\(\int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx\) [149]

Optimal result

Integrand size = 49, antiderivative size = 213 \[ \int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=\frac {C x \sqrt {a d+(b d+a e) x^2+b e x^4}}{2 b e \sqrt {d+e x^2}}-\frac {(2 b C d-2 b B e+a C e) \text {arctanh}\left (\frac {\sqrt {b} x \sqrt {d+e x^2}}{\sqrt {a d+(b d+a e) x^2+b e x^4}}\right )}{2 b^{3/2} e^2}+\frac {\left (C d^2-B d e+A e^2\right ) \text {arctanh}\left (\frac {\sqrt {b d-a e} x \sqrt {d+e x^2}}{\sqrt {d} \sqrt {a d+(b d+a e) x^2+b e x^4}}\right )}{\sqrt {d} e^2 \sqrt {b d-a e}} \] Output:

1/2*C*x*(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2)/b/e/(e*x^2+d)^(1/2)-1/2*(-2*B*b* 
e+C*a*e+2*C*b*d)*arctanh(b^(1/2)*x*(e*x^2+d)^(1/2)/(a*d+(a*e+b*d)*x^2+b*e* 
x^4)^(1/2))/b^(3/2)/e^2+(A*e^2-B*d*e+C*d^2)*arctanh((-a*e+b*d)^(1/2)*x*(e* 
x^2+d)^(1/2)/d^(1/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2))/d^(1/2)/e^2/(-a*e+ 
b*d)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.59 (sec) , antiderivative size = 486, normalized size of antiderivative = 2.28 \[ \int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=\frac {\sqrt {d+e x^2} \left (\sqrt {b} C e x \left (a+b x^2\right )+\frac {2 \sqrt {b} \left (-i \sqrt {a} \sqrt {e}+\sqrt {b d-a e}\right ) \sqrt {-b d+2 a e-2 i \sqrt {a} \sqrt {e} \sqrt {b d-a e}} \left (C d^2+e (-B d+A e)\right ) \sqrt {a+b x^2} \arctan \left (\frac {\sqrt {-b d+2 a e-2 i \sqrt {a} \sqrt {e} \sqrt {b d-a e}} x}{\sqrt {d} \left (\sqrt {a}-\sqrt {a+b x^2}\right )}\right )}{d^{3/2} \sqrt {b d-a e}}+\frac {2 \sqrt {b} \left (i \sqrt {a} \sqrt {e}+\sqrt {b d-a e}\right ) \sqrt {-b d+2 a e+2 i \sqrt {a} \sqrt {e} \sqrt {b d-a e}} \left (C d^2+e (-B d+A e)\right ) \sqrt {a+b x^2} \arctan \left (\frac {\sqrt {-b d+2 a e+2 i \sqrt {a} \sqrt {e} \sqrt {b d-a e}} x}{\sqrt {d} \left (\sqrt {a}-\sqrt {a+b x^2}\right )}\right )}{d^{3/2} \sqrt {b d-a e}}+2 (2 b C d-2 b B e+a C e) \sqrt {a+b x^2} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}-\sqrt {a+b x^2}}\right )\right )}{2 b^{3/2} e^2 \sqrt {\left (a+b x^2\right ) \left (d+e x^2\right )}} \] Input:

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

Output:

