\(\int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2} (e-f x^2)^{3/2}} \, dx\) [45]

Optimal result

Integrand size = 47, antiderivative size = 496 \[ \int \frac {A+B x^2+C x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e-f x^2\right )^{3/2}} \, dx=\frac {\left (C e^2+f (B e+A f)\right ) x \sqrt {c+d x^2}}{e f (d e+c f) \sqrt {a+b x^2} \sqrt {e-f x^2}}-\frac {\sqrt {a} \left (C e^2+f (B e+A f)\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b e+a f} x}{\sqrt {a} \sqrt {e-f x^2}}\right )|\frac {(b c-a d) e}{c (b e+a f)}\right )}{e f \sqrt {b e+a f} (d e+c f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {\sqrt {a} (a C e+A b f) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b e+a f} x}{\sqrt {a} \sqrt {e-f x^2}}\right ),\frac {(b c-a d) e}{c (b e+a f)}\right )}{b c e f \sqrt {b e+a f} \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} C \sqrt {c+d x^2} \operatorname {EllipticPi}\left (\frac {b e}{b e+a f},\arctan \left (\frac {\sqrt {b e+a f} x}{\sqrt {a} \sqrt {e-f x^2}}\right ),\frac {(b c-a d) e}{c (b e+a f)}\right )}{b c f \sqrt {b e+a f} \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

(C*e^2+f*(A*f+B*e))*x*(d*x^2+c)^(1/2)/e/f/(c*f+d*e)/(b*x^2+a)^(1/2)/(-f*x^ 
2+e)^(1/2)-a^(1/2)*(C*e^2+f*(A*f+B*e))*(d*x^2+c)^(1/2)*EllipticE((a*f+b*e) 
^(1/2)*x/a^(1/2)/(-f*x^2+e)^(1/2)/(1+(a*f+b*e)*x^2/a/(-f*x^2+e))^(1/2),((- 
a*d+b*c)*e/c/(a*f+b*e))^(1/2))/e/f/(a*f+b*e)^(1/2)/(c*f+d*e)/(b*x^2+a)^(1/ 
2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)+a^(1/2)*(A*b*f+C*a*e)*(d*x^2+c)^(1/2)*I 
nverseJacobiAM(arctan((a*f+b*e)^(1/2)*x/a^(1/2)/(-f*x^2+e)^(1/2)),((-a*d+b 
*c)*e/c/(a*f+b*e))^(1/2))/b/c/e/f/(a*f+b*e)^(1/2)/(b*x^2+a)^(1/2)/(a*(d*x^ 
2+c)/c/(b*x^2+a))^(1/2)-a^(3/2)*C*(d*x^2+c)^(1/2)*EllipticPi((a*f+b*e)^(1/ 
2)*x/a^(1/2)/(-f*x^2+e)^(1/2)/(1+(a*f+b*e)*x^2/a/(-f*x^2+e))^(1/2),b*e/(a* 
f+b*e),((-a*d+b*c)*e/c/(a*f+b*e))^(1/2))/b/c/f/(a*f+b*e)^(1/2)/(b*x^2+a)^( 
1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(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

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

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