\(\int \frac {A+B x^2}{a-\sqrt {a c} x^2+c x^4} \, dx\) [111]

Optimal result

Integrand size = 29, antiderivative size = 414 \[ \int \frac {A+B x^2}{a-\sqrt {a c} x^2+c x^4} \, dx=-\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \arctan \left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}-2 \sqrt {c} x}{\sqrt {2 \sqrt {a} \sqrt {c}-\sqrt {a c}}}\right )}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-\sqrt {a c}}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \arctan \left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}+2 \sqrt {c} x}{\sqrt {2 \sqrt {a} \sqrt {c}-\sqrt {a c}}}\right )}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-\sqrt {a c}}}-\frac {\left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \log \left (\sqrt {a}-\sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}} x+\sqrt {c} x^2\right )}{4 \sqrt {a} \sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}}+\frac {\left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \log \left (\sqrt {a}+\sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}} x+\sqrt {c} x^2\right )}{4 \sqrt {a} \sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1183, 648, 632, 210, 642} \[ \int \frac {A+B x^2}{a-\sqrt {a c} x^2+c x^4} \, dx=-\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \arctan \left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}-2 \sqrt {c} x}{\sqrt {2 \sqrt {a} \sqrt {c}-\sqrt {a c}}}\right )}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-\sqrt {a c}}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \arctan \left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}+2 \sqrt {c} x}{\sqrt {2 \sqrt {a} \sqrt {c}-\sqrt {a c}}}\right )}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-\sqrt {a c}}}-\frac {\left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \log \left (-x \sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {a} \sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}}+\frac {\left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \log \left (x \sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {a} \sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}} \]

[In]

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

[Out]

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

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 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 {\frac {A \sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}}{\sqrt {c}}-\left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) x}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}} x}{\sqrt {c}}+x^2} \, dx}{2 \sqrt {a} \sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}}+\frac {\int \frac {\frac {A \sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}}{\sqrt {c}}+\left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) x}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}} x}{\sqrt {c}}+x^2} \, dx}{2 \sqrt {a} \sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}} \\ & = \frac {\left (B+\frac {A \sqrt {c}}{\sqrt {a}}\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}} x}{\sqrt {c}}+x^2} \, dx}{4 c}+\frac {\left (B+\frac {A \sqrt {c}}{\sqrt {a}}\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}} x}{\sqrt {c}}+x^2} \, dx}{4 c}-\frac {\left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \int \frac {-\frac {\sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}}{\sqrt {c}}+2 x}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}} x}{\sqrt {c}}+x^2} \, dx}{4 \sqrt {a} \sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}}+\frac {\left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \int \frac {\frac {\sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}}{\sqrt {c}}+2 x}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}} x}{\sqrt {c}}+x^2} \, dx}{4 \sqrt {a} \sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}} \\ & = -\frac {\left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \log \left (\sqrt {a}-\sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}} x+\sqrt {c} x^2\right )}{4 \sqrt {a} \sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}}+\frac {\left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \log \left (\sqrt {a}+\sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}} x+\sqrt {c} x^2\right )}{4 \sqrt {a} \sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}}-\frac {\left (B+\frac {A \sqrt {c}}{\sqrt {a}}\right ) \text {Subst}\left (\int \frac {1}{-\frac {2 \sqrt {a} \sqrt {c}-\sqrt {a c}}{c}-x^2} \, dx,x,-\frac {\sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}}{\sqrt {c}}+2 x\right )}{2 c}-\frac {\left (B+\frac {A \sqrt {c}}{\sqrt {a}}\right ) \text {Subst}\left (\int \frac {1}{-\frac {2 \sqrt {a} \sqrt {c}-\sqrt {a c}}{c}-x^2} \, dx,x,\frac {\sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}}{\sqrt {c}}+2 x\right )}{2 c} \\ & = -\frac {\left (B+\frac {A \sqrt {c}}{\sqrt {a}}\right ) \tan ^{-1}\left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}-2 \sqrt {c} x}{\sqrt {2 \sqrt {a} \sqrt {c}-\sqrt {a c}}}\right )}{2 \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-\sqrt {a c}}}+\frac {\left (B+\frac {A \sqrt {c}}{\sqrt {a}}\right ) \tan ^{-1}\left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}+2 \sqrt {c} x}{\sqrt {2 \sqrt {a} \sqrt {c}-\sqrt {a c}}}\right )}{2 \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-\sqrt {a c}}}-\frac {\left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \log \left (\sqrt {a}-\sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}} x+\sqrt {c} x^2\right )}{4 \sqrt {a} \sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}}+\frac {\left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \log \left (\sqrt {a}+\sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}} x+\sqrt {c} x^2\right )}{4 \sqrt {a} \sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.10 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.34 \[ \int \frac {A+B x^2}{a-\sqrt {a c} x^2+c x^4} \, dx=\frac {\text {RootSum}\left [a^2+a c \text {$\#$1}^4+c^2 \text {$\#$1}^8\&,\frac {a A \log (x-\text {$\#$1})+a B \log (x-\text {$\#$1}) \text {$\#$1}^2+A \sqrt {a c} \log (x-\text {$\#$1}) \text {$\#$1}^2+A c \log (x-\text {$\#$1}) \text {$\#$1}^4+B \sqrt {a c} \log (x-\text {$\#$1}) \text {$\#$1}^4+B c \log (x-\text {$\#$1}) \text {$\#$1}^6}{a \text {$\#$1}^3+2 c \text {$\#$1}^7}\&\right ]}{4 c} \]

