\(\int \frac {A+B x^2}{\sqrt {d-e x^2} \sqrt {d e+(d^2-e^2) x^2-d e x^4}} \, dx\) [143]

Optimal result

Integrand size = 48, antiderivative size = 146 \[ \int \frac {A+B x^2}{\sqrt {d-e x^2} \sqrt {d e+\left (d^2-e^2\right ) x^2-d e x^4}} \, dx=-\frac {B \text {arctanh}\left (\frac {\sqrt {d} x \sqrt {d-e x^2}}{\sqrt {d e+\left (d^2-e^2\right ) x^2-d e x^4}}\right )}{\sqrt {d} e}+\frac {(B d+A e) \text {arctanh}\left (\frac {\sqrt {d^2+e^2} x \sqrt {d-e x^2}}{\sqrt {d} \sqrt {d e+\left (d^2-e^2\right ) x^2-d e x^4}}\right )}{\sqrt {d} e \sqrt {d^2+e^2}} \] Output:

-B*arctanh(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)*arctanh((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 [A] (verified)

Time = 11.71 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.29 \[ \int \frac {A+B x^2}{\sqrt {d-e x^2} \sqrt {d e+\left (d^2-e^2\right ) x^2-d e x^4}} \, dx=\frac {\frac {(B d+A e) \sqrt {d^2 x^2-e^2 x^2+d \left (e-e x^4\right )} \text {arctanh}\left (\frac {\sqrt {d^2+e^2} x}{\sqrt {d} \sqrt {e+d x^2}}\right )}{\sqrt {d^2+e^2} \sqrt {e+d x^2} \sqrt {d-e x^2}}+B \left (\log \left (d-e x^2\right )-\log \left (d^2 x-d e x^3+\sqrt {d} \sqrt {d-e x^2} \sqrt {d e+d^2 x^2-e^2 x^2-d e x^4}\right )\right )}{\sqrt {d} e} \] 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[d^2*x^2 - e^2*x^2 + d*(e - e*x^4)]*ArcTanh[(Sqrt[d^2 + 
e^2]*x)/(Sqrt[d]*Sqrt[e + d*x^2])])/(Sqrt[d^2 + e^2]*Sqrt[e + d*x^2]*Sqrt[ 
d - e*x^2]) + B*(Log[d - e*x^2] - Log[d^2*x - d*e*x^3 + Sqrt[d]*Sqrt[d - e 
*x^2]*Sqrt[d*e + d^2*x^2 - e^2*x^2 - d*e*x^4]]))/(Sqrt[d]*e)
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1395, 398, 224, 219, 291, 221}

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

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

\(\Big \downarrow \) 398

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

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*ArcTanh[(Sqrt[d]*x)/Sqrt[e + d*x^2] 
])/(Sqrt[d]*e)) + ((B*d + A*e)*ArcTanh[(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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[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. \(806\) vs. \(2(126)=252\).

Time = 0.19 (sec) , antiderivative size = 807, normalized size of antiderivative = 5.53

