\(\int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} (a+b x^2+c x^4)^2} \, dx\) [163]

Optimal result

Integrand size = 38, antiderivative size = 833 \[ \int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} \left (a+b x^2+c x^4\right )^2} \, dx=\frac {x \sqrt {d+e x^2} \left (A \left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e\right )-a \left (b B c d-2 a c C d-b^2 B e+2 a B c e+a b C e\right )+c \left (A \left (b c d-b^2 e+2 a c e\right )-a (2 B c d-b C d-b B e+2 a C e)\right ) x^2\right )}{2 a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x^2+c x^4\right )}+\frac {\left ((A c-a C) (b d-2 a e) (c d-b e)-a (B c-b C) \left (2 c d^2-e (3 b d-4 a e)\right )-\frac {A c \left (b^3 d e-12 a b c d e-b^2 \left (c d^2+2 a e^2\right )+4 a c \left (3 c d^2+4 a e^2\right )\right )-a \left (2 b^3 C d e-4 a c^2 d (C d-B e)+4 b B c \left (c d^2+a e^2\right )-b^2 \left (2 a C e^2+c d (C d+5 B e)\right )\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{2 a \left (b^2-4 a c\right ) \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d^2-e (b d-a e)\right )}+\frac {\left ((A c-a C) (b d-2 a e) (c d-b e)-a (B c-b C) \left (2 c d^2-e (3 b d-4 a e)\right )+\frac {A c \left (b^3 d e-12 a b c d e-b^2 \left (c d^2+2 a e^2\right )+4 a c \left (3 c d^2+4 a e^2\right )\right )-a \left (2 b^3 C d e-4 a c^2 d (C d-B e)+4 b B c \left (c d^2+a e^2\right )-b^2 \left (2 a C e^2+c d (C d+5 B e)\right )\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{2 a \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d^2-e (b d-a e)\right )} \] Output:

1/2*x*(e*x^2+d)^(1/2)*(A*(3*a*b*c*e-2*a*c^2*d-b^3*e+b^2*c*d)-a*(2*B*a*c*e- 
B*b^2*e+B*b*c*d+C*a*b*e-2*C*a*c*d)+c*(A*(2*a*c*e-b^2*e+b*c*d)-a*(-B*b*e+2* 
B*c*d+2*C*a*e-C*b*d))*x^2)/a/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)/(c*x^4+b*x^2 
+a)+1/2*((A*c-C*a)*(-2*a*e+b*d)*(-b*e+c*d)-a*(B*c-C*b)*(2*c*d^2-e*(-4*a*e+ 
3*b*d))-(A*c*(b^3*d*e-12*a*b*c*d*e-b^2*(2*a*e^2+c*d^2)+4*a*c*(4*a*e^2+3*c* 
d^2))-a*(2*b^3*C*d*e-4*a*c^2*d*(-B*e+C*d)+4*b*B*c*(a*e^2+c*d^2)-b^2*(2*C*a 
*e^2+c*d*(5*B*e+C*d))))/(-4*a*c+b^2)^(1/2))*arctan((2*c*d-(b-(-4*a*c+b^2)^ 
(1/2))*e)^(1/2)*x/(b-(-4*a*c+b^2)^(1/2))^(1/2)/(e*x^2+d)^(1/2))/a/(-4*a*c+ 
b^2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)^(1/2)/( 
c*d^2-e*(-a*e+b*d))+1/2*((A*c-C*a)*(-2*a*e+b*d)*(-b*e+c*d)-a*(B*c-C*b)*(2* 
c*d^2-e*(-4*a*e+3*b*d))+(A*c*(b^3*d*e-12*a*b*c*d*e-b^2*(2*a*e^2+c*d^2)+4*a 
*c*(4*a*e^2+3*c*d^2))-a*(2*b^3*C*d*e-4*a*c^2*d*(-B*e+C*d)+4*b*B*c*(a*e^2+c 
*d^2)-b^2*(2*C*a*e^2+c*d*(5*B*e+C*d))))/(-4*a*c+b^2)^(1/2))*arctan((2*c*d- 
(b+(-4*a*c+b^2)^(1/2))*e)^(1/2)*x/(b+(-4*a*c+b^2)^(1/2))^(1/2)/(e*x^2+d)^( 
1/2))/a/(-4*a*c+b^2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)/(2*c*d-(b+(-4*a*c+b^2)^( 
1/2))*e)^(1/2)/(c*d^2-e*(-a*e+b*d))
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(37781\) vs. \(2(833)=1666\).