(Sqrt[d + e*x^2]*(Sqrt[b]*C*e*x*(a + b*x^2) + (2*Sqrt[b]*((-I)*Sqrt[a]*Sqr 
t[e] + Sqrt[b*d - a*e])*Sqrt[-(b*d) + 2*a*e - (2*I)*Sqrt[a]*Sqrt[e]*Sqrt[b 
*d - a*e]]*(C*d^2 + e*(-(B*d) + A*e))*Sqrt[a + b*x^2]*ArcTan[(Sqrt[-(b*d) 
+ 2*a*e - (2*I)*Sqrt[a]*Sqrt[e]*Sqrt[b*d - a*e]]*x)/(Sqrt[d]*(Sqrt[a] - Sq 
rt[a + b*x^2]))])/(d^(3/2)*Sqrt[b*d - a*e]) + (2*Sqrt[b]*(I*Sqrt[a]*Sqrt[e 
] + Sqrt[b*d - a*e])*Sqrt[-(b*d) + 2*a*e + (2*I)*Sqrt[a]*Sqrt[e]*Sqrt[b*d 
- a*e]]*(C*d^2 + e*(-(B*d) + A*e))*Sqrt[a + b*x^2]*ArcTan[(Sqrt[-(b*d) + 2 
*a*e + (2*I)*Sqrt[a]*Sqrt[e]*Sqrt[b*d - a*e]]*x)/(Sqrt[d]*(Sqrt[a] - Sqrt[ 
a + b*x^2]))])/(d^(3/2)*Sqrt[b*d - a*e]) + 2*(2*b*C*d - 2*b*B*e + a*C*e)*S 
qrt[a + b*x^2]*ArcTanh[(Sqrt[b]*x)/(Sqrt[a] - Sqrt[a + b*x^2])]))/(2*b^(3/ 
2)*e^2*Sqrt[(a + b*x^2)*(d + e*x^2)])
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {1395, 7276, 2009}

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.

\(\displaystyle \int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} \sqrt {x^2 (a e+b d)+a d+b e x^4}} \, dx\)

\(\Big \downarrow \) 1395

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {a+b x^2} \sqrt {d+e x^2} \left (\frac {\left (A e^2-B d e+C d^2\right ) \text {arctanh}\left (\frac {x \sqrt {b d-a e}}{\sqrt {d} \sqrt {a+b x^2}}\right )}{\sqrt {d} e^2 \sqrt {b d-a e}}-\frac {a C \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{3/2} e}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (C d-B e)}{\sqrt {b} e^2}+\frac {C x \sqrt {a+b x^2}}{2 b e}\right )}{\sqrt {x^2 (a e+b d)+a d+b e x^4}}\)

Input:

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

Output:

(Sqrt[a + b*x^2]*Sqrt[d + e*x^2]*((C*x*Sqrt[a + b*x^2])/(2*b*e) - (a*C*Arc 
Tanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(3/2)*e) - ((C*d - B*e)*ArcTanh[(S 
qrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*e^2) + ((C*d^2 - B*d*e + A*e^2)*ArcTa 
nh[(Sqrt[b*d - a*e]*x)/(Sqrt[d]*Sqrt[a + b*x^2])])/(Sqrt[d]*e^2*Sqrt[b*d - 
 a*e])))/Sqrt[a*d + (b*d + a*e)*x^2 + b*e*x^4]
 

Defintions of rubi rules used

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])
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(464\) vs. \(2(185)=370\).

Time = 0.29 (sec) , antiderivative size = 465, normalized size of antiderivative = 2.18

method	result	size
risch	\(\frac {C x \left (b \,x^{2}+a \right ) \sqrt {e \,x^{2}+d}}{2 e b \sqrt {\left (e \,x^{2}+d \right ) \left (b \,x^{2}+a \right )}}+\frac {\left (\frac {\left (2 B b e -C a e -2 C b d \right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{e \sqrt {b}}-\frac {b \left (A \,e^{2}-B d e +C \,d^{2}\right ) \ln \left (\frac {\frac {2 a e -2 b d}{e}+\frac {2 b \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right )}{e}+2 \sqrt {\frac {a e -b d}{e}}\, \sqrt {\left (x -\frac {\sqrt {-d e}}{e}\right )^{2} b +\frac {2 b \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right )}{e}+\frac {a e -b d}{e}}}{x -\frac {\sqrt {-d e}}{e}}\right )}{\sqrt {-d e}\, e \sqrt {\frac {a e -b d}{e}}}+\frac {b \left (A \,e^{2}-B d e +C \,d^{2}\right ) \ln \left (\frac {\frac {2 a e -2 b d}{e}-\frac {2 b \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right )}{e}+2 \sqrt {\frac {a e -b d}{e}}\, \sqrt {\left (x +\frac {\sqrt {-d e}}{e}\right )^{2} b -\frac {2 b \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right )}{e}+\frac {a e -b d}{e}}}{x +\frac {\sqrt {-d e}}{e}}\right )}{\sqrt {-d e}\, e \sqrt {\frac {a e -b d}{e}}}\right ) \sqrt {b \,x^{2}+a}\, \sqrt {e \,x^{2}+d}}{2 e b \sqrt {\left (e \,x^{2}+d \right ) \left (b \,x^{2}+a \right )}}\)	\(465\)
default	\(\text {Expression too large to display}\)	\(1399\)

