\(\int \frac {A+B x^2}{\sqrt {d+e x^2} \sqrt {d e+(d^2+e^2) x^2+d e x^4}} \, dx\) [138]

Optimal result

Integrand size = 44, antiderivative size = 143 \[ \int \frac {A+B x^2}{\sqrt {d+e x^2} \sqrt {d e+\left (d^2+e^2\right ) x^2+d e x^4}} \, dx=\frac {B \text {arctanh}\left (\frac {\sqrt {d} x \sqrt {d+e x^2}}{\sqrt {d e+\left (d^2+e^2\right ) x^2+d e x^4}}\right )}{\sqrt {d} e}-\frac {(B d-A e) \text {arctanh}\left (\frac {\sqrt {d^2-e^2} x \sqrt {d+e x^2}}{\sqrt {d} \sqrt {d e+\left (d^2+e^2\right ) x^2+d e x^4}}\right )}{\sqrt {d} e \sqrt {d^2-e^2}} \] 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 = 0.55 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.23 \[ \int \frac {A+B x^2}{\sqrt {d+e x^2} \sqrt {d e+\left (d^2+e^2\right ) x^2+d e x^4}} \, dx=\frac {\sqrt {e+d x^2} \sqrt {d+e x^2} \left ((B d-A e) \sqrt {-d^2+e^2} \arctan \left (\frac {d^{3/2}+\sqrt {d} e x^2-e x \sqrt {e+d x^2}}{\sqrt {d} \sqrt {-d^2+e^2}}\right )+B \left (d^2-e^2\right ) \log \left (-\sqrt {d} x+\sqrt {e+d x^2}\right )\right )}{\sqrt {d} e (-d+e) (d+e) \sqrt {d^2 x^2+e^2 x^2+d e \left (1+x^4\right )}} \] 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:

(Sqrt[e + d*x^2]*Sqrt[d + e*x^2]*((B*d - A*e)*Sqrt[-d^2 + e^2]*ArcTan[(d^( 
3/2) + Sqrt[d]*e*x^2 - e*x*Sqrt[e + d*x^2])/(Sqrt[d]*Sqrt[-d^2 + e^2])] + 
B*(d^2 - e^2)*Log[-(Sqrt[d]*x) + Sqrt[e + d*x^2]]))/(Sqrt[d]*e*(-d + e)*(d 
 + e)*Sqrt[d^2*x^2 + e^2*x^2 + d*e*(1 + x^4)])
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, 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

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

\(\Big \downarrow \) 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

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

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. \(841\) vs. \(2(123)=246\).

Time = 0.40 (sec) , antiderivative size = 842, normalized size of antiderivative = 5.89

method	result	size
default	\(-\frac {\sqrt {d e \,x^{4}+d^{2} x^{2}+e^{2} x^{2}+d e}\, \left (2 A d \sqrt {-\frac {d^{2}-e^{2}}{e}}\, \sqrt {-d e}\, \ln \left (\frac {\sqrt {d}\, \sqrt {d \,x^{2}+e}+d x}{\sqrt {d}}\right ) e -2 B \,d^{2} \sqrt {-\frac {d^{2}-e^{2}}{e}}\, \sqrt {-d e}\, \ln \left (\frac {\sqrt {d}\, \sqrt {d \,x^{2}+e}+d x}{\sqrt {d}}\right )+A \,d^{\frac {5}{2}} \ln \left (\frac {2 \sqrt {-\frac {d^{2}-e^{2}}{e}}\, \sqrt {d \,x^{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 {-\frac {d^{2}-e^{2}}{e}}\, \sqrt {d \,x^{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 {-\frac {d^{2}-e^{2}}{e}}\, \sqrt {d \,x^{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 {-\frac {d^{2}-e^{2}}{e}}\, \sqrt {d \,x^{2}+e}\, e -e^{2}\right )}{e x +\sqrt {-d e}}\right ) e^{3}-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 -B \,d^{\frac {7}{2}} \ln \left (\frac {2 \sqrt {-\frac {d^{2}-e^{2}}{e}}\, \sqrt {d \,x^{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 {-\frac {d^{2}-e^{2}}{e}}\, \sqrt {d \,x^{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 {-\frac {d^{2}-e^{2}}{e}}\, \sqrt {d \,x^{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 {-\frac {d^{2}-e^{2}}{e}}\, \sqrt {d \,x^{2}+e}\, e -e^{2}\right )}{e x +\sqrt {-d e}}\right ) e^{2}+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}\right )}{2 \sqrt {e \,x^{2}+d}\, \sqrt {d}\, \sqrt {d \,x^{2}+e}\, e \sqrt {-d e}\, \left (d +e \right ) \left (d -e \right ) \sqrt {-\frac {d^{2}-e^{2}}{e}}}\)	\(842\)

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

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 804, normalized size of antiderivative = 5.62 \[ \int \frac {A+B x^2}{\sqrt {d+e x^2} \sqrt {d e+\left (d^2+e^2\right ) x^2+d e x^4}} \, dx =\text {Too large to display} \] 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, a 
lgorithm="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)*sqr 
t(e*x^2 + d)*sqrt(d)*x + d*e)/(e*x^2 + d)))/(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), -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), (sqrt(-d^3 + d*e^2)*(B*d - A*e)*arctan(sqr 
t(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(s 
qrt(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, a 
lgorithm="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, a 
lgorithm="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 = 709, normalized size of antiderivative = 4.96 \[ \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^{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)*( - 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 - s 
qrt(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)*sqr 
t( - 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 - s 
qrt(d**2 - e**2)*log((sqrt(e)*sqrt( - d**2 + e**2)*sqrt(d**2 - e**2)*d - s 
qrt(e)*sqrt( - d**2 + e**2)*d**2 + sqrt(e)*sqrt( - d**2 + e**2)*e**2 + sqr 
t(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) + sqr 
t(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))