\(\int \frac {A+B x^2+C x^4}{\sqrt {a-b x^2} \sqrt {c-d x^2} (e+f x^2)^{3/2}} \, dx\) [44]

Optimal result

Integrand size = 48, antiderivative size = 521 \[ \int \frac {A+B x^2+C x^4}{\sqrt {a-b x^2} \sqrt {c-d x^2} \left (e+f x^2\right )^{3/2}} \, dx=\frac {c \left (C e^2-B e f+A f^2\right ) \sqrt {a-b x^2} \sqrt {\frac {e \left (c-d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e+c f)}{c (b e+a f)}\right )}{\sqrt {a} e f \sqrt {b e+a f} (d e+c f) \sqrt {c-d x^2} \sqrt {\frac {e \left (a-b x^2\right )}{a \left (e+f x^2\right )}}}+\frac {\left ((B c+A d) f^2-C e (d e+2 c f)\right ) \sqrt {a-b x^2} \sqrt {\frac {e \left (c-d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e+c f)}{c (b e+a f)}\right )}{\sqrt {a} f^2 \sqrt {b e+a f} (d e+c f) \sqrt {c-d x^2} \sqrt {\frac {e \left (a-b x^2\right )}{a \left (e+f x^2\right )}}}+\frac {C e \sqrt {a-b x^2} \sqrt {\frac {e \left (c-d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticPi}\left (\frac {a f}{b e+a f},\arcsin \left (\frac {\sqrt {b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e+c f)}{c (b e+a f)}\right )}{\sqrt {a} f^2 \sqrt {b e+a f} \sqrt {c-d x^2} \sqrt {\frac {e \left (a-b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:

c*(A*f^2-B*e*f+C*e^2)*(-b*x^2+a)^(1/2)*(e*(-d*x^2+c)/c/(f*x^2+e))^(1/2)*El 
lipticE((a*f+b*e)^(1/2)*x/a^(1/2)/(f*x^2+e)^(1/2),(a*(c*f+d*e)/c/(a*f+b*e) 
)^(1/2))/a^(1/2)/e/f/(a*f+b*e)^(1/2)/(c*f+d*e)/(-d*x^2+c)^(1/2)/(e*(-b*x^2 
+a)/a/(f*x^2+e))^(1/2)+((A*d+B*c)*f^2-C*e*(2*c*f+d*e))*(-b*x^2+a)^(1/2)*(e 
*(-d*x^2+c)/c/(f*x^2+e))^(1/2)*EllipticF((a*f+b*e)^(1/2)*x/a^(1/2)/(f*x^2+ 
e)^(1/2),(a*(c*f+d*e)/c/(a*f+b*e))^(1/2))/a^(1/2)/f^2/(a*f+b*e)^(1/2)/(c*f 
+d*e)/(-d*x^2+c)^(1/2)/(e*(-b*x^2+a)/a/(f*x^2+e))^(1/2)+C*e*(-b*x^2+a)^(1/ 
2)*(e*(-d*x^2+c)/c/(f*x^2+e))^(1/2)*EllipticPi((a*f+b*e)^(1/2)*x/a^(1/2)/( 
f*x^2+e)^(1/2),a*f/(a*f+b*e),(a*(c*f+d*e)/c/(a*f+b*e))^(1/2))/a^(1/2)/f^2/ 
(a*f+b*e)^(1/2)/(-d*x^2+c)^(1/2)/(e*(-b*x^2+a)/a/(f*x^2+e))^(1/2)
 

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 2009

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

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [F]

\[\int \frac {C \,x^{4}+x^{2} B +A}{\sqrt {-b \,x^{2}+a}\, \sqrt {-x^{2} d +c}\, \left (f \,x^{2}+e \right )^{\frac {3}{2}}}d x\]

Input:

int((C*x^4+B*x^2+A)/(-b*x^2+a)^(1/2)/(-d*x^2+c)^(1/2)/(f*x^2+e)^(3/2),x)
 

Output:

int((C*x^4+B*x^2+A)/(-b*x^2+a)^(1/2)/(-d*x^2+c)^(1/2)/(f*x^2+e)^(3/2),x)
 

Fricas [F(-1)]

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

integrate((C*x^4+B*x^2+A)/(-b*x^2+a)^(1/2)/(-d*x^2+c)^(1/2)/(f*x^2+e)^(3/2 
),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

integrate((C*x**4+B*x**2+A)/(-b*x**2+a)**(1/2)/(-d*x**2+c)**(1/2)/(f*x**2+ 
e)**(3/2),x)
 

