(1/2)*a)*a^(1/4)*3^(1/2))/a^(1/4)*arctan((2*x+a^(1/4)*3^(1/2))/a^(1/4)))+1/6/a^(5/4)*(1/2*(-A*a
^(1/2)*3^(1/2)+B*3^(1/2)*a)*ln(x^2-a^(1/4)*x*3^(1/2)+a^(1/2))+2*(3*A*a^(3/4)+1/2*(-A*a^(1/2)*3^(1/2)+B*3^(1/2)
*a)*a^(1/4)*3^(1/2))/a^(1/4)*arctan((2*x-a^(1/4)*3^(1/2))/a^(1/4)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1141 vs. \(2 (116) = 232\).

Time = 0.38 (sec) , antiderivative size = 1141, normalized size of antiderivative = 7.13 \[ \int \frac {A+B x^2}{a-\sqrt {a} x^2+x^4} \, dx=\text {Too large to display} \]

[In]

integrate((B*x^2+A)/(a+x^4-x^2*a^(1/2)),x, algorithm="fricas")

[Out]

1/2*sqrt(1/6)*sqrt(-(4*A*B*a + 3*sqrt(1/3)*a^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a + A^4)/a^3) + (B^2*a + A^2)*sqrt(a
))/a^2)*log(2*(B^6*a^3 - A^6)*x + 3*sqrt(1/6)*(A*B^4*a^3 - A^5*a - sqrt(1/3)*(2*B^3*a^4 + A^2*B*a^3)*sqrt(-(B^
4*a^2 - 2*A^2*B^2*a + A^4)/a^3) - (A^2*B^3*a^2 - A^4*B*a - sqrt(1/3)*(A*B^2*a^3 - A^3*a^2)*sqrt(-(B^4*a^2 - 2*
A^2*B^2*a + A^4)/a^3))*sqrt(a))*sqrt(-(4*A*B*a + 3*sqrt(1/3)*a^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a + A^4)/a^3) + (B
^2*a + A^2)*sqrt(a))/a^2)) - 1/2*sqrt(1/6)*sqrt(-(4*A*B*a + 3*sqrt(1/3)*a^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a + A^4
)/a^3) + (B^2*a + A^2)*sqrt(a))/a^2)*log(2*(B^6*a^3 - A^6)*x - 3*sqrt(1/6)*(A*B^4*a^3 - A^5*a - sqrt(1/3)*(2*B
^3*a^4 + A^2*B*a^3)*sqrt(-(B^4*a^2 - 2*A^2*B^2*a + A^4)/a^3) - (A^2*B^3*a^2 - A^4*B*a - sqrt(1/3)*(A*B^2*a^3 -
A^3*a^2)*sqrt(-(B^4*a^2 - 2*A^2*B^2*a + A^4)/a^3))*sqrt(a))*sqrt(-(4*A*B*a + 3*sqrt(1/3)*a^2*sqrt(-(B^4*a^2 -
2*A^2*B^2*a + A^4)/a^3) + (B^2*a + A^2)*sqrt(a))/a^2)) + 1/2*sqrt(1/6)*sqrt(-(4*A*B*a - 3*sqrt(1/3)*a^2*sqrt(
-(B^4*a^2 - 2*A^2*B^2*a + A^4)/a^3) + (B^2*a + A^2)*sqrt(a))/a^2)*log(2*(B^6*a^3 - A^6)*x + 3*sqrt(1/6)*(A*B^4
*a^3 - A^5*a + sqrt(1/3)*(2*B^3*a^4 + A^2*B*a^3)*sqrt(-(B^4*a^2 - 2*A^2*B^2*a + A^4)/a^3) - (A^2*B^3*a^2 - A^4
*B*a + sqrt(1/3)*(A*B^2*a^3 - A^3*a^2)*sqrt(-(B^4*a^2 - 2*A^2*B^2*a + A^4)/a^3))*sqrt(a))*sqrt(-(4*A*B*a - 3*s
qrt(1/3)*a^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a + A^4)/a^3) + (B^2*a + A^2)*sqrt(a))/a^2)) - 1/2*sqrt(1/6)*sqrt(-(4*
A*B*a - 3*sqrt(1/3)*a^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a + A^4)/a^3) + (B^2*a + A^2)*sqrt(a))/a^2)*log(2*(B^6*a^3
- A^6)*x - 3*sqrt(1/6)*(A*B^4*a^3 - A^5*a + sqrt(1/3)*(2*B^3*a^4 + A^2*B*a^3)*sqrt(-(B^4*a^2 - 2*A^2*B^2*a + A
^4)/a^3) - (A^2*B^3*a^2 - A^4*B*a + sqrt(1/3)*(A*B^2*a^3 - A^3*a^2)*sqrt(-(B^4*a^2 - 2*A^2*B^2*a + A^4)/a^3))*
sqrt(a))*sqrt(-(4*A*B*a - 3*sqrt(1/3)*a^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a + A^4)/a^3) + (B^2*a + A^2)*sqrt(a))/a^
2))