[In]

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

[Out]

RootSum[a^2 + a*c*#1^4 + c^2*#1^8 & , (a*A*Log[x - #1] + a*B*Log[x - #1]*#1^2 + A*Sqrt[a*c]*Log[x - #1]*#1^2 +
A*c*Log[x - #1]*#1^4 + B*Sqrt[a*c]*Log[x - #1]*#1^4 + B*c*Log[x - #1]*#1^6)/(a*#1^3 + 2*c*#1^7) & ]/(4*c)

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 347, normalized size of antiderivative = 0.84

method	result	size
default	\(\frac {-\frac {\left (-B \sqrt {3}\, \left (a c \right )^{\frac {3}{4}} a^{\frac {3}{2}}+A \sqrt {3}\, \left (a c \right )^{\frac {3}{4}} \sqrt {c}\, a \right ) \ln \left (x \sqrt {3}\, \left (a c \right )^{\frac {1}{4}}-x^{2} \sqrt {c}-\sqrt {a}\right )}{2 \sqrt {c}}+\frac {2 \left (-3 A c \,a^{2}+\frac {\left (-B \sqrt {3}\, \left (a c \right )^{\frac {3}{4}} a^{\frac {3}{2}}+A \sqrt {3}\, \left (a c \right )^{\frac {3}{4}} \sqrt {c}\, a \right ) \sqrt {3}\, \left (a c \right )^{\frac {1}{4}}}{2 \sqrt {c}}\right ) \arctan \left (\frac {\sqrt {3}\, \left (a c \right )^{\frac {1}{4}}-2 x \sqrt {c}}{\sqrt {4 \sqrt {a}\, \sqrt {c}-3 \sqrt {a c}}}\right )}{\sqrt {4 \sqrt {a}\, \sqrt {c}-3 \sqrt {a c}}}}{6 a^{\frac {5}{2}} c}+\frac {\frac {\left (-B \sqrt {3}\, \left (a c \right )^{\frac {3}{4}} a^{\frac {3}{2}}+A \sqrt {3}\, \left (a c \right )^{\frac {3}{4}} \sqrt {c}\, a \right ) \ln \left (x^{2} \sqrt {c}+x \sqrt {3}\, \left (a c \right )^{\frac {1}{4}}+\sqrt {a}\right )}{2 \sqrt {c}}+\frac {2 \left (3 A c \,a^{2}-\frac {\left (-B \sqrt {3}\, \left (a c \right )^{\frac {3}{4}} a^{\frac {3}{2}}+A \sqrt {3}\, \left (a c \right )^{\frac {3}{4}} \sqrt {c}\, a \right ) \sqrt {3}\, \left (a c \right )^{\frac {1}{4}}}{2 \sqrt {c}}\right ) \arctan \left (\frac {2 x \sqrt {c}+\sqrt {3}\, \left (a c \right )^{\frac {1}{4}}}{\sqrt {4 \sqrt {a}\, \sqrt {c}-3 \sqrt {a c}}}\right )}{\sqrt {4 \sqrt {a}\, \sqrt {c}-3 \sqrt {a c}}}}{6 a^{\frac {5}{2}} c}\)	\(347\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1457 vs. \(2 (289) = 578\).

