\(\int \frac {A+B x^2}{\sqrt {-d-e x^2} \sqrt {d e+(d^2+e^2) x^2+d e x^4}} \, dx\) [141]

Optimal result

Integrand size = 47, antiderivative size = 149 \[ \int \frac {A+B x^2}{\sqrt {-d-e x^2} \sqrt {d e+\left (d^2+e^2\right ) x^2+d e x^4}} \, dx=-\frac {B \arctan \left (\frac {\sqrt {d} x \sqrt {-d-e x^2}}{\sqrt {d e+\left (d^2+e^2\right ) x^2+d e x^4}}\right )}{\sqrt {d} e}+\frac {(B d-A e) \arctan \left (\frac {\sqrt {d^2-e^2} x \sqrt {-d-e x^2}}{\sqrt {d} \sqrt {d e+\left (d^2+e^2\right ) x^2+d e x^4}}\right )}{\sqrt {d} e \sqrt {d^2-e^2}} \] Output:

-B*arctan(d^(1/2)*x*(-e*x^2-d)^(1/2)/(d*e+(d^2+e^2)*x^2+d*e*x^4)^(1/2))/d^ 
(1/2)/e+(-A*e+B*d)*arctan((d^2-e^2)^(1/2)*x*(-e*x^2-d)^(1/2)/d^(1/2)/(d*e+ 
(d^2+e^2)*x^2+d*e*x^4)^(1/2))/d^(1/2)/e/(d^2-e^2)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 11.71 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.22 \[ \int \frac {A+B x^2}{\sqrt {-d-e x^2} \sqrt {d e+\left (d^2+e^2\right ) x^2+d e x^4}} \, dx=\frac {\frac {(B d-A e) \sqrt {-e-d x^2} \sqrt {-d-e x^2} \arctan \left (\frac {\sqrt {d^2-e^2} x}{\sqrt {d} \sqrt {-e-d x^2}}\right )}{\sqrt {d^2-e^2} \sqrt {d^2 x^2+e^2 x^2+d e \left (1+x^4\right )}}+i B \log \left (-2 i \sqrt {d} x-\frac {2 \sqrt {d^2 x^2+e^2 x^2+d e \left (1+x^4\right )}}{\sqrt {-d-e x^2}}\right )}{\sqrt {d} e} \] Input:

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

Output:

(((B*d - A*e)*Sqrt[-e - d*x^2]*Sqrt[-d - e*x^2]*ArcTan[(Sqrt[d^2 - e^2]*x) 
/(Sqrt[d]*Sqrt[-e - d*x^2])])/(Sqrt[d^2 - e^2]*Sqrt[d^2*x^2 + e^2*x^2 + d* 
e*(1 + x^4)]) + I*B*Log[(-2*I)*Sqrt[d]*x - (2*Sqrt[d^2*x^2 + e^2*x^2 + d*e 
*(1 + x^4)])/Sqrt[-d - e*x^2]])/(Sqrt[d]*e)
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.149, Rules used = {1395, 25, 398, 224, 216, 291, 218}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {A+B x^2}{\sqrt {-d-e x^2} \sqrt {x^2 \left (d^2+e^2\right )+d e x^4+d e}} \, dx\)

\(\Big \downarrow \) 1395

\(\displaystyle \frac {\sqrt {-d x^2-e} \sqrt {-d-e x^2} \int -\frac {B x^2+A}{\sqrt {-d x^2-e} \left (e x^2+d\right )}dx}{\sqrt {x^2 \left (d^2+e^2\right )+d e x^4+d e}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\sqrt {-d x^2-e} \sqrt {-d-e x^2} \int \frac {B x^2+A}{\sqrt {-d x^2-e} \left (e x^2+d\right )}dx}{\sqrt {x^2 \left (d^2+e^2\right )+d e x^4+d e}}\)

