\(\int \frac {A+B x^2+C x^4}{(d+e x^2)^{3/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx\) [150]

Optimal result

Integrand size = 49, antiderivative size = 246 \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{3/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=-\frac {\left (C d^2-B d e+A e^2\right ) x \sqrt {a d+(b d+a e) x^2+b e x^4}}{2 d e (b d-a e) \left (d+e x^2\right )^{3/2}}+\frac {C \text {arctanh}\left (\frac {\sqrt {b} x \sqrt {d+e x^2}}{\sqrt {a d+(b d+a e) x^2+b e x^4}}\right )}{\sqrt {b} e^2}-\frac {\left (2 b \left (C d^3-A d e^2\right )-a e \left (3 C d^2-e (B d+A e)\right )\right ) \text {arctanh}\left (\frac {\sqrt {b d-a e} x \sqrt {d+e x^2}}{\sqrt {d} \sqrt {a d+(b d+a e) x^2+b e x^4}}\right )}{2 d^{3/2} e^2 (b d-a e)^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.23 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.00 \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{3/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=\frac {-\frac {e \left (C d^2+e (-B d+A e)\right ) x \left (a+b x^2\right )}{d (b d-a e)}-\frac {\left (2 b \left (C d^3-A d e^2\right )+a e \left (-3 C d^2+e (B d+A e)\right )\right ) \sqrt {a+b x^2} \left (d+e x^2\right ) \arctan \left (\frac {-e x \sqrt {a+b x^2}+\sqrt {b} \left (d+e x^2\right )}{\sqrt {d} \sqrt {-b d+a e}}\right )}{d^{3/2} (-b d+a e)^{3/2}}-\frac {2 C \sqrt {a+b x^2} \left (d+e x^2\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{\sqrt {b}}}{2 e^2 \sqrt {d+e x^2} \sqrt {\left (a+b x^2\right ) \left (d+e x^2\right )}} \] Input:

Integrate[(A + B*x^2 + C*x^4)/((d + e*x^2)^(3/2)*Sqrt[a*d + (b*d + a*e)*x^ 
2 + b*e*x^4]),x]
 

Output:

(-((e*(C*d^2 + e*(-(B*d) + A*e))*x*(a + b*x^2))/(d*(b*d - a*e))) - ((2*b*( 
C*d^3 - A*d*e^2) + a*e*(-3*C*d^2 + e*(B*d + A*e)))*Sqrt[a + b*x^2]*(d + e* 
x^2)*ArcTan[(-(e*x*Sqrt[a + b*x^2]) + Sqrt[b]*(d + e*x^2))/(Sqrt[d]*Sqrt[- 
(b*d) + a*e])])/(d^(3/2)*(-(b*d) + a*e)^(3/2)) - (2*C*Sqrt[a + b*x^2]*(d + 
 e*x^2)*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/Sqrt[b])/(2*e^2*Sqrt[d + e*x^ 
2]*Sqrt[(a + b*x^2)*(d + e*x^2)])
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.13, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {1395, 7293, 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.

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

\(\Big \downarrow \) 1395

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

Int[(A + B*x^2 + C*x^4)/((d + e*x^2)^(3/2)*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]*(-1/2*((C*d^2 - B*d*e + A*e^2)*x*Sqrt[a + 
 b*x^2])/(d*e*(b*d - a*e)*(d + e*x^2)) + (C*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b 
*x^2]])/(Sqrt[b]*e^2) - ((2*C*d - B*e)*ArcTanh[(Sqrt[b*d - a*e]*x)/(Sqrt[d 
]*Sqrt[a + b*x^2])])/(Sqrt[d]*e^2*Sqrt[b*d - a*e]) + ((2*b*d - a*e)*(C*d^2 
 - B*d*e + A*e^2)*ArcTanh[(Sqrt[b*d - a*e]*x)/(Sqrt[d]*Sqrt[a + b*x^2])])/ 
(2*d^(3/2)*e^2*(b*d - a*e)^(3/2))))/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 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3430\) vs. \(2(218)=436\).

Time = 0.11 (sec) , antiderivative size = 3431, normalized size of antiderivative = 13.95

method	result	size
default	\(\text {Expression too large to display}\)	\(3431\)

Input:

int((C*x^4+B*x^2+A)/(e*x^2+d)^(3/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2),x,me 
thod=_RETURNVERBOSE)
 

Output:

-1/4*(3*A*ln(2*((b*x^2+a)^(1/2)*((a*e-b*d)/e)^(1/2)*e-(-d*e)^(1/2)*b*x+a*e 
)/(e*x+(-d*e)^(1/2)))*a*b^(3/2)*d*e^4*x^2-3*A*ln(2*((b*x^2+a)^(1/2)*((a*e- 
b*d)/e)^(1/2)*e+(-d*e)^(1/2)*b*x+a*e)/(e*x-(-d*e)^(1/2)))*a*b^(3/2)*d*e^4* 
x^2+B*ln(2*((b*x^2+a)^(1/2)*((a*e-b*d)/e)^(1/2)*e-(-d*e)^(1/2)*b*x+a*e)/(e 
*x+(-d*e)^(1/2)))*a*b^(3/2)*d^2*e^3*x^2-5*C*ln(2*((b*x^2+a)^(1/2)*((a*e-b* 
d)/e)^(1/2)*e-(-d*e)^(1/2)*b*x+a*e)/(e*x+(-d*e)^(1/2)))*a*b^(3/2)*d^3*e^2* 
x^2+5*C*ln(2*((b*x^2+a)^(1/2)*((a*e-b*d)/e)^(1/2)*e+(-d*e)^(1/2)*b*x+a*e)/ 
(e*x-(-d*e)^(1/2)))*a*b^(3/2)*d^3*e^2*x^2-B*ln(2*((b*x^2+a)^(1/2)*((a*e-b* 
d)/e)^(1/2)*e-(-d*e)^(1/2)*b*x+a*e)/(e*x+(-d*e)^(1/2)))*a^2*d*e^4*x^2*b^(1 
/2)+B*ln(2*((b*x^2+a)^(1/2)*((a*e-b*d)/e)^(1/2)*e+(-d*e)^(1/2)*b*x+a*e)/(e 
*x-(-d*e)^(1/2)))*a^2*d*e^4*x^2*b^(1/2)+3*C*ln(2*((b*x^2+a)^(1/2)*((a*e-b* 
d)/e)^(1/2)*e-(-d*e)^(1/2)*b*x+a*e)/(e*x+(-d*e)^(1/2)))*a^2*d^2*e^3*x^2*b^ 
(1/2)-3*C*ln(2*((b*x^2+a)^(1/2)*((a*e-b*d)/e)^(1/2)*e+(-d*e)^(1/2)*b*x+a*e 
)/(e*x-(-d*e)^(1/2)))*a^2*d^2*e^3*x^2*b^(1/2)-4*A*ln(((-(-b*x+(-b*a)^(1/2) 
)/b*(b*x+(-b*a)^(1/2)))^(1/2)*b^(1/2)+b*x)/b^(1/2))*b^2*d^2*e^2*((a*e-b*d) 
/e)^(1/2)*(-d*e)^(1/2)+4*A*ln(((b*x^2+a)^(1/2)*b^(1/2)+b*x)/b^(1/2))*b^2*d 
^2*e^2*((a*e-b*d)/e)^(1/2)*(-d*e)^(1/2)-B*ln(2*((b*x^2+a)^(1/2)*((a*e-b*d) 
/e)^(1/2)*e+(-d*e)^(1/2)*b*x+a*e)/(e*x-(-d*e)^(1/2)))*a*b^(3/2)*d^3*e^2-4* 
C*ln(((-(-b*x+(-b*a)^(1/2))/b*(b*x+(-b*a)^(1/2)))^(1/2)*b^(1/2)+b*x)/b^(1/ 
2))*a^2*d^2*e^2*((a*e-b*d)/e)^(1/2)*(-d*e)^(1/2)-2*A*ln(2*((b*x^2+a)^(1...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 598 vs. \(2 (217) = 434\).

Time = 2.01 (sec) , antiderivative size = 2501, normalized size of antiderivative = 10.17 \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{3/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=\text {Too large to display} \] Input:

integrate((C*x^4+B*x^2+A)/(e*x^2+d)^(3/2)/(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^4*e - A*a*b*d*e^4 - (C*a*b + B*b^2)*d^3*e^2 + (B*a*b + A 
*b^2)*d^2*e^3)*sqrt(b*e*x^4 + (b*d + a*e)*x^2 + a*d)*sqrt(e*x^2 + d)*x + ( 
2*C*b^2*d^5 - 3*C*a*b*d^4*e + A*a*b*d^2*e^3 + (B*a*b - 2*A*b^2)*d^3*e^2 + 
(2*C*b^2*d^3*e^2 - 3*C*a*b*d^2*e^3 + A*a*b*e^5 + (B*a*b - 2*A*b^2)*d*e^4)* 
x^4 + 2*(2*C*b^2*d^4*e - 3*C*a*b*d^3*e^2 + A*a*b*d*e^4 + (B*a*b - 2*A*b^2) 
*d^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)*sqrt(b*d^2 - a*d*e)*sq 
rt(e*x^2 + d)*x)/(e^2*x^4 + 2*d*e*x^2 + d^2)) - 2*(C*b^2*d^6 - 2*C*a*b*d^5 
*e + C*a^2*d^4*e^2 + (C*b^2*d^4*e^2 - 2*C*a*b*d^3*e^3 + C*a^2*d^2*e^4)*x^4 
 + 2*(C*b^2*d^5*e - 2*C*a*b*d^4*e^2 + C*a^2*d^3*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^6*e^2 - 2*a*b^2*d^5*e^3 + 
 a^2*b*d^4*e^4 + (b^3*d^4*e^4 - 2*a*b^2*d^3*e^5 + a^2*b*d^2*e^6)*x^4 + 2*( 
b^3*d^5*e^3 - 2*a*b^2*d^4*e^4 + a^2*b*d^3*e^5)*x^2), -1/2*((C*b^2*d^4*e - 
A*a*b*d*e^4 - (C*a*b + B*b^2)*d^3*e^2 + (B*a*b + A*b^2)*d^2*e^3)*sqrt(b*e* 
x^4 + (b*d + a*e)*x^2 + a*d)*sqrt(e*x^2 + d)*x - (2*C*b^2*d^5 - 3*C*a*b*d^ 
4*e + A*a*b*d^2*e^3 + (B*a*b - 2*A*b^2)*d^3*e^2 + (2*C*b^2*d^3*e^2 - 3*C*a 
*b*d^2*e^3 + A*a*b*e^5 + (B*a*b - 2*A*b^2)*d*e^4)*x^4 + 2*(2*C*b^2*d^4*e - 
 3*C*a*b*d^3*e^2 + A*a*b*d*e^4 + (B*a*b - 2*A*b^2)*d^2*e^3)*x^2)*sqrt(-b*d 
^2 + a*d*e)*arctan(sqrt(b*e*x^4 + (b*d + a*e)*x^2 + a*d)*sqrt(-b*d^2 + ...
 