Sympy [F(-2)]

Exception generated. \[ \int \frac {A+B x^2}{a-\sqrt {a} x^2+x^4} \, dx=\text {Exception raised: PolynomialError} \]

[In]

integrate((B*x**2+A)/(a+x**4-x**2*a**(1/2)),x)

[Out]

Exception raised: PolynomialError >> 1/(64*_t**4*a - 16*_t**2*B**2*sqrt(a) + B**4) contains an element of the
set of generators.

Maxima [F]

\[ \int \frac {A+B x^2}{a-\sqrt {a} x^2+x^4} \, dx=\int { \frac {B x^{2} + A}{x^{4} - \sqrt {a} x^{2} + a} \,d x } \]

[In]

integrate((B*x^2+A)/(a+x^4-x^2*a^(1/2)),x, algorithm="maxima")

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {A+B x^2}{a-\sqrt {a} x^2+x^4} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((B*x^2+A)/(a+x^4-x^2*a^(1/2)),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [B] (verification not implemented)

Time = 14.47 (sec) , antiderivative size = 1155, normalized size of antiderivative = 7.22 \[ \int \frac {A+B x^2}{a-\sqrt {a} x^2+x^4} \, dx=-2\,\mathrm {atanh}\left (\frac {6\,A^2\,x\,\sqrt {\frac {B^2\,\sqrt {-27\,a^3}}{72\,a^2}-\frac {B^2}{24\,\sqrt {a}}-\frac {A^2\,\sqrt {-27\,a^3}}{72\,a^3}-\frac {A^2}{24\,a^{3/2}}-\frac {A\,B}{6\,a}}}{2\,A^2\,B-2\,B^3\,a+\frac {A^3}{\sqrt {a}}-A\,B^2\,\sqrt {a}+\frac {A^3\,\sqrt {-27\,a^3}}{3\,a^2}-\frac {A\,B^2\,\sqrt {-27\,a^3}}{3\,a}}-\frac {6\,B^2\,a\,x\,\sqrt {\frac {B^2\,\sqrt {-27\,a^3}}{72\,a^2}-\frac {B^2}{24\,\sqrt {a}}-\frac {A^2\,\sqrt {-27\,a^3}}{72\,a^3}-\frac {A^2}{24\,a^{3/2}}-\frac {A\,B}{6\,a}}}{2\,A^2\,B-2\,B^3\,a+\frac {A^3}{\sqrt {a}}-A\,B^2\,\sqrt {a}+\frac {A^3\,\sqrt {-27\,a^3}}{3\,a^2}-\frac {A\,B^2\,\sqrt {-27\,a^3}}{3\,a}}-\frac {2\,A^2\,x\,\sqrt {-27\,a^3}\,\sqrt {\frac {B^2\,\sqrt {-27\,a^3}}{72\,a^2}-\frac {B^2}{24\,\sqrt {a}}-\frac {A^2\,\sqrt {-27\,a^3}}{72\,a^3}-\frac {A^2}{24\,a^{3/2}}-\frac {A\,B}{6\,a}}}{3\,a^{3/2}\,\left (2\,A^2\,B-2\,B^3\,a+\frac {A^3}{\sqrt {a}}-A\,B^2\,\sqrt {a}+\frac {A^3\,\sqrt {-27\,a^3}}{3\,a^2}-\frac {A\,B^2\,\sqrt {-27\,a^3}}{3\,a}\right )}+\frac {2\,B^2\,x\,\sqrt {-27\,a^3}\,\sqrt {\frac {B^2\,\sqrt {-27\,a^3}}{72\,a^2}-\frac {B^2}{24\,\sqrt {a}}-\frac {A^2\,\sqrt {-27\,a^3}}{72\,a^3}-\frac {A^2}{24\,a^{3/2}}-\frac {A\,B}{6\,a}}}{3\,\sqrt {a}\,\left (2\,A^2\,B-2\,B^3\,a+\frac {A^3}{\sqrt {a}}-A\,B^2\,\sqrt {a}+\frac {A^3\,\sqrt {-27\,a^3}}{3\,a^2}-\frac {A\,B^2\,\sqrt {-27\,a^3}}{3\,a}\right )}\right )\,\sqrt {\frac {B^2\,\sqrt {-27\,a^3}}{72\,a^2}-\frac {B^2}{24\,\sqrt {a}}-\frac {A^2\,\sqrt {-27\,a^3}}{72\,a^3}-\frac {A^2}{24\,a^{3/2}}-\frac {A\,B}{6\,a}}-2\,\mathrm {atanh}\left (\frac {6\,A^2\,x\,\sqrt {\frac {A^2\,\sqrt {-27\,a^3}}{72\,a^3}-\frac {B^2}{24\,\sqrt {a}}-\frac {A^2}{24\,a^{3/2}}-\frac {B^2\,\sqrt {-27\,a^3}}{72\,a^2}-\frac {A\,B}{6\,a}}}{2\,A^2\,B-2\,B^3\,a+\frac {A^3}{\sqrt {a}}-A\,B^2\,\sqrt {a}-\frac {A^3\,\sqrt {-27\,a^3}}{3\,a^2}+\frac {A\,B^2\,\sqrt {-27\,a^3}}{3\,a}}-\frac {6\,B^2\,a\,x\,\sqrt {\frac {A^2\,\sqrt {-27\,a^3}}{72\,a^3}-\frac {B^2}{24\,\sqrt {a}}-\frac {A^2}{24\,a^{3/2}}-\frac {B^2\,\sqrt {-27\,a^3}}{72\,a^2}-\frac {A\,B}{6\,a}}}{2\,A^2\,B-2\,B^3\,a+\frac {A^3}{\sqrt {a}}-A\,B^2\,\sqrt {a}-\frac {A^3\,\sqrt {-27\,a^3}}{3\,a^2}+\frac {A\,B^2\,\sqrt {-27\,a^3}}{3\,a}}+\frac {2\,A^2\,x\,\sqrt {-27\,a^3}\,\sqrt {\frac {A^2\,\sqrt {-27\,a^3}}{72\,a^3}-\frac {B^2}{24\,\sqrt {a}}-\frac {A^2}{24\,a^{3/2}}-\frac {B^2\,\sqrt {-27\,a^3}}{72\,a^2}-\frac {A\,B}{6\,a}}}{3\,a^{3/2}\,\left (2\,A^2\,B-2\,B^3\,a+\frac {A^3}{\sqrt {a}}-A\,B^2\,\sqrt {a}-\frac {A^3\,\sqrt {-27\,a^3}}{3\,a^2}+\frac {A\,B^2\,\sqrt {-27\,a^3}}{3\,a}\right )}-\frac {2\,B^2\,x\,\sqrt {-27\,a^3}\,\sqrt {\frac {A^2\,\sqrt {-27\,a^3}}{72\,a^3}-\frac {B^2}{24\,\sqrt {a}}-\frac {A^2}{24\,a^{3/2}}-\frac {B^2\,\sqrt {-27\,a^3}}{72\,a^2}-\frac {A\,B}{6\,a}}}{3\,\sqrt {a}\,\left (2\,A^2\,B-2\,B^3\,a+\frac {A^3}{\sqrt {a}}-A\,B^2\,\sqrt {a}-\frac {A^3\,\sqrt {-27\,a^3}}{3\,a^2}+\frac {A\,B^2\,\sqrt {-27\,a^3}}{3\,a}\right )}\right )\,\sqrt {\frac {A^2\,\sqrt {-27\,a^3}}{72\,a^3}-\frac {B^2}{24\,\sqrt {a}}-\frac {A^2}{24\,a^{3/2}}-\frac {B^2\,\sqrt {-27\,a^3}}{72\,a^2}-\frac {A\,B}{6\,a}} \]

[In]

int((A + B*x^2)/(a + x^4 - a^(1/2)*x^2),x)

[Out]