Time = 0.60 (sec) , antiderivative size = 1457, normalized size of antiderivative = 3.52 \[ \int \frac {A+B x^2}{a-\sqrt {a c} x^2+c x^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Sympy [F(-2)]

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

[In]

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

[Out]

Exception raised: PolynomialError >> 1/(64*_t**4*a*c**3 - 16*_t**2*B**2*c*sqrt(a*c) + B**4) contains an elemen
t of the set of generators.

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

integrate((B*x^2+A)/(a+c*x^4-x^2*(a*c)^(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 = 14.58 (sec) , antiderivative size = 3285, normalized size of antiderivative = 7.93 \[ \int \frac {A+B x^2}{a-\sqrt {a c} x^2+c x^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

- atan(((((12*A*a)/c^2 - (2*x*(4*c*(a*c)^(3/2) - 16*a*c^2*(a*c)^(1/2))*(-(B^2*a*(-27*a^3*c^3)^(1/2) - A^2*c*(-
27*a^3*c^3)^(1/2) - B^2*a*(a*c)^(3/2) - A^2*c*(a*c)^(3/2) + 12*A*B*a^2*c^2 + 4*A^2*a*c^2*(a*c)^(1/2) + 4*B^2*a
^2*c*(a*c)^(1/2))/(72*a^3*c^3))^(1/2))/c^4)*(-(B^2*a*(-27*a^3*c^3)^(1/2) - A^2*c*(-27*a^3*c^3)^(1/2) - B^2*a*(
a*c)^(3/2) - A^2*c*(a*c)^(3/2) + 12*A*B*a^2*c^2 + 4*A^2*a*c^2*(a*c)^(1/2) + 4*B^2*a^2*c*(a*c)^(1/2))/(72*a^3*c
^3))^(1/2) + (2*x*(2*A^2*c^2 - B^2*a*c + 2*A*B*c*(a*c)^(1/2)))/c^4)*(-(B^2*a*(-27*a^3*c^3)^(1/2) - A^2*c*(-27*
a^3*c^3)^(1/2) - B^2*a*(a*c)^(3/2) - A^2*c*(a*c)^(3/2) + 12*A*B*a^2*c^2 + 4*A^2*a*c^2*(a*c)^(1/2) + 4*B^2*a^2*
c*(a*c)^(1/2))/(72*a^3*c^3))^(1/2)*1i - (((12*A*a)/c^2 + (2*x*(4*c*(a*c)^(3/2) - 16*a*c^2*(a*c)^(1/2))*(-(B^2*
a*(-27*a^3*c^3)^(1/2) - A^2*c*(-27*a^3*c^3)^(1/2) - B^2*a*(a*c)^(3/2) - A^2*c*(a*c)^(3/2) + 12*A*B*a^2*c^2 + 4