Output:

Integral((A + B*x**2 + C*x**4)/(sqrt(a - b*x**2)*sqrt(c - d*x**2)*(e + f*x 
**2)**(3/2)), x)
 

Maxima [F]

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

integrate((C*x^4+B*x^2+A)/(-b*x^2+a)^(1/2)/(-d*x^2+c)^(1/2)/(f*x^2+e)^(3/2 
),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((C*x^4+B*x^2+A)/(-b*x^2+a)^(1/2)/(-d*x^2+c)^(1/2)/(f*x^2+e)^(3/2 
),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int((A + B*x^2 + C*x^4)/((a - b*x^2)^(1/2)*(c - d*x^2)^(1/2)*(e + f*x^2)^( 
3/2)),x)
 

Output:

int((A + B*x^2 + C*x^4)/((a - b*x^2)^(1/2)*(c - d*x^2)^(1/2)*(e + f*x^2)^( 
3/2)), x)
 

Reduce [F]

\[ \int \frac {A+B x^2+C x^4}{\sqrt {a-b x^2} \sqrt {c-d x^2} \left (e+f x^2\right )^{3/2}} \, dx=\left (\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {-d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}\, x^{4}}{b d \,f^{2} x^{8}-a d \,f^{2} x^{6}-b c \,f^{2} x^{6}+2 b d e f \,x^{6}+a c \,f^{2} x^{4}-2 a d e f \,x^{4}-2 b c e f \,x^{4}+b d \,e^{2} x^{4}+2 a c e f \,x^{2}-a d \,e^{2} x^{2}-b c \,e^{2} x^{2}+a c \,e^{2}}d x \right ) c +\left (\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {-d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{b d \,f^{2} x^{8}-a d \,f^{2} x^{6}-b c \,f^{2} x^{6}+2 b d e f \,x^{6}+a c \,f^{2} x^{4}-2 a d e f \,x^{4}-2 b c e f \,x^{4}+b d \,e^{2} x^{4}+2 a c e f \,x^{2}-a d \,e^{2} x^{2}-b c \,e^{2} x^{2}+a c \,e^{2}}d x \right ) b +\left (\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {-d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}}{b d \,f^{2} x^{8}-a d \,f^{2} x^{6}-b c \,f^{2} x^{6}+2 b d e f \,x^{6}+a c \,f^{2} x^{4}-2 a d e f \,x^{4}-2 b c e f \,x^{4}+b d \,e^{2} x^{4}+2 a c e f \,x^{2}-a d \,e^{2} x^{2}-b c \,e^{2} x^{2}+a c \,e^{2}}d x \right ) a \] Input:

int((C*x^4+B*x^2+A)/(-b*x^2+a)^(1/2)/(-d*x^2+c)^(1/2)/(f*x^2+e)^(3/2),x)
 

Output:

int((sqrt(e + f*x**2)*sqrt(c - d*x**2)*sqrt(a - b*x**2)*x**4)/(a*c*e**2 + 
2*a*c*e*f*x**2 + a*c*f**2*x**4 - a*d*e**2*x**2 - 2*a*d*e*f*x**4 - a*d*f**2 
*x**6 - b*c*e**2*x**2 - 2*b*c*e*f*x**4 - b*c*f**2*x**6 + b*d*e**2*x**4 + 2 
*b*d*e*f*x**6 + b*d*f**2*x**8),x)*c + int((sqrt(e + f*x**2)*sqrt(c - d*x** 
2)*sqrt(a - b*x**2)*x**2)/(a*c*e**2 + 2*a*c*e*f*x**2 + a*c*f**2*x**4 - a*d 
*e**2*x**2 - 2*a*d*e*f*x**4 - a*d*f**2*x**6 - b*c*e**2*x**2 - 2*b*c*e*f*x* 
*4 - b*c*f**2*x**6 + b*d*e**2*x**4 + 2*b*d*e*f*x**6 + b*d*f**2*x**8),x)*b 
+ int((sqrt(e + f*x**2)*sqrt(c - d*x**2)*sqrt(a - b*x**2))/(a*c*e**2 + 2*a 
*c*e*f*x**2 + a*c*f**2*x**4 - a*d*e**2*x**2 - 2*a*d*e*f*x**4 - a*d*f**2*x* 
*6 - b*c*e**2*x**2 - 2*b*c*e*f*x**4 - b*c*f**2*x**6 + b*d*e**2*x**4 + 2*b* 
d*e*f*x**6 + b*d*f**2*x**8),x)*a