\(\int \frac {A+B x^2}{a-\sqrt {a} \sqrt {c} x^2+c x^4} \, dx\) [112]

Optimal result

Integrand size = 32, antiderivative size = 234 \[ \int \frac {A+B x^2}{a-\sqrt {a} \sqrt {c} x^2+c x^4} \, dx=-\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \arctan \left (\sqrt {3}-\frac {2 \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} c^{3/4}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \arctan \left (\sqrt {3}+\frac {2 \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} c^{3/4}}-\frac {\left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \log \left (\sqrt {a}-\sqrt {3} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {3} a^{3/4} \sqrt [4]{c}}+\frac {\left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \log \left (\sqrt {a}+\sqrt {3} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {3} a^{3/4} \sqrt [4]{c}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {1183, 648, 631, 210, 642} \[ \int \frac {A+B x^2}{a-\sqrt {a} \sqrt {c} x^2+c x^4} \, dx=-\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \arctan \left (\sqrt {3}-\frac {2 \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} c^{3/4}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \arctan \left (\frac {2 \sqrt [4]{c} x}{\sqrt [4]{a}}+\sqrt {3}\right )}{2 a^{3/4} c^{3/4}}-\frac {\left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \log \left (-\sqrt {3} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {3} a^{3/4} \sqrt [4]{c}}+\frac {\left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \log \left (\sqrt {3} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {3} a^{3/4} \sqrt [4]{c}} \]

[In]

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

[Out]

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

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.13 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.70 \[ \int \frac {A+B x^2}{a-\sqrt {a} \sqrt {c} x^2+c x^4} \, dx=\frac {\sqrt [4]{-1} \left (\frac {\left (\left (-i+\sqrt {3}\right ) \sqrt {a} B-2 i A \sqrt {c}\right ) \arctan \left (\frac {(1+i) \sqrt [4]{c} x}{\sqrt {-i+\sqrt {3}} \sqrt [4]{a}}\right )}{\sqrt {-i+\sqrt {3}}}-\frac {\left (\left (i+\sqrt {3}\right ) \sqrt {a} B+2 i A \sqrt {c}\right ) \text {arctanh}\left (\frac {(1+i) \sqrt [4]{c} x}{\sqrt {i+\sqrt {3}} \sqrt [4]{a}}\right )}{\sqrt {i+\sqrt {3}}}\right )}{\sqrt {6} a^{3/4} c^{3/4}} \]

[In]

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

[Out]

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

Maple [C] (verified)

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

Time = 0.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.32

method	result	size
risch	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}-\textit {\_Z}^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}-a , \operatorname {index} =1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-c , \operatorname {index} =1\right )+a \right )}{\sum }\frac {\left (-B \,\textit {\_R}^{2}-A \right ) \ln \left (x -\textit {\_R} \right )}{-2 c \,\textit {\_R}^{3}+\textit {\_R} \sqrt {a}\, \sqrt {c}}\right )}{2}\)	\(75\)
default	\(\frac {\frac {\left (A \sqrt {a}\, \sqrt {3}\, c -B \sqrt {c}\, \sqrt {3}\, a \right ) \ln \left (a^{\frac {1}{4}} c^{\frac {1}{4}} x \sqrt {3}+\sqrt {a}+x^{2} \sqrt {c}\right )}{2 \sqrt {c}}+\frac {2 \left (3 A \,c^{\frac {3}{4}} a^{\frac {3}{4}}-\frac {\left (A \sqrt {a}\, \sqrt {3}\, c -B \sqrt {c}\, \sqrt {3}\, a \right ) a^{\frac {1}{4}} \sqrt {3}}{2 c^{\frac {1}{4}}}\right ) \arctan \left (\frac {a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {3}+2 x \sqrt {c}}{\sqrt {\sqrt {a}\, \sqrt {c}}}\right )}{\sqrt {\sqrt {a}\, \sqrt {c}}}}{6 c^{\frac {3}{4}} a^{\frac {5}{4}}}+\frac {\frac {\left (-A \sqrt {a}\, \sqrt {3}\, c +B \sqrt {c}\, \sqrt {3}\, a \right ) \ln \left (-a^{\frac {1}{4}} c^{\frac {1}{4}} x \sqrt {3}+\sqrt {a}+x^{2} \sqrt {c}\right )}{2 \sqrt {c}}+\frac {2 \left (3 A \,c^{\frac {3}{4}} a^{\frac {3}{4}}+\frac {\left (-A \sqrt {a}\, \sqrt {3}\, c +B \sqrt {c}\, \sqrt {3}\, a \right ) a^{\frac {1}{4}} \sqrt {3}}{2 c^{\frac {1}{4}}}\right ) \arctan \left (\frac {-a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {3}+2 x \sqrt {c}}{\sqrt {\sqrt {a}\, \sqrt {c}}}\right )}{\sqrt {\sqrt {a}\, \sqrt {c}}}}{6 c^{\frac {3}{4}} a^{\frac {5}{4}}}\)	\(278\)