\(\Big \downarrow \) 398

\(\displaystyle -\frac {\sqrt {-d x^2-e} \sqrt {-d-e x^2} \left (\frac {B \int \frac {1}{\sqrt {-d x^2-e}}dx}{e}-\frac {(B d-A e) \int \frac {1}{\sqrt {-d x^2-e} \left (e x^2+d\right )}dx}{e}\right )}{\sqrt {x^2 \left (d^2+e^2\right )+d e x^4+d e}}\)

\(\Big \downarrow \) 224

\(\displaystyle -\frac {\sqrt {-d x^2-e} \sqrt {-d-e x^2} \left (\frac {B \int \frac {1}{\frac {d x^2}{-d x^2-e}+1}d\frac {x}{\sqrt {-d x^2-e}}}{e}-\frac {(B d-A e) \int \frac {1}{\sqrt {-d x^2-e} \left (e x^2+d\right )}dx}{e}\right )}{\sqrt {x^2 \left (d^2+e^2\right )+d e x^4+d e}}\)

\(\Big \downarrow \) 216

\(\displaystyle -\frac {\sqrt {-d x^2-e} \sqrt {-d-e x^2} \left (\frac {B \arctan \left (\frac {\sqrt {d} x}{\sqrt {-d x^2-e}}\right )}{\sqrt {d} e}-\frac {(B d-A e) \int \frac {1}{\sqrt {-d x^2-e} \left (e x^2+d\right )}dx}{e}\right )}{\sqrt {x^2 \left (d^2+e^2\right )+d e x^4+d e}}\)

\(\Big \downarrow \) 291

\(\displaystyle -\frac {\sqrt {-d x^2-e} \sqrt {-d-e x^2} \left (\frac {B \arctan \left (\frac {\sqrt {d} x}{\sqrt {-d x^2-e}}\right )}{\sqrt {d} e}-\frac {(B d-A e) \int \frac {1}{d-\frac {\left (e^2-d^2\right ) x^2}{-d x^2-e}}d\frac {x}{\sqrt {-d x^2-e}}}{e}\right )}{\sqrt {x^2 \left (d^2+e^2\right )+d e x^4+d e}}\)

\(\Big \downarrow \) 218

\(\displaystyle -\frac {\sqrt {-d x^2-e} \sqrt {-d-e x^2} \left (\frac {B \arctan \left (\frac {\sqrt {d} x}{\sqrt {-d x^2-e}}\right )}{\sqrt {d} e}-\frac {(B d-A e) \arctan \left (\frac {x \sqrt {d^2-e^2}}{\sqrt {d} \sqrt {-d x^2-e}}\right )}{\sqrt {d} e \sqrt {d^2-e^2}}\right )}{\sqrt {x^2 \left (d^2+e^2\right )+d e x^4+d e}}\)

Input:

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

Output:

-((Sqrt[-e - d*x^2]*Sqrt[-d - e*x^2]*((B*ArcTan[(Sqrt[d]*x)/Sqrt[-e - d*x^ 
2]])/(Sqrt[d]*e) - ((B*d - A*e)*ArcTan[(Sqrt[d^2 - e^2]*x)/(Sqrt[d]*Sqrt[- 
e - d*x^2])])/(Sqrt[d]*e*Sqrt[d^2 - e^2])))/Sqrt[d*e + (d^2 + e^2)*x^2 + d 
*e*x^4])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 291

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst 
[Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, 
d}, x] && NeQ[b*c - a*d, 0]
 

rule 398

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) 
, x_Symbol] :> Simp[f/b   Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ 
b   Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} 
, x]
 

rule 1395

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_)*((d_) + (e_.)*( 
x_)^(n_))^(q_.), x_Symbol] :> Simp[(a + b*x^n + c*x^(2*n))^FracPart[p]/((d 
+ e*x^n)^FracPart[p]*(a/d + c*(x^n/e))^FracPart[p])   Int[u*(d + e*x^n)^(p 
+ q)*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && E 
qQ[n2, 2*n] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] &&  !(EqQ[q, 
1] && EqQ[n, 2])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(836\) vs. \(2(129)=258\).