Time = 22.16 (sec) , antiderivative size = 37781, normalized size of antiderivative = 45.36 \[ \int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} \left (a+b x^2+c x^4\right )^2} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

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.

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

\(\Big \downarrow \) 2256

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(A c-a C) \int \frac {1}{\sqrt {e x^2+d} \left (c x^4+b x^2+a\right )^2}dx}{c}+\frac {(B c-b C) \int \frac {x^2}{\sqrt {e x^2+d} \left (c x^4+b x^2+a\right )^2}dx}{c}+\frac {2 C \arctan \left (\frac {x \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {2 C \arctan \left (\frac {x \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {d+e x^2}}\right )}{\sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\)

Input:

Int[(A + B*x^2 + C*x^4)/(Sqrt[d + e*x^2]*(a + b*x^2 + 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 2256

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

Maple [A] (verified)

Time = 8.88 (sec) , antiderivative size = 872, normalized size of antiderivative = 1.05

method	result	size
pseudoelliptic	\(-\frac {\left (\frac {3 \sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, \left (\frac {2 \left (C c d -e \left (\frac {b C}{2}+c \left (C \,x^{2}+B \right )\right )\right ) a^{2}}{3}+\left (-\frac {2 c \left (\frac {\left (-C \,x^{2}+B \right ) b}{2}+c \left (B \,x^{2}+A \right )\right ) d}{3}+e \left (\frac {B \,b^{2}}{3}+c \left (\frac {B \,x^{2}}{3}+A \right ) b +\frac {2 A \,c^{2} x^{2}}{3}\right )\right ) a -\frac {A b \left (c \,x^{2}+b \right ) \left (e b -c d \right )}{3}\right ) \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, x \sqrt {e \,x^{2}+d}}{4}+\left (c \,x^{4}+b \,x^{2}+a \right ) \left (\left (\frac {C c \,d^{2}}{4}-\frac {e \left (B c +\frac {b C}{2}\right ) d}{4}+A c \,e^{2}\right ) a^{2}+\frac {\left (\left (-\frac {1}{2} b B c +3 A \,c^{2}\right ) d^{2}-\frac {5 b e \left (A c -\frac {B b}{5}\right ) d}{2}-A \,b^{2} e^{2}\right ) a}{4}+\frac {A d \,b^{2} \left (e b -c d \right )}{8}\right ) \left (\arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}}\right ) \sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}-\operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}}\right ) \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\right ) \sqrt {2}\right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}+\frac {3 \left (\frac {\left (C c d e -2 e^{2} \left (B c -\frac {b C}{2}\right )\right ) a^{3}}{3}+\left (-\frac {c \left (B c -b C \right ) d^{2}}{3}-\frac {\left (A \,c^{2}-b B c +\frac {5}{4} b^{2} C \right ) e d}{3}+A b c \,e^{2}\right ) a^{2}-\frac {b \left (\left (-4 A \,c^{2}+\frac {1}{2} b B c \right ) d^{2}+\frac {7 b e \left (A c -\frac {B b}{7}\right ) d}{2}+A \,b^{2} e^{2}\right ) a}{6}+\frac {A \,b^{3} d \left (e b -c d \right )}{12}\right ) \left (c \,x^{4}+b \,x^{2}+a \right ) d \left (\arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}}\right ) \sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}+\operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}}\right ) \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\right ) \sqrt {2}}{2}}{2 \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\, \left (a c -\frac {b^{2}}{4}\right ) \left (c \,x^{4}+b \,x^{2}+a \right ) \sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, a}\)	\(872\)
default	\(\text {Expression too large to display}\)	\(1181\)

