\(\int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \sqrt {c-d x^2} (e+f x^2)^{3/2}} \, dx\) [43]

Optimal result

Integrand size = 47, antiderivative size = 533 \[ \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \sqrt {c-d x^2} \left (e+f x^2\right )^{3/2}} \, dx=-\frac {\sqrt {c} \left (C e^2-B e f+A f^2\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c-d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e+c f} x}{\sqrt {c} \sqrt {e+f x^2}}\right )|-\frac {c (b e-a f)}{a (d e+c f)}\right )}{e f (b e-a f) \sqrt {d e+c f} \sqrt {c-d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}+\frac {\sqrt {c} \left (a f (2 C e-B f)-b \left (C e^2-A f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c-d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d e+c f} x}{\sqrt {c} \sqrt {e+f x^2}}\right ),-\frac {c (b e-a f)}{a (d e+c f)}\right )}{a f^2 (b e-a f) \sqrt {d e+c f} \sqrt {c-d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}+\frac {\sqrt {c} C e \sqrt {a+b x^2} \sqrt {\frac {e \left (c-d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticPi}\left (\frac {c f}{d e+c f},\arcsin \left (\frac {\sqrt {d e+c f} x}{\sqrt {c} \sqrt {e+f x^2}}\right ),-\frac {c (b e-a f)}{a (d e+c f)}\right )}{a f^2 \sqrt {d e+c f} \sqrt {c-d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:

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

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

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