\(\int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} (a+c x^4)^2} \, dx\) [22]

Optimal result

Integrand size = 33, antiderivative size = 557 \[ \int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} \left (a+c x^4\right )^2} \, dx=\frac {x \sqrt {d+e x^2} \left (A c d-a C d+a B e+(B c d-A c e+a C e) x^2\right )}{4 a \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac {\sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (d \left (B c d^2+A c d e-a C d e+2 a B e^2\right )-\left (e-\frac {\sqrt {c d^2+a e^2}}{\sqrt {a}}\right ) \left (a d (C d-B e)+A \left (3 c d^2+4 a e^2\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e+\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{8 \sqrt {2} a^{5/4} \sqrt {c} d \left (c d^2+a e^2\right )^{3/2}}+\frac {\sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (d \left (B c d^2+A c d e-a C d e+2 a B e^2\right )-\left (e+\frac {\sqrt {c d^2+a e^2}}{\sqrt {a}}\right ) \left (a d (C d-B e)+A \left (3 c d^2+4 a e^2\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e-\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{8 \sqrt {2} a^{5/4} \sqrt {c} d \left (c d^2+a e^2\right )^{3/2}} \] Output:

1/4*x*(e*x^2+d)^(1/2)*(A*c*d-C*a*d+B*a*e+(-A*c*e+B*c*d+C*a*e)*x^2)/a/(a*e^ 
2+c*d^2)/(c*x^4+a)+1/16*(a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*(d*(A*c*d*e+ 
2*B*a*e^2+B*c*d^2-C*a*d*e)-(e-(a*e^2+c*d^2)^(1/2)/a^(1/2))*(a*d*(-B*e+C*d) 
+A*(4*a*e^2+3*c*d^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^(5/4)/c^(1/2)/d/(a*e^2+c*d^2)^(3/2)+1/16*(-a^(1/2)* 
e+(a*e^2+c*d^2)^(1/2))^(1/2)*(d*(A*c*d*e+2*B*a*e^2+B*c*d^2-C*a*d*e)-(e+(a* 
e^2+c*d^2)^(1/2)/a^(1/2))*(a*d*(-B*e+C*d)+A*(4*a*e^2+3*c*d^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^(5/4)/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 = 1.67 (sec) , antiderivative size = 1010, normalized size of antiderivative = 1.81 \[ \int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} \left (a+c x^4\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(x*Sqrt[d + e*x^2]*(A*c*d - a*C*d + a*B*e + B*c*d*x^2 - A*c*e*x^2 + a*C*e* 
x^2))/(4*a*(c*d^2 + a*e^2)*(a + c*x^4)) + (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^2*d^4*Lo 
g[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + A*c^2*d^3*e*Log[d + 2* 
e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - a*c*C*d^3*e*Log[d + 2*e*x^2 - 
2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 62*a*B*c*d^2*e^2*Log[d + 2*e*x^2 - 2*S 
qrt[e]*x*Sqrt[d + e*x^2] - #1] - 64*a^2*B*e^4*Log[d + 2*e*x^2 - 2*Sqrt[e]* 
x*Sqrt[d + e*x^2] - #1] - 2*B*c^2*d^3*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d 
 + e*x^2] - #1]*#1 + 10*A*c^2*d^2*e*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + 
 e*x^2] - #1]*#1 - 10*a*c*C*d^2*e*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e 
*x^2] - #1]*#1 - 8*a*B*c*d*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^ 
2] - #1]*#1 + 16*a*A*c*e^3*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - 
 #1]*#1 - 16*a^2*C*e^3*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] 
*#1 + B*c^2*d^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1^2 + 
 A*c^2*d*e*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1^2 - a*c* 
C*d*e*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1^2 + 2*a*B*c*e 
^2*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) & ])/(8*a*c*(c*d^2 + a*e^2)) - 
(2*e^(3/2)*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 & , (4*B*e*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^...
 

Rubi [F]

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.