Time = 0.09 (sec) , antiderivative size = 837, normalized size of antiderivative = 5.62

method	result	size
default	\(\frac {\sqrt {-e \,x^{2}-d}\, \sqrt {d e \,x^{4}+d^{2} x^{2}+e^{2} x^{2}+d e}\, \sqrt {d}\, \left (2 A \sqrt {-d e}\, \arctan \left (\frac {\sqrt {d}\, x}{\sqrt {-d \,x^{2}-e}}\right ) d \sqrt {\frac {d^{2}-e^{2}}{e}}\, e -2 B \sqrt {-d e}\, \arctan \left (\frac {\sqrt {d}\, x}{\sqrt {-d \,x^{2}-e}}\right ) d^{2} \sqrt {\frac {d^{2}-e^{2}}{e}}-2 A \sqrt {-d e}\, \arctan \left (\frac {\sqrt {d}\, x}{\sqrt {\frac {\left (-d x +\sqrt {-d e}\right ) \left (d x +\sqrt {-d e}\right )}{d}}}\right ) \sqrt {\frac {d^{2}-e^{2}}{e}}\, d e +A \ln \left (\frac {2 \sqrt {-d e}\, d x -2 \sqrt {-d \,x^{2}-e}\, \sqrt {\frac {d^{2}-e^{2}}{e}}\, e +2 e^{2}}{-e x +\sqrt {-d e}}\right ) d^{\frac {5}{2}} e -A \ln \left (\frac {2 \sqrt {-d e}\, d x -2 \sqrt {-d \,x^{2}-e}\, \sqrt {\frac {d^{2}-e^{2}}{e}}\, e +2 e^{2}}{-e x +\sqrt {-d e}}\right ) \sqrt {d}\, e^{3}-A \ln \left (\frac {2 \sqrt {-d e}\, d x +2 \sqrt {-d \,x^{2}-e}\, \sqrt {\frac {d^{2}-e^{2}}{e}}\, e -2 e^{2}}{e x +\sqrt {-d e}}\right ) d^{\frac {5}{2}} e +A \ln \left (\frac {2 \sqrt {-d e}\, d x +2 \sqrt {-d \,x^{2}-e}\, \sqrt {\frac {d^{2}-e^{2}}{e}}\, e -2 e^{2}}{e x +\sqrt {-d e}}\right ) \sqrt {d}\, e^{3}+2 B \sqrt {-d e}\, \arctan \left (\frac {\sqrt {d}\, x}{\sqrt {\frac {\left (-d x +\sqrt {-d e}\right ) \left (d x +\sqrt {-d e}\right )}{d}}}\right ) \sqrt {\frac {d^{2}-e^{2}}{e}}\, e^{2}-B \ln \left (\frac {2 \sqrt {-d e}\, d x -2 \sqrt {-d \,x^{2}-e}\, \sqrt {\frac {d^{2}-e^{2}}{e}}\, e +2 e^{2}}{-e x +\sqrt {-d e}}\right ) d^{\frac {7}{2}}+B \ln \left (\frac {2 \sqrt {-d e}\, d x -2 \sqrt {-d \,x^{2}-e}\, \sqrt {\frac {d^{2}-e^{2}}{e}}\, e +2 e^{2}}{-e x +\sqrt {-d e}}\right ) d^{\frac {3}{2}} e^{2}+B \ln \left (\frac {2 \sqrt {-d e}\, d x +2 \sqrt {-d \,x^{2}-e}\, \sqrt {\frac {d^{2}-e^{2}}{e}}\, e -2 e^{2}}{e x +\sqrt {-d e}}\right ) d^{\frac {7}{2}}-B \ln \left (\frac {2 \sqrt {-d e}\, d x +2 \sqrt {-d \,x^{2}-e}\, \sqrt {\frac {d^{2}-e^{2}}{e}}\, e -2 e^{2}}{e x +\sqrt {-d e}}\right ) d^{\frac {3}{2}} e^{2}\right )}{2 \left (e \,x^{2}+d \right ) \sqrt {-d \,x^{2}-e}\, \left (-d e \right )^{\frac {3}{2}} \left (d +e \right ) \left (d -e \right ) \sqrt {\frac {d^{2}-e^{2}}{e}}}\)	\(837\)