[In]

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

[Out]

1/2*sum((-B*_R^2-A)/(-2*c*_R^3+_R*a^(1/2)*c^(1/2))*ln(x-_R),_R=RootOf(c*_Z^4-_Z^2*RootOf(_Z^2-a,index=1)*RootO
f(_Z^2-c,index=1)+a))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1469 vs. \(2 (160) = 320\).

Time = 0.98 (sec) , antiderivative size = 1469, normalized size of antiderivative = 6.28 \[ \int \frac {A+B x^2}{a-\sqrt {a} \sqrt {c} x^2+c x^4} \, dx=\text {Too large to display} \]

[In]

integrate((B*x^2+A)/(a+c*x^4-x^2*a^(1/2)*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)*sqrt(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
- (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)*sqrt(c) - 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)))*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)*sqrt(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)*sqrt(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 - (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)*sqrt(c) - 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)))*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)*sqrt(c))/(a^2*c^2))) - 1/2*s
qrt(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)*sqrt(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 - (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)*sqrt(c) + 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)))*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)*sqrt(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)*sqrt(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 - (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)*sqrt(c) + 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)))*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)*sqrt(c))/(a^2*c^2)))

Sympy [F(-2)]

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

[In]

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

[Out]

Exception raised: PolynomialError >> 1/(64*_t**4*a*c**5 - 16*_t**2*B**2*sqrt(a)*c**(7/2) + B**4*c**2) contains
an element of the set of generators.

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

integrate((B*x^2+A)/(a+c*x^4-x^2*a^(1/2)*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 = 14.72 (sec) , antiderivative size = 1575, normalized size of antiderivative = 6.73 \[ \int \frac {A+B x^2}{a-\sqrt {a} \sqrt {c} x^2+c x^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