Input:

int((C*x^4+B*x^2+A)/(e*x^2+d)^(1/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2),x,me 
thod=_RETURNVERBOSE)
 

Output:

1/2*C/e/b*x*(b*x^2+a)/((e*x^2+d)*(b*x^2+a))^(1/2)*(e*x^2+d)^(1/2)+1/2/e/b* 
(1/e*(2*B*b*e-C*a*e-2*C*b*d)*ln(b^(1/2)*x+(b*x^2+a)^(1/2))/b^(1/2)-b*(A*e^ 
2-B*d*e+C*d^2)/(-d*e)^(1/2)/e/((a*e-b*d)/e)^(1/2)*ln((2*(a*e-b*d)/e+2*b*(- 
d*e)^(1/2)/e*(x-(-d*e)^(1/2)/e)+2*((a*e-b*d)/e)^(1/2)*((x-(-d*e)^(1/2)/e)^ 
2*b+2*b*(-d*e)^(1/2)/e*(x-(-d*e)^(1/2)/e)+(a*e-b*d)/e)^(1/2))/(x-(-d*e)^(1 
/2)/e))+b*(A*e^2-B*d*e+C*d^2)/(-d*e)^(1/2)/e/((a*e-b*d)/e)^(1/2)*ln((2*(a* 
e-b*d)/e-2*b*(-d*e)^(1/2)/e*(x+(-d*e)^(1/2)/e)+2*((a*e-b*d)/e)^(1/2)*((x+( 
-d*e)^(1/2)/e)^2*b-2*b*(-d*e)^(1/2)/e*(x+(-d*e)^(1/2)/e)+(a*e-b*d)/e)^(1/2 
))/(x+(-d*e)^(1/2)/e)))*(b*x^2+a)^(1/2)/((e*x^2+d)*(b*x^2+a))^(1/2)*(e*x^2 
+d)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 401 vs. \(2 (185) = 370\).

