\(\int \frac {A+B x^2+C x^4}{(d+e x^2)^{3/2} (a+c x^4)} \, dx\) [17]

Optimal result

Integrand size = 33, antiderivative size = 481 \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{3/2} \left (a+c x^4\right )} \, dx=\frac {\left (C d^2-B d e+A e^2\right ) x}{d \left (c d^2+a e^2\right ) \sqrt {d+e x^2}}+\frac {\sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (d (B c d-A c e+a C e)-(A c d-a C d+a B e) \left (e-\frac {\sqrt {c d^2+a e^2}}{\sqrt {a}}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e+\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c} d \left (c d^2+a e^2\right )^{3/2}}+\frac {\sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (d (B c d-A c e+a C e)-(A c d-a C d+a B e) \left (e+\frac {\sqrt {c d^2+a e^2}}{\sqrt {a}}\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e-\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c} d \left (c d^2+a e^2\right )^{3/2}} \] Output:

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.67 (sec) , antiderivative size = 549, normalized size of antiderivative = 1.14 \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{3/2} \left (a+c x^4\right )} \, dx=\frac {\frac {2 \left (C d^2+e (-B d+A e)\right ) x}{d \sqrt {d+e x^2}}+\sqrt {e} \text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2+16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {B c d^3 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )-A c d^2 e \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+a C d^2 e \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )-2 B c d^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+6 A c d e \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}-6 a C d e \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+4 a B e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+B c d \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-A c e \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+a C e \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{c d^3-3 c d^2 \text {$\#$1}-8 a e^2 \text {$\#$1}+3 c d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]}{2 \left (c d^2+a e^2\right )} \] Input:

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

Output:

((2*(C*d^2 + e*(-(B*d) + A*e))*x)/(d*Sqrt[d + e*x^2]) + Sqrt[e]*RootSum[c* 
d^4 - 4*c*d^3*#1 + 6*c*d^2*#1^2 + 16*a*e^2*#1^2 - 4*c*d*#1^3 + c*#1^4 & , 
(B*c*d^3*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - A*c*d^2*e*L 
og[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + a*C*d^2*e*Log[d + 2*e 
*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 2*B*c*d^2*Log[d + 2*e*x^2 - 2*S 
qrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 6*A*c*d*e*Log[d + 2*e*x^2 - 2*Sqrt[e]* 
x*Sqrt[d + e*x^2] - #1]*#1 - 6*a*C*d*e*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[ 
d + e*x^2] - #1]*#1 + 4*a*B*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x 
^2] - #1]*#1 + B*c*d*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*# 
1^2 - A*c*e*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1^2 + a*C 
*e*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1^2)/(c*d^3 - 3*c* 
d^2*#1 - 8*a*e^2*#1 + 3*c*d*#1^2 - c*#1^3) & ])/(2*(c*d^2 + a*e^2))
 

Rubi [A] (verified)

Time = 1.00 (sec) , antiderivative size = 356, normalized size of antiderivative = 0.74, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {2257, 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.

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

\(\Big \downarrow \) 2257

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

\(\Big \downarrow \) 2009

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

Input:

Int[(A + B*x^2 + C*x^4)/((d + e*x^2)^(3/2)*(a + c*x^4)),x]
 

Output:

(C*x)/(c*d*Sqrt[d + e*x^2]) - ((Sqrt[-a]*B*Sqrt[c] + A*c - a*C)*e*x)/(2*c* 
d*(Sqrt[-a]*Sqrt[c]*d - a*e)*Sqrt[d + e*x^2]) - ((Sqrt[-a]*B*Sqrt[c] - A*c 
 + a*C)*e*x)/(2*c*d*(Sqrt[-a]*Sqrt[c]*d + a*e)*Sqrt[d + e*x^2]) + ((Sqrt[- 
a]*B*Sqrt[c] - A*c + a*C)*ArcTan[(Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*x)/((-a)^(1 
/4)*Sqrt[d + e*x^2])])/(2*(-a)^(3/4)*Sqrt[c]*(Sqrt[c]*d - Sqrt[-a]*e)^(3/2 
)) - ((Sqrt[-a]*B*Sqrt[c] + A*c - a*C)*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[-a]* 
e]*x)/((-a)^(1/4)*Sqrt[d + e*x^2])])/(2*(-a)^(3/4)*Sqrt[c]*(Sqrt[c]*d + Sq 
rt[-a]*e)^(3/2))
 

Defintions of rubi rules used

rule 2009

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

rule 2257

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol 
] :> Int[ExpandIntegrand[Px*(d + e*x^2)^q*(a + c*x^4)^p, x], x] /; FreeQ[{a 
, c, d, e, q}, x] && PolyQ[Px, x] && IntegerQ[p]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(945\) vs. \(2(399)=798\).

