\(\int \frac {A+B x^2+C x^4}{(d+e x^2)^{5/2} (a+c x^4)^2} \, dx\) [24]

Optimal result

Integrand size = 33, antiderivative size = 928 \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{5/2} \left (a+c x^4\right )^2} \, dx=\frac {e \left (3 B c d^3-6 A c d^2 e+10 a C d^2 e-7 a B d e^2+4 a A e^3\right ) x}{12 a d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )^{3/2}}+\frac {e \left (a C d^2 e \left (41 c d^2-19 a e^2\right )+B \left (3 c^2 d^5-53 a c d^3 e^2+4 a^2 d e^4\right )-A \left (9 c^2 d^4 e-59 a c d^2 e^3-8 a^2 e^5\right )\right ) x}{12 a d^2 \left (c d^2+a e^2\right )^3 \sqrt {d+e x^2}}+\frac {x \left (A c d-a C d+a B e+(B c d-A c e+a C e) x^2\right )}{4 a \left (c d^2+a e^2\right ) \left (d+e x^2\right )^{3/2} \left (a+c x^4\right )}-\frac {\sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (\left (e-\frac {\sqrt {c d^2+a e^2}}{\sqrt {a}}\right ) \left (A c \left (3 c^2 d^4+15 a c d^2 e^2-8 a^2 e^4\right )+a \left (4 a^2 C e^4-a c d e^2 (15 C d-17 B e)+c^2 d^3 (C d-3 B e)\right )\right )-c d \left (B \left (c^2 d^4+15 a c d^2 e^2-6 a^2 e^4\right )-d e \left (a C \left (7 c d^2-13 a e^2\right )+A c \left (c d^2+21 a e^2\right )\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e+\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{8 \sqrt {2} a^{5/4} \sqrt {c} d \left (c d^2+a e^2\right )^{7/2}}-\frac {\sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (\left (e+\frac {\sqrt {c d^2+a e^2}}{\sqrt {a}}\right ) \left (A c \left (3 c^2 d^4+15 a c d^2 e^2-8 a^2 e^4\right )+a \left (4 a^2 C e^4-a c d e^2 (15 C d-17 B e)+c^2 d^3 (C d-3 B e)\right )\right )-c d \left (B \left (c^2 d^4+15 a c d^2 e^2-6 a^2 e^4\right )-d e \left (a C \left (7 c d^2-13 a e^2\right )+A c \left (c d^2+21 a e^2\right )\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e-\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{8 \sqrt {2} a^{5/4} \sqrt {c} d \left (c d^2+a e^2\right )^{7/2}} \] Output:

1/12*e*(4*A*a*e^3-6*A*c*d^2*e-7*B*a*d*e^2+3*B*c*d^3+10*C*a*d^2*e)*x/a/d/(a 
*e^2+c*d^2)^2/(e*x^2+d)^(3/2)+1/12*e*(a*C*d^2*e*(-19*a*e^2+41*c*d^2)+B*(4* 
a^2*d*e^4-53*a*c*d^3*e^2+3*c^2*d^5)-A*(-8*a^2*e^5-59*a*c*d^2*e^3+9*c^2*d^4 
*e))*x/a/d^2/(a*e^2+c*d^2)^3/(e*x^2+d)^(1/2)+1/4*x*(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)/(e*x^2+d)^(3/2)/(c*x^4+a)-1/16*(a^(1 
/2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*((e-(a*e^2+c*d^2)^(1/2)/a^(1/2))*(A*c*(-8 
*a^2*e^4+15*a*c*d^2*e^2+3*c^2*d^4)+a*(4*a^2*C*e^4-a*c*d*e^2*(-17*B*e+15*C* 
d)+c^2*d^3*(-3*B*e+C*d)))-c*d*(B*(-6*a^2*e^4+15*a*c*d^2*e^2+c^2*d^4)-d*e*( 
a*C*(-13*a*e^2+7*c*d^2)+A*c*(21*a*e^2+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)^(7/2)-1/16*(-a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*((e+(a*e^2+c*d^2)^(1 
/2)/a^(1/2))*(A*c*(-8*a^2*e^4+15*a*c*d^2*e^2+3*c^2*d^4)+a*(4*a^2*C*e^4-a*c 
*d*e^2*(-17*B*e+15*C*d)+c^2*d^3*(-3*B*e+C*d)))-c*d*(B*(-6*a^2*e^4+15*a*c*d 
^2*e^2+c^2*d^4)-d*e*(a*C*(-13*a*e^2+7*c*d^2)+A*c*(21*a*e^2+c*d^2))))*arcta 
nh(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)^(7/2)
 

Mathematica [C] (verified)

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

Time = 4.90 (sec) , antiderivative size = 2576, normalized size of antiderivative = 2.78 \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{5/2} \left (a+c x^4\right )^2} \, dx=\text {Result too large to show} \] Input:

Integrate[(A + B*x^2 + C*x^4)/((d + e*x^2)^(5/2)*(a + c*x^4)^2),x]
 

Output:

((2*x*(A*(3*c^3*d^4*(d - 3*e*x^2)*(d + e*x^2)^2 + 4*a^3*e^6*(3*d + 2*e*x^2 
) + 4*a^2*c*e^4*(15*d^3 + 14*d^2*e*x^2 + 3*d*e^2*x^4 + 2*e^3*x^6) + a*c^2* 
d^2*e^2*(-9*d^3 - 15*d^2*e*x^2 + 57*d*e^2*x^4 + 59*e^3*x^6)) + d*(3*B*c^3* 
d^4*x^2*(d + e*x^2)^2 + 4*a^3*e^4*(B*e^2*x^2 - C*d*(3*d + 4*e*x^2)) + a^2* 
c*e^2*(C*d*(45*d^3 + 47*d^2*e*x^2 - 9*d*e^2*x^4 - 19*e^3*x^6) + B*e*(-51*d 
^3 - 50*d^2*e*x^2 - 3*d*e^2*x^4 + 4*e^3*x^6)) + a*c^2*d^2*(B*e*(9*d^3 + 9* 
d^2*e*x^2 - 57*d*e^2*x^4 - 53*e^3*x^6) + C*d*(-3*d^3 + 3*d^2*e*x^2 + 51*d* 
e^2*x^4 + 41*e^3*x^6)))))/(a*d^2*(d + e*x^2)^(3/2)*(a + c*x^4)) - 12*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 & , (2*c^2*C*d^5*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - 
#1] - 19*B*c^2*d^4*e*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 
 36*A*c^2*d^3*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 34 
*a*c*C*d^3*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + a*B*c 
*d^2*e^3*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 32*a*A*c*d* 
e^4*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 32*a^2*C*d*e^4*L 
og[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 16*a^2*B*e^5*Log[d + 
2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 8*c^2*C*d^4*Log[d + 2*e*x^2 
- 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 14*B*c^2*d^3*e*Log[d + 2*e*x^2 - 
2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 - 20*A*c^2*d^2*e^2*Log[d + 2*e*x^2 - 
2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 16*a*c*C*d^2*e^2*Log[d + 2*e*x^2...
 

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.

\(\displaystyle \int \frac {A+B x^2+C x^4}{\left (a+c x^4\right )^2 \left (d+e x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 2257

\(\displaystyle \int \left (\frac {-a C+A c+B c x^2}{c \left (a+c x^4\right )^2 \left (d+e x^2\right )^{5/2}}+\frac {C}{c \left (a+c x^4\right ) \left (d+e x^2\right )^{5/2}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(A c-a C) \int \frac {1}{\left (e x^2+d\right )^{5/2} \left (c x^4+a\right )^2}dx}{c}+B \int \frac {x^2}{\left (e x^2+d\right )^{5/2} \left (c x^4+a\right )^2}dx-\frac {C \left (\sqrt {-a} e+\sqrt {c} d\right ) \arctan \left (\frac {x \sqrt {\sqrt {c} d-\sqrt {-a} e}}{\sqrt [4]{-a} \sqrt {d+e x^2}}\right )}{2 (-a)^{3/4} \left (\sqrt {c} d-\sqrt {-a} e\right )^{3/2} \left (a e^2+c d^2\right )}-\frac {C \left (\sqrt {c} d-\sqrt {-a} e\right ) \text {arctanh}\left (\frac {x \sqrt {\sqrt {-a} e+\sqrt {c} d}}{\sqrt [4]{-a} \sqrt {d+e x^2}}\right )}{2 (-a)^{3/4} \left (\sqrt {-a} e+\sqrt {c} d\right )^{3/2} \left (a e^2+c d^2\right )}+\frac {2 C e^2 x}{3 c d^2 \sqrt {d+e x^2} \left (a e^2+c d^2\right )}+\frac {C e^2 x}{3 c d \left (d+e x^2\right )^{3/2} \left (a e^2+c d^2\right )}-\frac {C e x \left (\sqrt {c} d-\sqrt {-a} e\right )}{2 \sqrt {c} d \sqrt {d+e x^2} \left (\sqrt {-a} \sqrt {c} d-a e\right ) \left (a e^2+c d^2\right )}+\frac {C e x \left (\sqrt {-a} e+\sqrt {c} d\right )}{2 \sqrt {c} d \sqrt {d+e x^2} \left (\sqrt {-a} \sqrt {c} d+a e\right ) \left (a e^2+c d^2\right )}\)

Input:

Int[(A + B*x^2 + C*x^4)/((d + e*x^2)^(5/2)*(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. \(2296\) vs. \(2(836)=1672\).