- 2*atanh((6*A^2*x*((B^2*(-27*a^3)^(1/2))/(72*a^2) - B^2/(24*a^(1/2)) - (A^2*(-27*a^3)^(1/2))/(72*a^3) - A^2/(
24*a^(3/2)) - (A*B)/(6*a))^(1/2))/(2*A^2*B - 2*B^3*a + A^3/a^(1/2) - A*B^2*a^(1/2) + (A^3*(-27*a^3)^(1/2))/(3*
a^2) - (A*B^2*(-27*a^3)^(1/2))/(3*a)) - (6*B^2*a*x*((B^2*(-27*a^3)^(1/2))/(72*a^2) - B^2/(24*a^(1/2)) - (A^2*(
-27*a^3)^(1/2))/(72*a^3) - A^2/(24*a^(3/2)) - (A*B)/(6*a))^(1/2))/(2*A^2*B - 2*B^3*a + A^3/a^(1/2) - A*B^2*a^(
1/2) + (A^3*(-27*a^3)^(1/2))/(3*a^2) - (A*B^2*(-27*a^3)^(1/2))/(3*a)) - (2*A^2*x*(-27*a^3)^(1/2)*((B^2*(-27*a^
3)^(1/2))/(72*a^2) - B^2/(24*a^(1/2)) - (A^2*(-27*a^3)^(1/2))/(72*a^3) - A^2/(24*a^(3/2)) - (A*B)/(6*a))^(1/2)
)/(3*a^(3/2)*(2*A^2*B - 2*B^3*a + A^3/a^(1/2) - A*B^2*a^(1/2) + (A^3*(-27*a^3)^(1/2))/(3*a^2) - (A*B^2*(-27*a^
3)^(1/2))/(3*a))) + (2*B^2*x*(-27*a^3)^(1/2)*((B^2*(-27*a^3)^(1/2))/(72*a^2) - B^2/(24*a^(1/2)) - (A^2*(-27*a^
3)^(1/2))/(72*a^3) - A^2/(24*a^(3/2)) - (A*B)/(6*a))^(1/2))/(3*a^(1/2)*(2*A^2*B - 2*B^3*a + A^3/a^(1/2) - A*B^
2*a^(1/2) + (A^3*(-27*a^3)^(1/2))/(3*a^2) - (A*B^2*(-27*a^3)^(1/2))/(3*a))))*((B^2*(-27*a^3)^(1/2))/(72*a^2) -
B^2/(24*a^(1/2)) - (A^2*(-27*a^3)^(1/2))/(72*a^3) - A^2/(24*a^(3/2)) - (A*B)/(6*a))^(1/2) - 2*atanh((6*A^2*x*
((A^2*(-27*a^3)^(1/2))/(72*a^3) - B^2/(24*a^(1/2)) - A^2/(24*a^(3/2)) - (B^2*(-27*a^3)^(1/2))/(72*a^2) - (A*B)
/(6*a))^(1/2))/(2*A^2*B - 2*B^3*a + A^3/a^(1/2) - A*B^2*a^(1/2) - (A^3*(-27*a^3)^(1/2))/(3*a^2) + (A*B^2*(-27*
a^3)^(1/2))/(3*a)) - (6*B^2*a*x*((A^2*(-27*a^3)^(1/2))/(72*a^3) - B^2/(24*a^(1/2)) - A^2/(24*a^(3/2)) - (B^2*(
-27*a^3)^(1/2))/(72*a^2) - (A*B)/(6*a))^(1/2))/(2*A^2*B - 2*B^3*a + A^3/a^(1/2) - A*B^2*a^(1/2) - (A^3*(-27*a^
3)^(1/2))/(3*a^2) + (A*B^2*(-27*a^3)^(1/2))/(3*a)) + (2*A^2*x*(-27*a^3)^(1/2)*((A^2*(-27*a^3)^(1/2))/(72*a^3)
- B^2/(24*a^(1/2)) - A^2/(24*a^(3/2)) - (B^2*(-27*a^3)^(1/2))/(72*a^2) - (A*B)/(6*a))^(1/2))/(3*a^(3/2)*(2*A^2
*B - 2*B^3*a + A^3/a^(1/2) - A*B^2*a^(1/2) - (A^3*(-27*a^3)^(1/2))/(3*a^2) + (A*B^2*(-27*a^3)^(1/2))/(3*a))) -
(2*B^2*x*(-27*a^3)^(1/2)*((A^2*(-27*a^3)^(1/2))/(72*a^3) - B^2/(24*a^(1/2)) - A^2/(24*a^(3/2)) - (B^2*(-27*a^
3)^(1/2))/(72*a^2) - (A*B)/(6*a))^(1/2))/(3*a^(1/2)*(2*A^2*B - 2*B^3*a + A^3/a^(1/2) - A*B^2*a^(1/2) - (A^3*(-
27*a^3)^(1/2))/(3*a^2) + (A*B^2*(-27*a^3)^(1/2))/(3*a))))*((A^2*(-27*a^3)^(1/2))/(72*a^3) - B^2/(24*a^(1/2)) -
A^2/(24*a^(3/2)) - (B^2*(-27*a^3)^(1/2))/(72*a^2) - (A*B)/(6*a))^(1/2)