*A^2*a*c^2*(a*c)^(1/2) + 4*B^2*a^2*c*(a*c)^(1/2))/(72*a^3*c^3))^(1/2))/c^4)*(-(B^2*a*(-27*a^3*c^3)^(1/2) - A^2
*c*(-27*a^3*c^3)^(1/2) - B^2*a*(a*c)^(3/2) - A^2*c*(a*c)^(3/2) + 12*A*B*a^2*c^2 + 4*A^2*a*c^2*(a*c)^(1/2) + 4*
B^2*a^2*c*(a*c)^(1/2))/(72*a^3*c^3))^(1/2) - (2*x*(2*A^2*c^2 - B^2*a*c + 2*A*B*c*(a*c)^(1/2)))/c^4)*(-(B^2*a*(
-27*a^3*c^3)^(1/2) - A^2*c*(-27*a^3*c^3)^(1/2) - B^2*a*(a*c)^(3/2) - A^2*c*(a*c)^(3/2) + 12*A*B*a^2*c^2 + 4*A^
2*a*c^2*(a*c)^(1/2) + 4*B^2*a^2*c*(a*c)^(1/2))/(72*a^3*c^3))^(1/2)*1i)/((((12*A*a)/c^2 - (2*x*(4*c*(a*c)^(3/2)
- 16*a*c^2*(a*c)^(1/2))*(-(B^2*a*(-27*a^3*c^3)^(1/2) - A^2*c*(-27*a^3*c^3)^(1/2) - B^2*a*(a*c)^(3/2) - A^2*c*
(a*c)^(3/2) + 12*A*B*a^2*c^2 + 4*A^2*a*c^2*(a*c)^(1/2) + 4*B^2*a^2*c*(a*c)^(1/2))/(72*a^3*c^3))^(1/2))/c^4)*(-
(B^2*a*(-27*a^3*c^3)^(1/2) - A^2*c*(-27*a^3*c^3)^(1/2) - B^2*a*(a*c)^(3/2) - A^2*c*(a*c)^(3/2) + 12*A*B*a^2*c^
2 + 4*A^2*a*c^2*(a*c)^(1/2) + 4*B^2*a^2*c*(a*c)^(1/2))/(72*a^3*c^3))^(1/2) + (2*x*(2*A^2*c^2 - B^2*a*c + 2*A*B
*c*(a*c)^(1/2)))/c^4)*(-(B^2*a*(-27*a^3*c^3)^(1/2) - A^2*c*(-27*a^3*c^3)^(1/2) - B^2*a*(a*c)^(3/2) - A^2*c*(a*
c)^(3/2) + 12*A*B*a^2*c^2 + 4*A^2*a*c^2*(a*c)^(1/2) + 4*B^2*a^2*c*(a*c)^(1/2))/(72*a^3*c^3))^(1/2) + (((12*A*a
)/c^2 + (2*x*(4*c*(a*c)^(3/2) - 16*a*c^2*(a*c)^(1/2))*(-(B^2*a*(-27*a^3*c^3)^(1/2) - A^2*c*(-27*a^3*c^3)^(1/2)
- B^2*a*(a*c)^(3/2) - A^2*c*(a*c)^(3/2) + 12*A*B*a^2*c^2 + 4*A^2*a*c^2*(a*c)^(1/2) + 4*B^2*a^2*c*(a*c)^(1/2))
/(72*a^3*c^3))^(1/2))/c^4)*(-(B^2*a*(-27*a^3*c^3)^(1/2) - A^2*c*(-27*a^3*c^3)^(1/2) - B^2*a*(a*c)^(3/2) - A^2*
c*(a*c)^(3/2) + 12*A*B*a^2*c^2 + 4*A^2*a*c^2*(a*c)^(1/2) + 4*B^2*a^2*c*(a*c)^(1/2))/(72*a^3*c^3))^(1/2) - (2*x
*(2*A^2*c^2 - B^2*a*c + 2*A*B*c*(a*c)^(1/2)))/c^4)*(-(B^2*a*(-27*a^3*c^3)^(1/2) - A^2*c*(-27*a^3*c^3)^(1/2) -