Time = 20.85 (sec) , antiderivative size = 2297, normalized size of antiderivative = 2.48

method	result	size
pseudoelliptic	\(\text {Expression too large to display}\)	\(2297\)
default	\(\text {Expression too large to display}\)	\(9630\)

Input:

int((C*x^4+B*x^2+A)/(e*x^2+d)^(5/2)/(c*x^4+a)^2,x,method=_RETURNVERBOSE)
 

Output:

1/2/(e*x^2+d)^(3/2)/a^(5/2)*(-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)*(2*(a*(a*e^2+c*d^2))^(1/2)+2*a*e)^(1/2)*(e*x 
^2+d)^(3/2)*((-2*(c*x^4+a)*(a^2*(A*c-1/2*C*a)*e^4-17/8*B*a^2*c*d*e^3-15/8* 
a*c*d^2*(A*c-C*a)*e^2+3/8*B*a*c^2*d^3*e-3/8*c^2*d^4*(A*c+1/3*C*a))*(a*e^2+ 
c*d^2)^(1/2)+c^3*(9*A*e^3*x^4-9/2*B*e^2*d*x^4+d^2*(2*C*x^4+A)*e-1/4*B*d^3) 
*d^2*a^(3/2)+9*c^2*(-2/9*A*e^4*x^4+23/36*B*d*e^3*x^4+d^2*(-7/9*C*x^4+A)*e^ 
2-1/2*B*d^3*e+2/9*C*d^4)*e*a^(5/2)-2*c*e^3*((-1/2*C*x^4+A)*e^2-23/8*B*d*e+ 
7/2*C*d^2)*a^(7/2)+a^(9/2)*C*e^5+a^(1/2)*c^4*d^4*x^4*(-1/4*B*d+A*e))*(a*(a 
*e^2+c*d^2))^(1/2)-e*(-2*(c*x^4+a)*(a^2*(A*c-1/2*C*a)*e^4-17/8*B*a^2*c*d*e 
^3-15/8*a*c*d^2*(A*c-C*a)*e^2+3/8*B*a*c^2*d^3*e-3/8*c^2*d^4*(A*c+1/3*C*a)) 
*a*(a*e^2+c*d^2)^(1/2)+c^4*d^4*x^4*(-1/4*B*d+A*e)*a^(3/2)+c^3*(9*A*e^3*x^4 
-9/2*B*e^2*d*x^4+d^2*(2*C*x^4+A)*e-1/4*B*d^3)*d^2*a^(5/2)+9*e*(c^2*(-2/9*A 
*e^4*x^4+23/36*B*d*e^3*x^4+d^2*(-7/9*C*x^4+A)*e^2-1/2*B*d^3*e+2/9*C*d^4)*a 
^(7/2)-2/9*(c*((-1/2*C*x^4+A)*e^2-23/8*B*d*e+7/2*C*d^2)*a^(9/2)-1/2*a^(11/ 
2)*C*e^2)*e^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)+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)*(2*(a*(a*e^2+c*d^2))^ 
(1/2)+2*a*e)^(1/2)*(e*x^2+d)^(3/2)*((-2*(c*x^4+a)*(a^2*(A*c-1/2*C*a)*e^4-1 
7/8*B*a^2*c*d*e^3-15/8*a*c*d^2*(A*c-C*a)*e^2+3/8*B*a*c^2*d^3*e-3/8*c^2*d^4 
*(A*c+1/3*C*a))*(a*e^2+c*d^2)^(1/2)+c^3*(9*A*e^3*x^4-9/2*B*e^2*d*x^4+d^...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{5/2} \left (a+c x^4\right )^2} \, dx=\text {Timed out} \] Input:

integrate((C*x^4+B*x^2+A)/(e*x^2+d)^(5/2)/(c*x^4+a)^2,x, algorithm="fricas 
")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{5/2} \left (a+c x^4\right )^2} \, dx=\text {Timed out} \] Input:

integrate((C*x**4+B*x**2+A)/(e*x**2+d)**(5/2)/(c*x**4+a)**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{5/2} \left (a+c x^4\right )^2} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{{\left (c x^{4} + a\right )}^{2} {\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((C*x^4+B*x^2+A)/(e*x^2+d)^(5/2)/(c*x^4+a)^2,x, algorithm="maxima 
")
 

Output:

integrate((C*x^4 + B*x^2 + A)/((c*x^4 + a)^2*(e*x^2 + d)^(5/2)), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{5/2} \left (a+c x^4\right )^2} \, dx=\text {Timed out} \] Input:

integrate((C*x^4+B*x^2+A)/(e*x^2+d)^(5/2)/(c*x^4+a)^2,x, algorithm="giac")
 

Output:

Timed out
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{5/2} \left (a+c x^4\right )^2} \, dx=\int \frac {C\,x^4+B\,x^2+A}{{\left (c\,x^4+a\right )}^2\,{\left (e\,x^2+d\right )}^{5/2}} \,d x \] Input:

int((A + B*x^2 + C*x^4)/((a + c*x^4)^2*(d + e*x^2)^(5/2)),x)
 

Output:

int((A + B*x^2 + C*x^4)/((a + c*x^4)^2*(d + e*x^2)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^{5/2} \left (a+c x^4\right )^2} \, dx=\int \frac {C \,x^{4}+B \,x^{2}+A}{\left (e \,x^{2}+d \right )^{\frac {5}{2}} \left (c \,x^{4}+a \right )^{2}}d x \] Input:

int((C*x^4+B*x^2+A)/(e*x^2+d)^(5/2)/(c*x^4+a)^2,x)
 

Output:

int((C*x^4+B*x^2+A)/(e*x^2+d)^(5/2)/(c*x^4+a)^2,x)