method	result	size
default	\(-\frac {\sqrt {-d e \,x^{4}+d^{2} x^{2}-e^{2} x^{2}+d e}\, \sqrt {d}\, \left (2 A \sqrt {\frac {d^{2}+e^{2}}{e}}\, d \sqrt {d e}\, \ln \left (\frac {\sqrt {d}\, \sqrt {d \,x^{2}+e}+d x}{\sqrt {d}}\right ) e +2 B \sqrt {\frac {d^{2}+e^{2}}{e}}\, d^{2} \sqrt {d e}\, \ln \left (\frac {\sqrt {d}\, \sqrt {d \,x^{2}+e}+d x}{\sqrt {d}}\right )-2 A \sqrt {\frac {d^{2}+e^{2}}{e}}\, \sqrt {d e}\, \ln \left (\frac {\sqrt {d}\, \sqrt {-\frac {\left (-d x +\sqrt {-d e}\right ) \left (d x +\sqrt {-d e}\right )}{d}}+d x}{\sqrt {d}}\right ) d e -A \,d^{\frac {5}{2}} \ln \left (\frac {2 \sqrt {d \,x^{2}+e}\, \sqrt {\frac {d^{2}+e^{2}}{e}}\, e +2 \sqrt {d e}\, d x +2 e^{2}}{e x -\sqrt {d e}}\right ) e -A \sqrt {d}\, \ln \left (\frac {2 \sqrt {d \,x^{2}+e}\, \sqrt {\frac {d^{2}+e^{2}}{e}}\, e +2 \sqrt {d e}\, d x +2 e^{2}}{e x -\sqrt {d e}}\right ) e^{3}+A \,d^{\frac {5}{2}} \ln \left (-\frac {2 \left (\sqrt {d e}\, d x -\sqrt {d \,x^{2}+e}\, \sqrt {\frac {d^{2}+e^{2}}{e}}\, e -e^{2}\right )}{e x +\sqrt {d e}}\right ) e +A \sqrt {d}\, \ln \left (-\frac {2 \left (\sqrt {d e}\, d x -\sqrt {d \,x^{2}+e}\, \sqrt {\frac {d^{2}+e^{2}}{e}}\, e -e^{2}\right )}{e x +\sqrt {d e}}\right ) e^{3}+2 B \sqrt {\frac {d^{2}+e^{2}}{e}}\, \sqrt {d e}\, \ln \left (\frac {\sqrt {d}\, \sqrt {-\frac {\left (-d x +\sqrt {-d e}\right ) \left (d x +\sqrt {-d e}\right )}{d}}+d x}{\sqrt {d}}\right ) e^{2}-B \,d^{\frac {7}{2}} \ln \left (\frac {2 \sqrt {d \,x^{2}+e}\, \sqrt {\frac {d^{2}+e^{2}}{e}}\, e +2 \sqrt {d e}\, d x +2 e^{2}}{e x -\sqrt {d e}}\right )-B \,d^{\frac {3}{2}} \ln \left (\frac {2 \sqrt {d \,x^{2}+e}\, \sqrt {\frac {d^{2}+e^{2}}{e}}\, e +2 \sqrt {d e}\, d x +2 e^{2}}{e x -\sqrt {d e}}\right ) e^{2}+B \,d^{\frac {7}{2}} \ln \left (-\frac {2 \left (\sqrt {d e}\, d x -\sqrt {d \,x^{2}+e}\, \sqrt {\frac {d^{2}+e^{2}}{e}}\, e -e^{2}\right )}{e x +\sqrt {d e}}\right )+B \,d^{\frac {3}{2}} \ln \left (-\frac {2 \left (\sqrt {d e}\, d x -\sqrt {d \,x^{2}+e}\, \sqrt {\frac {d^{2}+e^{2}}{e}}\, e -e^{2}\right )}{e x +\sqrt {d e}}\right ) e^{2}\right )}{2 \sqrt {-e \,x^{2}+d}\, \sqrt {d \,x^{2}+e}\, \left (\sqrt {d e}\, d -\sqrt {-d e}\, e \right ) \left (\sqrt {-d e}\, e +\sqrt {d e}\, d \right ) \sqrt {d e}\, \sqrt {\frac {d^{2}+e^{2}}{e}}}\)	\(807\)