B^2*a*(a*c)^(3/2) - A^2*c*(a*c)^(3/2) + 12*A*B*a^2*c^2 + 4*A^2*a*c^2*(a*c)^(1/2) + 4*B^2*a^2*c*(a*c)^(1/2))/(7
2*a^3*c^3))^(1/2) + (2*(B^3*a + A^2*B*c + A*B^2*(a*c)^(1/2)))/c^4))*(-(B^2*a*(-27*a^3*c^3)^(1/2) - A^2*c*(-27*
a^3*c^3)^(1/2) - B^2*a*(a*c)^(3/2) - A^2*c*(a*c)^(3/2) + 12*A*B*a^2*c^2 + 4*A^2*a*c^2*(a*c)^(1/2) + 4*B^2*a^2*
c*(a*c)^(1/2))/(72*a^3*c^3))^(1/2)*2i - atan(((((12*A*a)/c^2 - (2*x*(4*c*(a*c)^(3/2) - 16*a*c^2*(a*c)^(1/2))*(
-(A^2*c*(-27*a^3*c^3)^(1/2) - B^2*a*(-27*a^3*c^3)^(1/2) - B^2*a*(a*c)^(3/2) - A^2*c*(a*c)^(3/2) + 12*A*B*a^2*c
^2 + 4*A^2*a*c^2*(a*c)^(1/2) + 4*B^2*a^2*c*(a*c)^(1/2))/(72*a^3*c^3))^(1/2))/c^4)*(-(A^2*c*(-27*a^3*c^3)^(1/2)
- B^2*a*(-27*a^3*c^3)^(1/2) - B^2*a*(a*c)^(3/2) - A^2*c*(a*c)^(3/2) + 12*A*B*a^2*c^2 + 4*A^2*a*c^2*(a*c)^(1/2
) + 4*B^2*a^2*c*(a*c)^(1/2))/(72*a^3*c^3))^(1/2) + (2*x*(2*A^2*c^2 - B^2*a*c + 2*A*B*c*(a*c)^(1/2)))/c^4)*(-(A
^2*c*(-27*a^3*c^3)^(1/2) - B^2*a*(-27*a^3*c^3)^(1/2) - B^2*a*(a*c)^(3/2) - A^2*c*(a*c)^(3/2) + 12*A*B*a^2*c^2
+ 4*A^2*a*c^2*(a*c)^(1/2) + 4*B^2*a^2*c*(a*c)^(1/2))/(72*a^3*c^3))^(1/2)*1i - (((12*A*a)/c^2 + (2*x*(4*c*(a*c)
^(3/2) - 16*a*c^2*(a*c)^(1/2))*(-(A^2*c*(-27*a^3*c^3)^(1/2) - B^2*a*(-27*a^3*c^3)^(1/2) - B^2*a*(a*c)^(3/2) -
A^2*c*(a*c)^(3/2) + 12*A*B*a^2*c^2 + 4*A^2*a*c^2*(a*c)^(1/2) + 4*B^2*a^2*c*(a*c)^(1/2))/(72*a^3*c^3))^(1/2))/c
^4)*(-(A^2*c*(-27*a^3*c^3)^(1/2) - B^2*a*(-27*a^3*c^3)^(1/2) - B^2*a*(a*c)^(3/2) - A^2*c*(a*c)^(3/2) + 12*A*B*
a^2*c^2 + 4*A^2*a*c^2*(a*c)^(1/2) + 4*B^2*a^2*c*(a*c)^(1/2))/(72*a^3*c^3))^(1/2) - (2*x*(2*A^2*c^2 - B^2*a*c +
2*A*B*c*(a*c)^(1/2)))/c^4)*(-(A^2*c*(-27*a^3*c^3)^(1/2) - B^2*a*(-27*a^3*c^3)^(1/2) - B^2*a*(a*c)^(3/2) - A^2