Sympy [F]

\[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{3/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=\int \frac {A + B x^{2} + C x^{4}}{\sqrt {\left (a + b x^{2}\right ) \left (d + e x^{2}\right )} \left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((C*x**4+B*x**2+A)/(e*x**2+d)**(3/2)/(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))*(d + e*x** 
2)**(3/2)), x)
 

Maxima [F]

\[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{3/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{\sqrt {b e x^{4} + {\left (b d + a e\right )} x^{2} + a d} {\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((C*x^4+B*x^2+A)/(e*x^2+d)^(3/2)/(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)*(e*x^ 
2 + d)^(3/2)), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 469 vs. \(2 (217) = 434\).

Time = 0.38 (sec) , antiderivative size = 469, normalized size of antiderivative = 1.91 \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{3/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=\frac {{\left (2 \, C b^{\frac {3}{2}} d^{3} - 3 \, C a \sqrt {b} d^{2} e + B a \sqrt {b} d e^{2} - 2 \, A b^{\frac {3}{2}} d e^{2} + A a \sqrt {b} e^{3}\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} e + 2 \, b d - a e}{2 \, \sqrt {-b^{2} d^{2} + a b d e}}\right )}{2 \, {\left (b d^{2} e^{2} - a d e^{3}\right )} \sqrt {-b^{2} d^{2} + a b d e}} - \frac {C \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{\sqrt {b} e^{2}} - \frac {2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} C b^{\frac {3}{2}} d^{3} - {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} C a \sqrt {b} d^{2} e - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B b^{\frac {3}{2}} d^{2} e + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a \sqrt {b} d e^{2} + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A b^{\frac {3}{2}} d e^{2} - {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a \sqrt {b} e^{3} + C a^{2} \sqrt {b} d^{2} e - B a^{2} \sqrt {b} d e^{2} + A a^{2} \sqrt {b} e^{3}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} e + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b d - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a e + a^{2} e\right )} {\left (b d^{2} e^{2} - a d e^{3}\right )}} \] Input:

integrate((C*x^4+B*x^2+A)/(e*x^2+d)^(3/2)/(a*d+(a*e+b*d)*x^2+b*e*x^4)^(1/2 
),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{3/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx=\int \frac {C\,x^4+B\,x^2+A}{{\left (e\,x^2+d\right )}^{3/2}\,\sqrt {b\,e\,x^4+\left (a\,e+b\,d\right )\,x^2+a\,d}} \,d x \] Input:

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

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 1328, normalized size of antiderivative = 5.40 \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{3/2} \sqrt {a d+(b d+a e) x^2+b e x^4}} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 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**2*b*d*e**3 - 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**2*b*e**4*x**2 + sqrt(d)*sqrt(a*e - b*d)*atan((s 
qrt(a*e - b*d) - sqrt(e)*sqrt(a + b*x**2) - sqrt(e)*sqrt(b)*x)/(sqrt(d)*sq 
rt(b)))*a*b**2*d**2*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)))*a*b**2*d 
*e**3*x**2 + 3*sqrt(d)*sqrt(a*e - b*d)*atan((sqrt(a*e - b*d) - sqrt(e)*sqr 
t(a + b*x**2) - sqrt(e)*sqrt(b)*x)/(sqrt(d)*sqrt(b)))*a*b*c*d**3*e + 3*sqr 
t(d)*sqrt(a*e - b*d)*atan((sqrt(a*e - b*d) - sqrt(e)*sqrt(a + b*x**2) - sq 
rt(e)*sqrt(b)*x)/(sqrt(d)*sqrt(b)))*a*b*c*d**2*e**2*x**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**2*c*d**4 - 2*sqrt(d)*sqrt(a*e - b*d)*atan((sq 
rt(a*e - b*d) - sqrt(e)*sqrt(a + b*x**2) - sqrt(e)*sqrt(b)*x)/(sqrt(d)*sqr 
t(b)))*b**2*c*d**3*e*x**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**2*b* 
d*e**3 - 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**2*b*e**4*x**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*d**2*e**2 + sqrt(d)*sqrt(a*e - b...