Input:

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

Output:

1/2*(-e*x^2-d)^(1/2)*(d*e*x^4+d^2*x^2+e^2*x^2+d*e)^(1/2)*d^(1/2)*(2*A*(-d* 
e)^(1/2)*arctan(d^(1/2)*x/(-d*x^2-e)^(1/2))*d*((d^2-e^2)/e)^(1/2)*e-2*B*(- 
d*e)^(1/2)*arctan(d^(1/2)*x/(-d*x^2-e)^(1/2))*d^2*((d^2-e^2)/e)^(1/2)-2*A* 
(-d*e)^(1/2)*arctan(d^(1/2)*x/(1/d*(-d*x+(-d*e)^(1/2))*(d*x+(-d*e)^(1/2))) 
^(1/2))*((d^2-e^2)/e)^(1/2)*d*e+A*ln(2*((-d*e)^(1/2)*d*x-(-d*x^2-e)^(1/2)* 
((d^2-e^2)/e)^(1/2)*e+e^2)/(-e*x+(-d*e)^(1/2)))*d^(5/2)*e-A*ln(2*((-d*e)^( 
1/2)*d*x-(-d*x^2-e)^(1/2)*((d^2-e^2)/e)^(1/2)*e+e^2)/(-e*x+(-d*e)^(1/2)))* 
d^(1/2)*e^3-A*ln(2*((-d*e)^(1/2)*d*x+(-d*x^2-e)^(1/2)*((d^2-e^2)/e)^(1/2)* 
e-e^2)/(e*x+(-d*e)^(1/2)))*d^(5/2)*e+A*ln(2*((-d*e)^(1/2)*d*x+(-d*x^2-e)^( 
1/2)*((d^2-e^2)/e)^(1/2)*e-e^2)/(e*x+(-d*e)^(1/2)))*d^(1/2)*e^3+2*B*(-d*e) 
^(1/2)*arctan(d^(1/2)*x/(1/d*(-d*x+(-d*e)^(1/2))*(d*x+(-d*e)^(1/2)))^(1/2) 
)*((d^2-e^2)/e)^(1/2)*e^2-B*ln(2*((-d*e)^(1/2)*d*x-(-d*x^2-e)^(1/2)*((d^2- 
e^2)/e)^(1/2)*e+e^2)/(-e*x+(-d*e)^(1/2)))*d^(7/2)+B*ln(2*((-d*e)^(1/2)*d*x 
-(-d*x^2-e)^(1/2)*((d^2-e^2)/e)^(1/2)*e+e^2)/(-e*x+(-d*e)^(1/2)))*d^(3/2)* 
e^2+B*ln(2*((-d*e)^(1/2)*d*x+(-d*x^2-e)^(1/2)*((d^2-e^2)/e)^(1/2)*e-e^2)/( 
e*x+(-d*e)^(1/2)))*d^(7/2)-B*ln(2*((-d*e)^(1/2)*d*x+(-d*x^2-e)^(1/2)*((d^2 
-e^2)/e)^(1/2)*e-e^2)/(e*x+(-d*e)^(1/2)))*d^(3/2)*e^2)/(e*x^2+d)/(-d*x^2-e 