- 2*atanh((6*A^2*x*((B^2*(-27*a^3*c^3)^(1/2))/(72*a^2*c^3) - B^2/(24*a^(1/2)*c^(3/2)) - (A*B)/(6*a*c) - (A^2*(
-27*a^3*c^3)^(1/2))/(72*a^3*c^2) - A^2/(24*a^(3/2)*c^(1/2)))^(1/2))/((2*A^2*B)/c - (2*B^3*a)/c^2 + A^3/(a^(1/2
)*c^(1/2)) + (A^3*(-27*a^3*c^3)^(1/2))/(3*a^2*c^2) - (A*B^2*a^(1/2))/c^(3/2) - (A*B^2*(-27*a^3*c^3)^(1/2))/(3*
a*c^3)) - (6*B^2*a*x*((B^2*(-27*a^3*c^3)^(1/2))/(72*a^2*c^3) - B^2/(24*a^(1/2)*c^(3/2)) - (A*B)/(6*a*c) - (A^2
*(-27*a^3*c^3)^(1/2))/(72*a^3*c^2) - A^2/(24*a^(3/2)*c^(1/2)))^(1/2))/(2*A^2*B - (2*B^3*a)/c + (A^3*c^(1/2))/a
^(1/2) + (A^3*(-27*a^3*c^3)^(1/2))/(3*a^2*c) - (A*B^2*a^(1/2))/c^(1/2) - (A*B^2*(-27*a^3*c^3)^(1/2))/(3*a*c^2)
) - (2*A^2*x*(-27*a^3*c^3)^(1/2)*((B^2*(-27*a^3*c^3)^(1/2))/(72*a^2*c^3) - B^2/(24*a^(1/2)*c^(3/2)) - (A*B)/(6
*a*c) - (A^2*(-27*a^3*c^3)^(1/2))/(72*a^3*c^2) - A^2/(24*a^(3/2)*c^(1/2)))^(1/2))/(3*a^(3/2)*c^(7/2)*((2*A^2*B
)/c^3 - (2*B^3*a)/c^4 + A^3/(a^(1/2)*c^(5/2)) + (A^3*(-27*a^3*c^3)^(1/2))/(3*a^2*c^4) - (A*B^2*a^(1/2))/c^(7/2
) - (A*B^2*(-27*a^3*c^3)^(1/2))/(3*a*c^5))) + (2*B^2*x*(-27*a^3*c^3)^(1/2)*((B^2*(-27*a^3*c^3)^(1/2))/(72*a^2*
c^3) - B^2/(24*a^(1/2)*c^(3/2)) - (A*B)/(6*a*c) - (A^2*(-27*a^3*c^3)^(1/2))/(72*a^3*c^2) - A^2/(24*a^(3/2)*c^(
1/2)))^(1/2))/(3*a^(1/2)*c^(9/2)*((2*A^2*B)/c^3 - (2*B^3*a)/c^4 + A^3/(a^(1/2)*c^(5/2)) + (A^3*(-27*a^3*c^3)^(
1/2))/(3*a^2*c^4) - (A*B^2*a^(1/2))/c^(7/2) - (A*B^2*(-27*a^3*c^3)^(1/2))/(3*a*c^5))))*((B^2*(-27*a^3*c^3)^(1/
2))/(72*a^2*c^3) - B^2/(24*a^(1/2)*c^(3/2)) - (A*B)/(6*a*c) - (A^2*(-27*a^3*c^3)^(1/2))/(72*a^3*c^2) - A^2/(24
*a^(3/2)*c^(1/2)))^(1/2) - 2*atanh((6*A^2*x*((A^2*(-27*a^3*c^3)^(1/2))/(72*a^3*c^2) - B^2/(24*a^(1/2)*c^(3/2))
- (A*B)/(6*a*c) - A^2/(24*a^(3/2)*c^(1/2)) - (B^2*(-27*a^3*c^3)^(1/2))/(72*a^2*c^3))^(1/2))/((2*A^2*B)/c - (2
*B^3*a)/c^2 + A^3/(a^(1/2)*c^(1/2)) - (A^3*(-27*a^3*c^3)^(1/2))/(3*a^2*c^2) - (A*B^2*a^(1/2))/c^(3/2) + (A*B^2
*(-27*a^3*c^3)^(1/2))/(3*a*c^3)) - (6*B^2*a*x*((A^2*(-27*a^3*c^3)^(1/2))/(72*a^3*c^2) - B^2/(24*a^(1/2)*c^(3/2
)) - (A*B)/(6*a*c) - A^2/(24*a^(3/2)*c^(1/2)) - (B^2*(-27*a^3*c^3)^(1/2))/(72*a^2*c^3))^(1/2))/(2*A^2*B - (2*B
^3*a)/c + (A^3*c^(1/2))/a^(1/2) - (A^3*(-27*a^3*c^3)^(1/2))/(3*a^2*c) - (A*B^2*a^(1/2))/c^(1/2) + (A*B^2*(-27*
a^3*c^3)^(1/2))/(3*a*c^2)) + (2*A^2*x*(-27*a^3*c^3)^(1/2)*((A^2*(-27*a^3*c^3)^(1/2))/(72*a^3*c^2) - B^2/(24*a^
(1/2)*c^(3/2)) - (A*B)/(6*a*c) - A^2/(24*a^(3/2)*c^(1/2)) - (B^2*(-27*a^3*c^3)^(1/2))/(72*a^2*c^3))^(1/2))/(3*
a^(3/2)*c^(7/2)*((2*A^2*B)/c^3 - (2*B^3*a)/c^4 + A^3/(a^(1/2)*c^(5/2)) - (A^3*(-27*a^3*c^3)^(1/2))/(3*a^2*c^4)
- (A*B^2*a^(1/2))/c^(7/2) + (A*B^2*(-27*a^3*c^3)^(1/2))/(3*a*c^5))) - (2*B^2*x*(-27*a^3*c^3)^(1/2)*((A^2*(-27
*a^3*c^3)^(1/2))/(72*a^3*c^2) - B^2/(24*a^(1/2)*c^(3/2)) - (A*B)/(6*a*c) - A^2/(24*a^(3/2)*c^(1/2)) - (B^2*(-2
7*a^3*c^3)^(1/2))/(72*a^2*c^3))^(1/2))/(3*a^(1/2)*c^(9/2)*((2*A^2*B)/c^3 - (2*B^3*a)/c^4 + A^3/(a^(1/2)*c^(5/2
)) - (A^3*(-27*a^3*c^3)^(1/2))/(3*a^2*c^4) - (A*B^2*a^(1/2))/c^(7/2) + (A*B^2*(-27*a^3*c^3)^(1/2))/(3*a*c^5)))
)*((A^2*(-27*a^3*c^3)^(1/2))/(72*a^3*c^2) - B^2/(24*a^(1/2)*c^(3/2)) - (A*B)/(6*a*c) - A^2/(24*a^(3/2)*c^(1/2)
) - (B^2*(-27*a^3*c^3)^(1/2))/(72*a^2*c^3))^(1/2)