\(\int \frac {A+B x^2}{\sqrt {a-b x^2} \sqrt {c-d x^2} (e-f x^2)^{3/2}} \, dx\) [40]

Optimal result

Integrand size = 44, antiderivative size = 344 \[ \int \frac {A+B x^2}{\sqrt {a-b x^2} \sqrt {c-d x^2} \left (e-f x^2\right )^{3/2}} \, dx=-\frac {c (B e+A f) \sqrt {a-b x^2} \sqrt {\frac {e \left (c-d x^2\right )}{c \left (e-f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {b e-a f} x}{\sqrt {a} \sqrt {e-f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{\sqrt {a} e \sqrt {b e-a f} (d e-c f) \sqrt {c-d x^2} \sqrt {\frac {e \left (a-b x^2\right )}{a \left (e-f x^2\right )}}}+\frac {(B c+A d) \sqrt {a-b x^2} \sqrt {\frac {e \left (c-d x^2\right )}{c \left (e-f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b e-a f} x}{\sqrt {a} \sqrt {e-f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{\sqrt {a} \sqrt {b e-a f} (d e-c f) \sqrt {c-d x^2} \sqrt {\frac {e \left (a-b x^2\right )}{a \left (e-f x^2\right )}}} \] Output:

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

Mathematica [F]

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

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

Output:

Integrate[(A + B*x^2)/(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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

Int[(A + B*x^2)/(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]

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

Input:

int((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((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]

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

integrate((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((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]

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

integrate((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)/(sqrt(a - b*x**2)*sqrt(c - d*x**2)*(e - f*x**2)**(3/ 
2)), x)
 

Maxima [F]

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

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

Giac [F]

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

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

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

Reduce [F]

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

int((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**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