)^(1/2)/(-d*e)^(3/2)/(d+e)/(d-e)/((d^2-e^2)/e)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 830, normalized size of antiderivative = 5.57 \[ \int \frac {A+B x^2}{\sqrt {-d-e x^2} \sqrt {d e+\left (d^2+e^2\right ) x^2+d e x^4}} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/2*(sqrt(-d^3 + d*e^2)*(B*d - A*e)*log((2*d^3*x^2 + (2*d^2*e - e^3)*x^4 
+ d^2*e - 2*sqrt(d*e*x^4 + (d^2 + e^2)*x^2 + d*e)*sqrt(-d^3 + d*e^2)*sqrt( 
-e*x^2 - d)*x)/(e^2*x^4 + 2*d*e*x^2 + d^2)) - (B*d^2 - B*e^2)*sqrt(-d)*log 
((2*d*e*x^4 + (2*d^2 + e^2)*x^2 - 2*sqrt(d*e*x^4 + (d^2 + e^2)*x^2 + d*e)* 
sqrt(-e*x^2 - d)*sqrt(-d)*x + d*e)/(e*x^2 + d)))/(d^3*e - d*e^3), -1/2*(2* 
(B*d^2 - B*e^2)*sqrt(d)*arctan(sqrt(-e*x^2 - d)*sqrt(d)*x/sqrt(d*e*x^4 + ( 
d^2 + e^2)*x^2 + d*e)) - sqrt(-d^3 + d*e^2)*(B*d - A*e)*log((2*d^3*x^2 + ( 
2*d^2*e - e^3)*x^4 + d^2*e - 2*sqrt(d*e*x^4 + (d^2 + e^2)*x^2 + d*e)*sqrt( 
-d^3 + d*e^2)*sqrt(-e*x^2 - d)*x)/(e^2*x^4 + 2*d*e*x^2 + d^2)))/(d^3*e - d 
*e^3), 1/2*(2*sqrt(d^3 - d*e^2)*(B*d - A*e)*arctan(sqrt(d*e*x^4 + (d^2 + e 
^2)*x^2 + d*e)*sqrt(d^3 - d*e^2)*sqrt(-e*x^2 - d)*x/(d^2*e*x^4 + d^2*e + ( 
d^3 + d*e^2)*x^2)) - (B*d^2 - B*e^2)*sqrt(-d)*log((2*d*e*x^4 + (2*d^2 + e^ 
2)*x^2 - 2*sqrt(d*e*x^4 + (d^2 + e^2)*x^2 + d*e)*sqrt(-e*x^2 - d)*sqrt(-d) 
*x + d*e)/(e*x^2 + d)))/(d^3*e - d*e^3), (sqrt(d^3 - d*e^2)*(B*d - A*e)*ar 
ctan(sqrt(d*e*x^4 + (d^2 + e^2)*x^2 + d*e)*sqrt(d^3 - d*e^2)*sqrt(-e*x^2 - 
 d)*x/(d^2*e*x^4 + d^2*e + (d^3 + d*e^2)*x^2)) - (B*d^2 - B*e^2)*sqrt(d)*a 
rctan(sqrt(-e*x^2 - d)*sqrt(d)*x/sqrt(d*e*x^4 + (d^2 + e^2)*x^2 + d*e)))/( 
d^3*e - d*e^3)]
 

