\(\int \frac {\sqrt {d+e x^2} (A+B x^2+C x^4)}{(a+c x^4)^2} \, dx\) [21]

Optimal result

Integrand size = 33, antiderivative size = 459 \[ \int \frac {\sqrt {d+e x^2} \left (A+B x^2+C x^4\right )}{\left (a+c x^4\right )^2} \, dx=\frac {x \left (A c-a C+B c x^2\right ) \sqrt {d+e x^2}}{4 a c \left (a+c x^4\right )}+\frac {\sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (a^{3/2} C e+\sqrt {a} c (B d-A e)+(3 A c+a C) \sqrt {c d^2+a e^2}\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e+\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{8 \sqrt {2} a^{7/4} c^{3/2} \sqrt {c d^2+a e^2}}+\frac {\sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (a^{3/2} C e+\sqrt {a} c (B d-A e)-(3 A c+a C) \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e-\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{8 \sqrt {2} a^{7/4} c^{3/2} \sqrt {c d^2+a e^2}} \] Output:

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

Mathematica [C] (verified)

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

Time = 1.61 (sec) , antiderivative size = 1059, normalized size of antiderivative = 2.31 \[ \int \frac {\sqrt {d+e x^2} \left (A+B x^2+C x^4\right )}{\left (a+c x^4\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(4*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 & , (c*C*d^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2 
] - #1] + 16*B*c*d*e*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 
 16*A*c*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 16*a*C*e 
^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 2*c*C*d*Log[d + 2 
*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 4*B*c*e*Log[d + 2*e*x^2 - 
2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + c*C*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*S 
qrt[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*x*(A*c - a*C + B*c*x^2)*Sqrt[d + e*x^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^3*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^ 
2] - #1] + 2*A*c^2*d^2*e*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - # 
1] - 2*a*c*C*d^2*e*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 6 
4*a*B*c*d*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 64*a*A 
*c*e^3*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 64*a^2*C*e^3* 
Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 2*B*c^2*d^2*Log[d + 
2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 8*A*c^2*d*e*Log[d + 2*e*x 
^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 - 8*a*c*C*d*e*Log[d + 2*e*x^2 - 
2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 - 16*a*B*c*e^2*Log[d + 2*e*x^2 - 2*Sq 
rt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + B*c^2*d*Log[d + 2*e*x^2 - 2*Sqrt[e]*...
 

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.

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

\(\Big \downarrow \) 2257

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

\(\Big \downarrow \) 2009

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

Input:

Int[(Sqrt[d + e*x^2]*(A + B*x^2 + C*x^4))/(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. \(977\) vs. \(2(371)=742\).

Time = 0.64 (sec) , antiderivative size = 978, normalized size of antiderivative = 2.13

method	result	size
pseudoelliptic	\(-\frac {\frac {\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 (3 \left (A c +\frac {C a}{3}\right ) \left (c \,x^{4}+a \right ) \sqrt {a \,e^{2}+c \,d^{2}}+c \left (-e C \,x^{4}+A e -B d \right ) a^{\frac {3}{2}}-a^{\frac {5}{2}} C e +\sqrt {a}\, c^{2} x^{4} \left (A e -B d \right )\right ) \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-\left (3 \left (A c +\frac {C a}{3}\right ) \left (c \,x^{4}+a \right ) a \sqrt {a \,e^{2}+c \,d^{2}}+c^{2} x^{4} \left (A e -B d \right ) a^{\frac {3}{2}}+c \left (-e C \,x^{4}+A e -B d \right ) a^{\frac {5}{2}}-C \,a^{\frac {7}{2}} e \right ) e \right ) \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 {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}}{4}+c d \left (\left (-2 x \left (c \left (B \,x^{2}+A \right ) a^{\frac {3}{2}}-a^{\frac {5}{2}} C \right ) \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}-3 \left (\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 )-\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 )\right ) d \left (A c +\frac {C a}{3}\right ) \left (c \,x^{4}+a \right ) a \right ) \sqrt {a \,e^{2}+c \,d^{2}}+\left (\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 )-\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 )\right ) d \left (c^{2} x^{4} \left (A e -B d \right ) a^{\frac {3}{2}}+c \left (-e C \,x^{4}+A e -B d \right ) a^{\frac {5}{2}}-C \,a^{\frac {7}{2}} e \right )\right )}{8 \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 {5}{2}} \sqrt {a \,e^{2}+c \,d^{2}}\, d \,c^{2} \left (c \,x^{4}+a \right )}\)	\(978\)
default	\(\text {Expression too large to display}\)	\(5584\)

Input:

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

Output:

-1/8/(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)*(1/4*(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))*((3*(A*c+1/3*C*a)*(c*x^4+a)*(a*e^2+c*d^2)^(1/2)+c*(-C*e*x^4+ 
A*e-B*d)*a^(3/2)-a^(5/2)*C*e+a^(1/2)*c^2*x^4*(A*e-B*d))*(a*(a*e^2+c*d^2))^ 
(1/2)-(3*(A*c+1/3*C*a)*(c*x^4+a)*a*(a*e^2+c*d^2)^(1/2)+c^2*x^4*(A*e-B*d)*a 
^(3/2)+c*(-C*e*x^4+A*e-B*d)*a^(5/2)-C*a^(7/2)*e)*e)*(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)+c*d*((-2*x*(c*(B*x^2+A)*a^(3/2)-a^(5/2)*C)*(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)-3*(ar 
ctan(((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))- 
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) 
))*d*(A*c+1/3*C*a)*(c*x^4+a)*a)*(a*e^2+c*d^2)^(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))-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)))*d*(c^2*x^4*(A...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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