Input:

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

Output:

-1/2/(-4*(a*c-1/4*b^2)*d^2)^(1/2)*((3/4*((2*a*e-b*d+(-4*(a*c-1/4*b^2)*d^2) 
^(1/2))*a)^(1/2)*(2/3*(C*c*d-e*(1/2*b*C+c*(C*x^2+B)))*a^2+(-2/3*c*(1/2*(-C 
*x^2+B)*b+c*(B*x^2+A))*d+e*(1/3*B*b^2+c*(1/3*B*x^2+A)*b+2/3*A*c^2*x^2))*a- 
1/3*A*b*(c*x^2+b)*(b*e-c*d))*((-2*a*e+b*d+(-4*(a*c-1/4*b^2)*d^2)^(1/2))*a) 
^(1/2)*x*(e*x^2+d)^(1/2)+(c*x^4+b*x^2+a)*((1/4*C*c*d^2-1/4*e*(B*c+1/2*b*C) 
*d+A*c*e^2)*a^2+1/4*((-1/2*b*B*c+3*A*c^2)*d^2-5/2*b*e*(A*c-1/5*B*b)*d-A*b^ 
2*e^2)*a+1/8*A*d*b^2*(b*e-c*d))*(arctan(a*(e*x^2+d)^(1/2)/x*2^(1/2)/((-2*a 
*e+b*d+(-4*(a*c-1/4*b^2)*d^2)^(1/2))*a)^(1/2))*((2*a*e-b*d+(-4*(a*c-1/4*b^ 
2)*d^2)^(1/2))*a)^(1/2)-arctanh(a*(e*x^2+d)^(1/2)/x*2^(1/2)/((2*a*e-b*d+(- 
4*(a*c-1/4*b^2)*d^2)^(1/2))*a)^(1/2))*((-2*a*e+b*d+(-4*(a*c-1/4*b^2)*d^2)^ 
(1/2))*a)^(1/2))*2^(1/2))*(-4*(a*c-1/4*b^2)*d^2)^(1/2)+3/2*(1/3*(C*c*d*e-2 
*e^2*(B*c-1/2*b*C))*a^3+(-1/3*c*(B*c-C*b)*d^2-1/3*(A*c^2-b*B*c+5/4*b^2*C)* 
e*d+A*b*c*e^2)*a^2-1/6*b*((-4*A*c^2+1/2*b*B*c)*d^2+7/2*b*e*(A*c-1/7*B*b)*d 
+A*b^2*e^2)*a+1/12*A*b^3*d*(b*e-c*d))*(c*x^4+b*x^2+a)*d*(arctan(a*(e*x^2+d 
)^(1/2)/x*2^(1/2)/((-2*a*e+b*d+(-4*(a*c-1/4*b^2)*d^2)^(1/2))*a)^(1/2))*((2 
*a*e-b*d+(-4*(a*c-1/4*b^2)*d^2)^(1/2))*a)^(1/2)+arctanh(a*(e*x^2+d)^(1/2)/ 
x*2^(1/2)/((2*a*e-b*d+(-4*(a*c-1/4*b^2)*d^2)^(1/2))*a)^(1/2))*((-2*a*e+b*d 
+(-4*(a*c-1/4*b^2)*d^2)^(1/2))*a)^(1/2))*2^(1/2))/(a*c-1/4*b^2)/(c*x^4+b*x 
^2+a)/((2*a*e-b*d+(-4*(a*c-1/4*b^2)*d^2)^(1/2))*a)^(1/2)/(a*e^2-b*d*e+c*d^ 
2)/((-2*a*e+b*d+(-4*(a*c-1/4*b^2)*d^2)^(1/2))*a)^(1/2)/a
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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