*c*(a*c)^(3/2) + 12*A*B*a^2*c^2 + 4*A^2*a*c^2*(a*c)^(1/2) + 4*B^2*a^2*c*(a*c)^(1/2))/(72*a^3*c^3))^(1/2)*1i)/(
(((12*A*a)/c^2 - (2*x*(4*c*(a*c)^(3/2) - 16*a*c^2*(a*c)^(1/2))*(-(A^2*c*(-27*a^3*c^3)^(1/2) - B^2*a*(-27*a^3*c
^3)^(1/2) - B^2*a*(a*c)^(3/2) - A^2*c*(a*c)^(3/2) + 12*A*B*a^2*c^2 + 4*A^2*a*c^2*(a*c)^(1/2) + 4*B^2*a^2*c*(a*
c)^(1/2))/(72*a^3*c^3))^(1/2))/c^4)*(-(A^2*c*(-27*a^3*c^3)^(1/2) - B^2*a*(-27*a^3*c^3)^(1/2) - B^2*a*(a*c)^(3/
2) - A^2*c*(a*c)^(3/2) + 12*A*B*a^2*c^2 + 4*A^2*a*c^2*(a*c)^(1/2) + 4*B^2*a^2*c*(a*c)^(1/2))/(72*a^3*c^3))^(1/
2) + (2*x*(2*A^2*c^2 - B^2*a*c + 2*A*B*c*(a*c)^(1/2)))/c^4)*(-(A^2*c*(-27*a^3*c^3)^(1/2) - B^2*a*(-27*a^3*c^3)
^(1/2) - B^2*a*(a*c)^(3/2) - A^2*c*(a*c)^(3/2) + 12*A*B*a^2*c^2 + 4*A^2*a*c^2*(a*c)^(1/2) + 4*B^2*a^2*c*(a*c)^
(1/2))/(72*a^3*c^3))^(1/2) + (((12*A*a)/c^2 + (2*x*(4*c*(a*c)^(3/2) - 16*a*c^2*(a*c)^(1/2))*(-(A^2*c*(-27*a^3*
c^3)^(1/2) - B^2*a*(-27*a^3*c^3)^(1/2) - B^2*a*(a*c)^(3/2) - A^2*c*(a*c)^(3/2) + 12*A*B*a^2*c^2 + 4*A^2*a*c^2*
(a*c)^(1/2) + 4*B^2*a^2*c*(a*c)^(1/2))/(72*a^3*c^3))^(1/2))/c^4)*(-(A^2*c*(-27*a^3*c^3)^(1/2) - B^2*a*(-27*a^3
*c^3)^(1/2) - B^2*a*(a*c)^(3/2) - A^2*c*(a*c)^(3/2) + 12*A*B*a^2*c^2 + 4*A^2*a*c^2*(a*c)^(1/2) + 4*B^2*a^2*c*(
a*c)^(1/2))/(72*a^3*c^3))^(1/2) - (2*x*(2*A^2*c^2 - B^2*a*c + 2*A*B*c*(a*c)^(1/2)))/c^4)*(-(A^2*c*(-27*a^3*c^3
)^(1/2) - B^2*a*(-27*a^3*c^3)^(1/2) - B^2*a*(a*c)^(3/2) - A^2*c*(a*c)^(3/2) + 12*A*B*a^2*c^2 + 4*A^2*a*c^2*(a*
c)^(1/2) + 4*B^2*a^2*c*(a*c)^(1/2))/(72*a^3*c^3))^(1/2) + (2*(B^3*a + A^2*B*c + A*B^2*(a*c)^(1/2)))/c^4))*(-(A
^2*c*(-27*a^3*c^3)^(1/2) - B^2*a*(-27*a^3*c^3)^(1/2) - B^2*a*(a*c)^(3/2) - A^2*c*(a*c)^(3/2) + 12*A*B*a^2*c^2
+ 4*A^2*a*c^2*(a*c)^(1/2) + 4*B^2*a^2*c*(a*c)^(1/2))/(72*a^3*c^3))^(1/2)*2i