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

Optimal. Leaf size=414 \[ -\frac {\left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \log \left (-x \sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {a} \sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}}+\frac {\left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \log \left (x \sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {a} \sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}}-\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}-2 \sqrt {c} x}{\sqrt {2 \sqrt {a} \sqrt {c}-\sqrt {a c}}}\right )}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-\sqrt {a c}}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}+2 \sqrt {c} x}{\sqrt {2 \sqrt {a} \sqrt {c}-\sqrt {a c}}}\right )}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-\sqrt {a c}}} \]

[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]  time = 0.45, antiderivative size = 414, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1169, 634, 618, 204, 628} \[ -\frac {\left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \log \left (-x \sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {a} \sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}}+\frac {\left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \log \left (x \sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {a} \sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}}-\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}-2 \sqrt {c} x}{\sqrt {2 \sqrt {a} \sqrt {c}-\sqrt {a c}}}\right )}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-\sqrt {a c}}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {2 \sqrt {a} \sqrt {c}+\sqrt {a c}}+2 \sqrt {c} x}{\sqrt {2 \sqrt {a} \sqrt {c}-\sqrt {a c}}}\right )}{2 \sqrt {a} \sqrt {c} \sqrt {2 \sqrt {a} \sqrt {c}-\sqrt {a c}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((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]]) + ((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*S
qrt[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] + Sqr
t[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 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 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]

Rubi steps

\begin {align*} \int \frac {A+B x^2}{a-\sqrt {a c} x^2+c x^4} \, dx &=\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 ) \operatorname {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 ) \operatorname {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]  time = 0.20, size = 247, normalized size = 0.60 \[ \frac {\frac {\left (\sqrt {3} \sqrt {a} B \sqrt {c}-i \left (B \sqrt {a c}+2 A c\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {-\sqrt {a c}-i \sqrt {3} \sqrt {a} \sqrt {c}}}\right )}{\sqrt {-\sqrt {a c}-i \sqrt {3} \sqrt {a} \sqrt {c}}}+\frac {\left (\sqrt {3} \sqrt {a} B \sqrt {c}+i \left (B \sqrt {a c}+2 A c\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {-\sqrt {a c}+i \sqrt {3} \sqrt {a} \sqrt {c}}}\right )}{\sqrt {-\sqrt {a c}+i \sqrt {3} \sqrt {a} \sqrt {c}}}}{\sqrt {6} \sqrt {a} c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.66, size = 1457, normalized size = 3.52 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)))

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[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,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

________________________________________________________________________________________

maple [A]  time = 0.06, size = 404, normalized size = 0.98 \[ -\frac {A \arctan \left (\frac {-2 \sqrt {c}\, x +\sqrt {3}\, \left (a c \right )^{\frac {1}{4}}}{\sqrt {4 \sqrt {a}\, \sqrt {c}-3 \sqrt {a c}}}\right )}{2 \sqrt {4 \sqrt {a}\, \sqrt {c}-3 \sqrt {a c}}\, \sqrt {a}}+\frac {A \arctan \left (\frac {2 \sqrt {c}\, x +\sqrt {3}\, \left (a c \right )^{\frac {1}{4}}}{\sqrt {4 \sqrt {a}\, \sqrt {c}-3 \sqrt {a c}}}\right )}{2 \sqrt {4 \sqrt {a}\, \sqrt {c}-3 \sqrt {a c}}\, \sqrt {a}}-\frac {B \arctan \left (\frac {-2 \sqrt {c}\, x +\sqrt {3}\, \left (a c \right )^{\frac {1}{4}}}{\sqrt {4 \sqrt {a}\, \sqrt {c}-3 \sqrt {a c}}}\right )}{2 \sqrt {4 \sqrt {a}\, \sqrt {c}-3 \sqrt {a c}}\, \sqrt {c}}+\frac {B \arctan \left (\frac {2 \sqrt {c}\, x +\sqrt {3}\, \left (a c \right )^{\frac {1}{4}}}{\sqrt {4 \sqrt {a}\, \sqrt {c}-3 \sqrt {a c}}}\right )}{2 \sqrt {4 \sqrt {a}\, \sqrt {c}-3 \sqrt {a c}}\, \sqrt {c}}+\frac {\sqrt {3}\, \left (a c \right )^{\frac {3}{4}} A \ln \left (\sqrt {c}\, x^{2}+\sqrt {3}\, \left (a c \right )^{\frac {1}{4}} x +\sqrt {a}\right )}{12 a^{\frac {3}{2}} c}-\frac {\sqrt {3}\, \left (a c \right )^{\frac {3}{4}} A \ln \left (-\sqrt {c}\, x^{2}+\sqrt {3}\, \left (a c \right )^{\frac {1}{4}} x -\sqrt {a}\right )}{12 a^{\frac {3}{2}} c}-\frac {\sqrt {3}\, \left (a c \right )^{\frac {3}{4}} B \ln \left (\sqrt {c}\, x^{2}+\sqrt {3}\, \left (a c \right )^{\frac {1}{4}} x +\sqrt {a}\right )}{12 a \,c^{\frac {3}{2}}}+\frac {\sqrt {3}\, \left (a c \right )^{\frac {3}{4}} B \ln \left (-\sqrt {c}\, x^{2}+\sqrt {3}\, \left (a c \right )^{\frac {1}{4}} x -\sqrt {a}\right )}{12 a \,c^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12/a/c^(3/2)*ln(x*3^(1/2)*(a*c)^(1/4)-c^(1/2)*x^2-a^(1/2))*B*3^(1/2)*(a*c)^(3/4)-1/12/a^(3/2)/c*ln(x*3^(1/2)
*(a*c)^(1/4)-c^(1/2)*x^2-a^(1/2))*A*3^(1/2)*(a*c)^(3/4)-1/2/a^(1/2)/(4*a^(1/2)*c^(1/2)-3*(a*c)^(1/2))^(1/2)*ar
ctan((3^(1/2)*(a*c)^(1/4)-2*c^(1/2)*x)/(4*a^(1/2)*c^(1/2)-3*(a*c)^(1/2))^(1/2))*A-1/2/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*c^(1/2)*x)/(4*a^(1/2)*c^(1/2)-3*(a*c)^(1/2))^(1/2))*B-1/
12/a/c^(3/2)*ln(c^(1/2)*x^2+x*3^(1/2)*(a*c)^(1/4)+a^(1/2))*B*3^(1/2)*(a*c)^(3/4)+1/12/a^(3/2)/c*ln(c^(1/2)*x^2
+x*3^(1/2)*(a*c)^(1/4)+a^(1/2))*A*3^(1/2)*(a*c)^(3/4)+1/2/a^(1/2)/(4*a^(1/2)*c^(1/2)-3*(a*c)^(1/2))^(1/2)*arct
an((2*c^(1/2)*x+3^(1/2)*(a*c)^(1/4))/(4*a^(1/2)*c^(1/2)-3*(a*c)^(1/2))^(1/2))*A+1/2/c^(1/2)/(4*a^(1/2)*c^(1/2)
-3*(a*c)^(1/2))^(1/2)*arctan((2*c^(1/2)*x+3^(1/2)*(a*c)^(1/4))/(4*a^(1/2)*c^(1/2)-3*(a*c)^(1/2))^(1/2))*B

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {B x^{2} + A}{c x^{4} - \sqrt {a c} x^{2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [B]  time = 5.22, size = 3285, normalized size = 7.93 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: PolynomialError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: PolynomialError

________________________________________________________________________________________