Sympy [F]

\[ \int \frac {A+B x^2}{\sqrt {-d-e x^2} \sqrt {d e+\left (d^2+e^2\right ) x^2+d e x^4}} \, dx=\int \frac {A + B x^{2}}{\sqrt {\left (d + e x^{2}\right ) \left (d x^{2} + e\right )} \sqrt {- d - e x^{2}}}\, dx \] Input:

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

Output:

Integral((A + B*x**2)/(sqrt((d + e*x**2)*(d*x**2 + e))*sqrt(-d - e*x**2)), 
 x)
 

Maxima [F]

\[ \int \frac {A+B x^2}{\sqrt {-d-e x^2} \sqrt {d e+\left (d^2+e^2\right ) x^2+d e x^4}} \, dx=\int { \frac {B x^{2} + A}{\sqrt {d e x^{4} + {\left (d^{2} + e^{2}\right )} x^{2} + d e} \sqrt {-e x^{2} - d}} \,d x } \] Input:

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

Output:

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

Giac [F(-1)]

Timed out. \[ \int \frac {A+B x^2}{\sqrt {-d-e x^2} \sqrt {d e+\left (d^2+e^2\right ) x^2+d e x^4}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2}{\sqrt {-d-e x^2} \sqrt {d e+\left (d^2+e^2\right ) x^2+d e x^4}} \, dx=\int \frac {B\,x^2+A}{\sqrt {-e\,x^2-d}\,\sqrt {d\,e\,x^4+\left (d^2+e^2\right )\,x^2+d\,e}} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 710, normalized size of antiderivative = 4.77 \[ \int \frac {A+B x^2}{\sqrt {-d-e x^2} \sqrt {d e+\left (d^2+e^2\right ) x^2+d e x^4}} \, dx=\frac {\sqrt {d}\, i \left (\sqrt {d^{2}-e^{2}}\, \mathrm {log}\left (2 \sqrt {d^{2}-e^{2}}\, d +2 \sqrt {d}\, \sqrt {d \,x^{2}+e}\, e x +2 d^{2}+2 d e \,x^{2}\right ) a e -\sqrt {d^{2}-e^{2}}\, \mathrm {log}\left (2 \sqrt {d^{2}-e^{2}}\, d +2 \sqrt {d}\, \sqrt {d \,x^{2}+e}\, e x +2 d^{2}+2 d e \,x^{2}\right ) b d -\sqrt {d^{2}-e^{2}}\, \mathrm {log}\left (\frac {-\sqrt {e}\, \sqrt {-d^{2}+e^{2}}\, \sqrt {d^{2}-e^{2}}\, d +\sqrt {e}\, \sqrt {-d^{2}+e^{2}}\, d^{2}-\sqrt {e}\, \sqrt {-d^{2}+e^{2}}\, e^{2}+\sqrt {d \,x^{2}+e}\, d^{2} e -\sqrt {d \,x^{2}+e}\, e^{3}+\sqrt {d}\, d^{2} e x -\sqrt {d}\, e^{3} x}{\sqrt {e}\, d^{2}-\sqrt {e}\, e^{2}}\right ) a e +\sqrt {d^{2}-e^{2}}\, \mathrm {log}\left (\frac {-\sqrt {e}\, \sqrt {-d^{2}+e^{2}}\, \sqrt {d^{2}-e^{2}}\, d +\sqrt {e}\, \sqrt {-d^{2}+e^{2}}\, d^{2}-\sqrt {e}\, \sqrt {-d^{2}+e^{2}}\, e^{2}+\sqrt {d \,x^{2}+e}\, d^{2} e -\sqrt {d \,x^{2}+e}\, e^{3}+\sqrt {d}\, d^{2} e x -\sqrt {d}\, e^{3} x}{\sqrt {e}\, d^{2}-\sqrt {e}\, e^{2}}\right ) b d -\sqrt {d^{2}-e^{2}}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {-d^{2}+e^{2}}\, \sqrt {d^{2}-e^{2}}\, d -\sqrt {e}\, \sqrt {-d^{2}+e^{2}}\, d^{2}+\sqrt {e}\, \sqrt {-d^{2}+e^{2}}\, e^{2}+\sqrt {d \,x^{2}+e}\, d^{2} e -\sqrt {d \,x^{2}+e}\, e^{3}+\sqrt {d}\, d^{2} e x -\sqrt {d}\, e^{3} x}{\sqrt {e}\, d^{2}-\sqrt {e}\, e^{2}}\right ) a e +\sqrt {d^{2}-e^{2}}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {-d^{2}+e^{2}}\, \sqrt {d^{2}-e^{2}}\, d -\sqrt {e}\, \sqrt {-d^{2}+e^{2}}\, d^{2}+\sqrt {e}\, \sqrt {-d^{2}+e^{2}}\, e^{2}+\sqrt {d \,x^{2}+e}\, d^{2} e -\sqrt {d \,x^{2}+e}\, e^{3}+\sqrt {d}\, d^{2} e x -\sqrt {d}\, e^{3} x}{\sqrt {e}\, d^{2}-\sqrt {e}\, e^{2}}\right ) b d -2 \,\mathrm {log}\left (\frac {\sqrt {d \,x^{2}+e}+\sqrt {d}\, x}{\sqrt {e}}\right ) b \,d^{2}+2 \,\mathrm {log}\left (\frac {\sqrt {d \,x^{2}+e}+\sqrt {d}\, x}{\sqrt {e}}\right ) b \,e^{2}\right )}{2 d e \left (d^{2}-e^{2}\right )} \] Input:

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

Output:

(sqrt(d)*i*(sqrt(d**2 - e**2)*log(2*sqrt(d**2 - e**2)*d + 2*sqrt(d)*sqrt(d 
*x**2 + e)*e*x + 2*d**2 + 2*d*e*x**2)*a*e - sqrt(d**2 - e**2)*log(2*sqrt(d 
**2 - e**2)*d + 2*sqrt(d)*sqrt(d*x**2 + e)*e*x + 2*d**2 + 2*d*e*x**2)*b*d 
- sqrt(d**2 - e**2)*log(( - sqrt(e)*sqrt( - d**2 + e**2)*sqrt(d**2 - e**2) 
*d + sqrt(e)*sqrt( - d**2 + e**2)*d**2 - sqrt(e)*sqrt( - d**2 + e**2)*e**2 
 + sqrt(d*x**2 + e)*d**2*e - sqrt(d*x**2 + e)*e**3 + sqrt(d)*d**2*e*x - sq 
rt(d)*e**3*x)/(sqrt(e)*d**2 - sqrt(e)*e**2))*a*e + sqrt(d**2 - e**2)*log(( 
 - sqrt(e)*sqrt( - d**2 + e**2)*sqrt(d**2 - e**2)*d + sqrt(e)*sqrt( - d**2 
 + e**2)*d**2 - sqrt(e)*sqrt( - d**2 + e**2)*e**2 + sqrt(d*x**2 + e)*d**2* 
e - sqrt(d*x**2 + e)*e**3 + sqrt(d)*d**2*e*x - sqrt(d)*e**3*x)/(sqrt(e)*d* 
*2 - sqrt(e)*e**2))*b*d - sqrt(d**2 - e**2)*log((sqrt(e)*sqrt( - d**2 + e* 
*2)*sqrt(d**2 - e**2)*d - sqrt(e)*sqrt( - d**2 + e**2)*d**2 + sqrt(e)*sqrt 
( - d**2 + e**2)*e**2 + sqrt(d*x**2 + e)*d**2*e - sqrt(d*x**2 + e)*e**3 + 
sqrt(d)*d**2*e*x - sqrt(d)*e**3*x)/(sqrt(e)*d**2 - sqrt(e)*e**2))*a*e + sq 
rt(d**2 - e**2)*log((sqrt(e)*sqrt( - d**2 + e**2)*sqrt(d**2 - e**2)*d - sq 
rt(e)*sqrt( - d**2 + e**2)*d**2 + sqrt(e)*sqrt( - d**2 + e**2)*e**2 + sqrt 
(d*x**2 + e)*d**2*e - sqrt(d*x**2 + e)*e**3 + sqrt(d)*d**2*e*x - sqrt(d)*e 
**3*x)/(sqrt(e)*d**2 - sqrt(e)*e**2))*b*d - 2*log((sqrt(d*x**2 + e) + sqrt 
(d)*x)/sqrt(e))*b*d**2 + 2*log((sqrt(d*x**2 + e) + sqrt(d)*x)/sqrt(e))*b*e 
**2))/(2*d*e*(d**2 - e**2))