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 446, normalized size of antiderivative = 3.05 \[ \int \frac {A+B x^2}{\sqrt {d-e x^2} \sqrt {d e+\left (d^2-e^2\right ) x^2-d e x^4}} \, dx=\left [\frac {\sqrt {d^{3} + d e^{2}} {\left (B d + A e\right )} \log \left (\frac {2 \, d^{3} x^{2} - {\left (2 \, d^{2} e + e^{3}\right )} x^{4} + d^{2} e + 2 \, \sqrt {-d e x^{4} + {\left (d^{2} - e^{2}\right )} x^{2} + d e} \sqrt {d^{3} + d e^{2}} \sqrt {-e x^{2} + d} x}{e^{2} x^{4} - 2 \, d e x^{2} + d^{2}}\right ) + {\left (B d^{2} + B e^{2}\right )} \sqrt {d} \log \left (\frac {2 \, d e x^{4} - {\left (2 \, d^{2} - e^{2}\right )} x^{2} + 2 \, \sqrt {-d e x^{4} + {\left (d^{2} - e^{2}\right )} x^{2} + d e} \sqrt {-e x^{2} + d} \sqrt {d} x - d e}{e x^{2} - d}\right )}{2 \, {\left (d^{3} e + d e^{3}\right )}}, \frac {\sqrt {-d^{3} - d e^{2}} {\left (B d + A e\right )} \arctan \left (\frac {\sqrt {-d e x^{4} + {\left (d^{2} - e^{2}\right )} x^{2} + d e} \sqrt {-d^{3} - d e^{2}} \sqrt {-e x^{2} + d} x}{d^{2} e x^{4} - d^{2} e - {\left (d^{3} - d e^{2}\right )} x^{2}}\right ) - {\left (B d^{2} + B e^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d e x^{4} + {\left (d^{2} - e^{2}\right )} x^{2} + d e} \sqrt {-e x^{2} + d} \sqrt {-d} x}{d e x^{4} - {\left (d^{2} - e^{2}\right )} x^{2} - d e}\right )}{d^{3} e + d e^{3}}\right ] \] 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)*s 
qrt(-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)*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)*arctan(sqrt(-d*e*x^4 + (d^2 - e^2)*x^2 + d*e)*sq 
rt(-e*x^2 + d)*sqrt(-d)*x/(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. \[ \int \frac {A+B x^2}{\sqrt {d-e x^2} \sqrt {d e+\left (d^2-e^2\right ) x^2-d e x^4}} \, dx=\int \frac {B\,x^2+A}{\sqrt {d-e\,x^2}\,\sqrt {-d\,e\,x^4+\left (d^2-e^2\right )\,x^2+d\,e}} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 370, normalized size of antiderivative = 2.53 \[ \int \frac {A+B x^2}{\sqrt {d-e x^2} \sqrt {d e+\left (d^2-e^2\right ) x^2-d e x^4}} \, dx=\frac {\sqrt {d}\, \left (\sqrt {d^{2}+e^{2}}\, \mathrm {log}\left (2 \sqrt {d^{2}+e^{2}}\, d +2 \sqrt {d}\, \sqrt {d \,x^{2}+e}\, e x -2 d^{2}+2 d e \,x^{2}\right ) a e +\sqrt {d^{2}+e^{2}}\, \mathrm {log}\left (2 \sqrt {d^{2}+e^{2}}\, d +2 \sqrt {d}\, \sqrt {d \,x^{2}+e}\, e x -2 d^{2}+2 d e \,x^{2}\right ) b d -\sqrt {d^{2}+e^{2}}\, \mathrm {log}\left (\frac {-\sqrt {e}\, \sqrt {d^{2}+e^{2}}+\sqrt {d \,x^{2}+e}\, e +\sqrt {d}\, e x -\sqrt {e}\, d}{\sqrt {e}}\right ) a e -\sqrt {d^{2}+e^{2}}\, \mathrm {log}\left (\frac {-\sqrt {e}\, \sqrt {d^{2}+e^{2}}+\sqrt {d \,x^{2}+e}\, e +\sqrt {d}\, e x -\sqrt {e}\, d}{\sqrt {e}}\right ) b d -\sqrt {d^{2}+e^{2}}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {d^{2}+e^{2}}+\sqrt {d \,x^{2}+e}\, e +\sqrt {d}\, e x +\sqrt {e}\, d}{\sqrt {e}}\right ) a e -\sqrt {d^{2}+e^{2}}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {d^{2}+e^{2}}+\sqrt {d \,x^{2}+e}\, e +\sqrt {d}\, e x +\sqrt {e}\, d}{\sqrt {e}}\right ) b d -2 \,\mathrm {log}\left (\frac {\sqrt {d \,x^{2}+e}+\sqrt {d}\, x}{\sqrt {e}}\right ) b \,d^{2}-2 \,\mathrm {log}\left (\frac {\sqrt {d \,x^{2}+e}+\sqrt {d}\, x}{\sqrt {e}}\right ) b \,e^{2}\right )}{2 d e \left (d^{2}+e^{2}\right )} \] 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)*(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*x**2 + e)*e + 
 sqrt(d)*e*x - sqrt(e)*d)/sqrt(e))*a*e - sqrt(d**2 + e**2)*log(( - sqrt(e) 
*sqrt(d**2 + e**2) + sqrt(d*x**2 + e)*e + sqrt(d)*e*x - sqrt(e)*d)/sqrt(e) 
)*b*d - sqrt(d**2 + e**2)*log((sqrt(e)*sqrt(d**2 + e**2) + sqrt(d*x**2 + e 
)*e + sqrt(d)*e*x + sqrt(e)*d)/sqrt(e))*a*e - sqrt(d**2 + e**2)*log((sqrt( 
e)*sqrt(d**2 + e**2) + sqrt(d*x**2 + e)*e + sqrt(d)*e*x + sqrt(e)*d)/sqrt( 
e))*b*d - 2*log((sqrt(d*x**2 + e) + sqrt(d)*x)/sqrt(e))*b*d**2 - 2*log((sq 
rt(d*x**2 + e) + sqrt(d)*x)/sqrt(e))*b*e**2))/(2*d*e*(d**2 + e**2))