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

\(\Big \downarrow \) 2257

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(A c-a C) \int \frac {1}{\sqrt {e x^2+d} \left (c x^4+a\right )^2}dx}{c}+B \int \frac {x^2}{\sqrt {e x^2+d} \left (c x^4+a\right )^2}dx-\frac {C \arctan \left (\frac {x \sqrt {\sqrt {c} d-\sqrt {-a} e}}{\sqrt [4]{-a} \sqrt {d+e x^2}}\right )}{2 (-a)^{3/4} c \sqrt {\sqrt {c} d-\sqrt {-a} e}}-\frac {C \text {arctanh}\left (\frac {x \sqrt {\sqrt {-a} e+\sqrt {c} d}}{\sqrt [4]{-a} \sqrt {d+e x^2}}\right )}{2 (-a)^{3/4} c \sqrt {\sqrt {-a} e+\sqrt {c} d}}\)

Input:

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

Output:

$Aborted
 

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. \(1178\) vs. \(2(473)=946\).

Time = 3.16 (sec) , antiderivative size = 1179, normalized size of antiderivative = 2.12

method	result	size
pseudoelliptic	\(\text {Expression too large to display}\)	\(1179\)
default	\(\text {Expression too large to display}\)	\(2750\)

Input:

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

Output:

-1/16/(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^(5/2)/(a*e^2+c*d^2)^(3/2)*(((2*(1/4*(3*A*c+C*a)*d^2-1/4*a*B*d*e+A*a*e^ 
2)*(c*x^4+a)*(a*e^2+c*d^2)^(1/2)+c*(-1/2*B*d^3+e*(C*x^4+A)*d^2-3/2*B*e^2*d 
*x^4+2*A*e^3*x^4)*a^(3/2)+(2*A*e^3-3/2*B*d*e^2+C*d^2*e)*a^(5/2)+c^2*d^2*x^ 
4*a^(1/2)*(A*e-1/2*B*d))*(a*(a*e^2+c*d^2))^(1/2)-e*(2*(1/4*(3*A*c+C*a)*d^2 
-1/4*a*B*d*e+A*a*e^2)*(c*x^4+a)*a*(a*e^2+c*d^2)^(1/2)+c*(-1/2*B*d^3+e*(C*x 
^4+A)*d^2-3/2*B*e^2*d*x^4+2*A*e^3*x^4)*a^(5/2)+c^2*d^2*x^4*(A*e-1/2*B*d)*a 
^(3/2)+2*(A*e^2-3/4*B*d*e+1/2*C*d^2)*e*a^(7/2)))*(ln((a^(1/2)*(e*x^2+d)-(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))/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))*(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)*(2*(a*(a*e^2+c*d^2))^(1/2)+2* 
a*e)^(1/2)-4*c*d^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)*((-C*d+(C*x^2+B)*e)*a^(5/2)+c*((B*x^2+A)*d-A*e*x^2)*a^( 
3/2))*(e*x^2+d)^(1/2)-2*(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)))*(1/4*(3*A*c+C*a)*d^2-1/4*a*B*d*e+A*a*e^2)*(c 
*x^4+a)*a)*(a*e^2+c*d^2)^(1/2)+(arctan((2*a^(1/2)*(e*x^2+d)^(1/2)+(2*(a...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} \left (a+c x^4\right )^2} \, dx=\left (\int \frac {x^{4}}{\sqrt {e \,x^{2}+d}\, a^{2}+2 \sqrt {e \,x^{2}+d}\, a c \,x^{4}+\sqrt {e \,x^{2}+d}\, c^{2} x^{8}}d x \right ) c +\left (\int \frac {x^{2}}{\sqrt {e \,x^{2}+d}\, a^{2}+2 \sqrt {e \,x^{2}+d}\, a c \,x^{4}+\sqrt {e \,x^{2}+d}\, c^{2} x^{8}}d x \right ) b +\left (\int \frac {1}{\sqrt {e \,x^{2}+d}\, a^{2}+2 \sqrt {e \,x^{2}+d}\, a c \,x^{4}+\sqrt {e \,x^{2}+d}\, c^{2} x^{8}}d x \right ) a \] Input:

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

Output:

int(x**4/(sqrt(d + e*x**2)*a**2 + 2*sqrt(d + e*x**2)*a*c*x**4 + sqrt(d + e 
*x**2)*c**2*x**8),x)*c + int(x**2/(sqrt(d + e*x**2)*a**2 + 2*sqrt(d + e*x* 
*2)*a*c*x**4 + sqrt(d + e*x**2)*c**2*x**8),x)*b + int(1/(sqrt(d + e*x**2)* 
a**2 + 2*sqrt(d + e*x**2)*a*c*x**4 + sqrt(d + e*x**2)*c**2*x**8),x)*a