Time = 0.70 (sec) , antiderivative size = 1716, normalized size of antiderivative = 8.06 \[ \int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/4*(2*(C*b^2*d^2*e - C*a*b*d*e^2)*sqrt(b*e*x^4 + (b*d + a*e)*x^2 + a*d)* 
sqrt(e*x^2 + d)*x + 2*(C*b^2*d^3 - B*b^2*d^2*e + A*b^2*d*e^2 + (C*b^2*d^2* 
e - B*b^2*d*e^2 + A*b^2*e^3)*x^2)*sqrt(b*d^2 - a*d*e)*log((2*b*d^2*x^2 + ( 
2*b*d*e - a*e^2)*x^4 + a*d^2 + 2*sqrt(b*e*x^4 + (b*d + a*e)*x^2 + a*d)*sqr 
t(b*d^2 - a*d*e)*sqrt(e*x^2 + d)*x)/(e^2*x^4 + 2*d*e*x^2 + d^2)) - (2*C*b^ 
2*d^4 - (C*a*b + 2*B*b^2)*d^3*e - (C*a^2 - 2*B*a*b)*d^2*e^2 + (2*C*b^2*d^3 
*e - (C*a*b + 2*B*b^2)*d^2*e^2 - (C*a^2 - 2*B*a*b)*d*e^3)*x^2)*sqrt(b)*log 
((2*b*e*x^4 + (2*b*d + a*e)*x^2 + 2*sqrt(b*e*x^4 + (b*d + a*e)*x^2 + a*d)* 
sqrt(e*x^2 + d)*sqrt(b)*x + a*d)/(e*x^2 + d)))/(b^3*d^3*e^2 - a*b^2*d^2*e^ 
3 + (b^3*d^2*e^3 - a*b^2*d*e^4)*x^2), 1/4*(2*(C*b^2*d^2*e - C*a*b*d*e^2)*s 
qrt(b*e*x^4 + (b*d + a*e)*x^2 + a*d)*sqrt(e*x^2 + d)*x - 4*(C*b^2*d^3 - B* 
b^2*d^2*e + A*b^2*d*e^2 + (C*b^2*d^2*e - B*b^2*d*e^2 + A*b^2*e^3)*x^2)*sqr 
t(-b*d^2 + a*d*e)*arctan(sqrt(b*e*x^4 + (b*d + a*e)*x^2 + a*d)*sqrt(-b*d^2 
 + a*d*e)*sqrt(e*x^2 + d)*x/(b*d*e*x^4 + a*d^2 + (b*d^2 + a*d*e)*x^2)) - ( 
2*C*b^2*d^4 - (C*a*b + 2*B*b^2)*d^3*e - (C*a^2 - 2*B*a*b)*d^2*e^2 + (2*C*b 
^2*d^3*e - (C*a*b + 2*B*b^2)*d^2*e^2 - (C*a^2 - 2*B*a*b)*d*e^3)*x^2)*sqrt( 
b)*log((2*b*e*x^4 + (2*b*d + a*e)*x^2 + 2*sqrt(b*e*x^4 + (b*d + a*e)*x^2 + 
 a*d)*sqrt(e*x^2 + d)*sqrt(b)*x + a*d)/(e*x^2 + d)))/(b^3*d^3*e^2 - a*b^2* 
d^2*e^3 + (b^3*d^2*e^3 - a*b^2*d*e^4)*x^2), 1/2*((C*b^2*d^2*e - C*a*b*d*e^ 
2)*sqrt(b*e*x^4 + (b*d + a*e)*x^2 + a*d)*sqrt(e*x^2 + d)*x + (2*C*b^2*d...
 

Sympy [F]

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

integrate((C*x**4+B*x**2+A)/(e*x**2+d)**(1/2)/(a*d+(a*e+b*d)*x**2+b*e*x**4 
)**(1/2),x)
 

Output:

Integral((A + B*x**2 + C*x**4)/(sqrt((a + b*x**2)*(d + e*x**2))*sqrt(d + e 
*x**2)), x)
 

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.70 \[ \int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=\frac {\sqrt {b x^{2} + a} C x}{2 \, b e} - \frac {{\left (C d^{2} - B d e + A e^{2}\right )} \sqrt {b} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} e + 2 \, b d - a e}{2 \, \sqrt {-b^{2} d^{2} + a b d e}}\right )}{\sqrt {-b^{2} d^{2} + a b d e} e^{2}} + \frac {{\left (2 \, C b d + C a e - 2 \, B b e\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, b^{\frac {3}{2}} e^{2}} \] Input:

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

Output:

1/2*sqrt(b*x^2 + a)*C*x/(b*e) - (C*d^2 - B*d*e + A*e^2)*sqrt(b)*arctan(1/2 
*((sqrt(b)*x - sqrt(b*x^2 + a))^2*e + 2*b*d - a*e)/sqrt(-b^2*d^2 + a*b*d*e 
))/(sqrt(-b^2*d^2 + a*b*d*e)*e^2) + 1/2*(2*C*b*d + C*a*e - 2*B*b*e)*log(ab 
s(-sqrt(b)*x + sqrt(b*x^2 + a)))/(b^(3/2)*e^2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 554, normalized size of antiderivative = 2.60 \[ \int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=\frac {-2 \sqrt {d}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {a e -b d}-\sqrt {e}\, \sqrt {b \,x^{2}+a}-\sqrt {e}\, \sqrt {b}\, x}{\sqrt {d}\, \sqrt {b}}\right ) a \,b^{2} e^{2}+2 \sqrt {d}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {a e -b d}-\sqrt {e}\, \sqrt {b \,x^{2}+a}-\sqrt {e}\, \sqrt {b}\, x}{\sqrt {d}\, \sqrt {b}}\right ) b^{3} d e -2 \sqrt {d}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {a e -b d}-\sqrt {e}\, \sqrt {b \,x^{2}+a}-\sqrt {e}\, \sqrt {b}\, x}{\sqrt {d}\, \sqrt {b}}\right ) b^{2} c \,d^{2}-2 \sqrt {d}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {a e -b d}+\sqrt {e}\, \sqrt {b \,x^{2}+a}+\sqrt {e}\, \sqrt {b}\, x}{\sqrt {d}\, \sqrt {b}}\right ) a \,b^{2} e^{2}+2 \sqrt {d}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {a e -b d}+\sqrt {e}\, \sqrt {b \,x^{2}+a}+\sqrt {e}\, \sqrt {b}\, x}{\sqrt {d}\, \sqrt {b}}\right ) b^{3} d e -2 \sqrt {d}\, \sqrt {a e -b d}\, \mathit {atan} \left (\frac {\sqrt {a e -b d}+\sqrt {e}\, \sqrt {b \,x^{2}+a}+\sqrt {e}\, \sqrt {b}\, x}{\sqrt {d}\, \sqrt {b}}\right ) b^{2} c \,d^{2}+\sqrt {b \,x^{2}+a}\, a b c d \,e^{2} x -\sqrt {b \,x^{2}+a}\, b^{2} c \,d^{2} e x -\sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} c d \,e^{2}+2 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} d \,e^{2}-\sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b c \,d^{2} e -2 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{3} d^{2} e +2 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c \,d^{3}}{2 b^{2} d \,e^{2} \left (a e -b d \right )} \] Input:

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

Output:

( - 2*sqrt(d)*sqrt(a*e - b*d)*atan((sqrt(a*e - b*d) - sqrt(e)*sqrt(a + b*x 
**2) - sqrt(e)*sqrt(b)*x)/(sqrt(d)*sqrt(b)))*a*b**2*e**2 + 2*sqrt(d)*sqrt( 
a*e - b*d)*atan((sqrt(a*e - b*d) - sqrt(e)*sqrt(a + b*x**2) - sqrt(e)*sqrt 
(b)*x)/(sqrt(d)*sqrt(b)))*b**3*d*e - 2*sqrt(d)*sqrt(a*e - b*d)*atan((sqrt( 
a*e - b*d) - sqrt(e)*sqrt(a + b*x**2) - sqrt(e)*sqrt(b)*x)/(sqrt(d)*sqrt(b 
)))*b**2*c*d**2 - 2*sqrt(d)*sqrt(a*e - b*d)*atan((sqrt(a*e - b*d) + sqrt(e 
)*sqrt(a + b*x**2) + sqrt(e)*sqrt(b)*x)/(sqrt(d)*sqrt(b)))*a*b**2*e**2 + 2 
*sqrt(d)*sqrt(a*e - b*d)*atan((sqrt(a*e - b*d) + sqrt(e)*sqrt(a + b*x**2) 
+ sqrt(e)*sqrt(b)*x)/(sqrt(d)*sqrt(b)))*b**3*d*e - 2*sqrt(d)*sqrt(a*e - b* 
d)*atan((sqrt(a*e - b*d) + sqrt(e)*sqrt(a + b*x**2) + sqrt(e)*sqrt(b)*x)/( 
sqrt(d)*sqrt(b)))*b**2*c*d**2 + sqrt(a + b*x**2)*a*b*c*d*e**2*x - sqrt(a + 
 b*x**2)*b**2*c*d**2*e*x - sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt 
(a))*a**2*c*d*e**2 + 2*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a)) 
*a*b**2*d*e**2 - sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a*b*c 
*d**2*e - 2*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*b**3*d**2* 
e + 2*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*b**2*c*d**3)/(2* 
b**2*d*e**2*(a*e - b*d))