Time = 0.87 (sec) , antiderivative size = 946, normalized size of antiderivative = 1.97

method	result	size
pseudoelliptic	\(-\frac {\sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}\, \sqrt {e \,x^{2}+d}\, \left (\ln \left (\frac {\sqrt {e \,x^{2}+d}\, \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x -x^{2} \sqrt {a \,e^{2}+c \,d^{2}}-\sqrt {a}\, \left (e \,x^{2}+d \right )}{x^{2}}\right )-\ln \left (\frac {\sqrt {a}\, \left (e \,x^{2}+d \right )+x^{2} \sqrt {a \,e^{2}+c \,d^{2}}+\sqrt {e \,x^{2}+d}\, \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x}{x^{2}}\right )\right ) \left (\left (\frac {\left (\left (A c -C a \right ) d +B a e \right ) \sqrt {a \,e^{2}+c \,d^{2}}}{2}+\left (\frac {1}{2} B \,e^{2}-C d e \right ) a^{\frac {3}{2}}+c d \sqrt {a}\, \left (A e -\frac {B d}{2}\right )\right ) \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-\left (\frac {\left (\left (A c -C a \right ) d +B a e \right ) a \sqrt {a \,e^{2}+c \,d^{2}}}{2}+c d \left (A e -\frac {B d}{2}\right ) a^{\frac {3}{2}}+\frac {a^{\frac {5}{2}} e \left (B e -2 C d \right )}{2}\right ) e \right ) \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}-4 c \left (\left (-\frac {d \left (\arctan \left (\frac {2 \sqrt {a}\, \sqrt {e \,x^{2}+d}+\sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x}{x \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}}\right )-\arctan \left (\frac {\sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x -2 \sqrt {a}\, \sqrt {e \,x^{2}+d}}{x \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}}\right )\right ) \left (\left (A c -C a \right ) d +B a e \right ) a \sqrt {e \,x^{2}+d}}{2}+\sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}\, a^{\frac {3}{2}} x \left (A \,e^{2}-B d e +C \,d^{2}\right )\right ) \sqrt {a \,e^{2}+c \,d^{2}}+\left (c d \left (A e -\frac {B d}{2}\right ) a^{\frac {3}{2}}+\frac {a^{\frac {5}{2}} e \left (B e -2 C d \right )}{2}\right ) d \left (\arctan \left (\frac {2 \sqrt {a}\, \sqrt {e \,x^{2}+d}+\sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x}{x \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}}\right )-\arctan \left (\frac {\sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x -2 \sqrt {a}\, \sqrt {e \,x^{2}+d}}{x \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}}\right )\right ) \sqrt {e \,x^{2}+d}\right ) d}{4 \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}\, a^{\frac {3}{2}} \sqrt {e \,x^{2}+d}\, \left (a \,e^{2}+c \,d^{2}\right )^{\frac {3}{2}} c \,d^{2}}\)	\(946\)
default	\(\text {Expression too large to display}\)	\(2196\)

Input:

int((C*x^4+B*x^2+A)/(e*x^2+d)^(3/2)/(c*x^4+a),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{3/2} \left (a+c x^4\right )} \, dx=\left (\int \frac {x^{4}}{\sqrt {e \,x^{2}+d}\, a d +\sqrt {e \,x^{2}+d}\, a e \,x^{2}+\sqrt {e \,x^{2}+d}\, c d \,x^{4}+\sqrt {e \,x^{2}+d}\, c e \,x^{6}}d x \right ) c +\left (\int \frac {x^{2}}{\sqrt {e \,x^{2}+d}\, a d +\sqrt {e \,x^{2}+d}\, a e \,x^{2}+\sqrt {e \,x^{2}+d}\, c d \,x^{4}+\sqrt {e \,x^{2}+d}\, c e \,x^{6}}d x \right ) b +\left (\int \frac {1}{\sqrt {e \,x^{2}+d}\, a d +\sqrt {e \,x^{2}+d}\, a e \,x^{2}+\sqrt {e \,x^{2}+d}\, c d \,x^{4}+\sqrt {e \,x^{2}+d}\, c e \,x^{6}}d x \right ) a \] Input:

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

Output:

int(x**4/(sqrt(d + e*x**2)*a*d + sqrt(d + e*x**2)*a*e*x**2 + sqrt(d + e*x* 
*2)*c*d*x**4 + sqrt(d + e*x**2)*c*e*x**6),x)*c + int(x**2/(sqrt(d + e*x**2 
)*a*d + sqrt(d + e*x**2)*a*e*x**2 + sqrt(d + e*x**2)*c*d*x**4 + sqrt(d + e 
*x**2)*c*e*x**6),x)*b + int(1/(sqrt(d + e*x**2)*a*d + sqrt(d + e*x**2)*a*e 
*x**2 + sqrt(d + e*x**2)*c*d*x**4 + sqrt(d + e*x**2)*c*e*x**6),x)*a