\(\int \frac {A+B x^2+C x^4}{\sqrt {a-b x^2} \sqrt {c+d x^2} (e+f x^2)^{3/2}} \, dx\) [42]

Optimal result

Integrand size = 47, antiderivative size = 525 \[ \int \frac {A+B x^2+C x^4}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx=-\frac {c \left (C e^2-B e f+A f^2\right ) \sqrt {a-b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|-\frac {a (d e-c f)}{c (b e+a f)}\right )}{\sqrt {a} e f \sqrt {b e+a f} (d e-c f) \sqrt {c+d x^2} \sqrt {\frac {e \left (a-b x^2\right )}{a \left (e+f x^2\right )}}}-\frac {\left ((B c-A d) f^2+C e (d e-2 c f)\right ) \sqrt {a-b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),-\frac {a (d e-c f)}{c (b e+a f)}\right )}{\sqrt {a} f^2 \sqrt {b e+a f} (d e-c f) \sqrt {c+d x^2} \sqrt {\frac {e \left (a-b x^2\right )}{a \left (e+f x^2\right )}}}+\frac {C e \sqrt {a-b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticPi}\left (\frac {a f}{b e+a f},\arcsin \left (\frac {\sqrt {b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),-\frac {a (d e-c f)}{c (b e+a f)}\right )}{\sqrt {a} f^2 \sqrt {b e+a f} \sqrt {c+d x^2} \sqrt {\frac {e \left (a-b x^2\right )}{a \left (e+f x^2\right )}}} \] 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

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

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]

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

Output:

integral(-(C*x^4 + B*x^2 + A)*sqrt(-b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(f*x^2 
+ e)/(b*d*f^2*x^8 + (2*b*d*e*f + (b*c - a*d)*f^2)*x^6 + (b*d*e^2 - a*c*f^2 
 + 2*(b*c - a*d)*e*f)*x^4 - a*c*e^2 - (2*a*c*e*f - (b*c - a*d)*e^2)*x^2), 
x)
 

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