\(\int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (A+B x^2+C x^4)}{x^2 \sqrt {e+f x^2}} \, dx\) [18]

Optimal result

Integrand size = 49, antiderivative size = 804 \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (A+B x^2+C x^4\right )}{x^2 \sqrt {e+f x^2}} \, dx=-\frac {a A \sqrt {c+d x^2} \sqrt {e+f x^2}}{e x \sqrt {a+b x^2}}+\frac {(3 a C d f-b (3 C d e-c C f-4 B d f)) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{8 d f^2 \sqrt {a+b x^2}}+\frac {b C x^3 \sqrt {c+d x^2} \sqrt {e+f x^2}}{4 f \sqrt {a+b x^2}}-\frac {\sqrt {b e-a f} (a C d e f+b (4 d f (B e+2 A f)-C e (3 d e-c f))) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )|\frac {(b c-a d) e}{c (b e-a f)}\right )}{8 b d \sqrt {e} f^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}-\frac {\sqrt {b e-a f} \left (a^2 C d e f-8 A b^2 c f^2+a b e (3 C d e-3 c C f-4 B d f)\right ) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right ),\frac {(b c-a d) e}{c (b e-a f)}\right )}{8 b^2 c \sqrt {e} f^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}-\frac {a \sqrt {e} \left (a^2 C d^2 f^2+2 a b d f (C d e-c C f-2 B d f)+b^2 \left (4 d f (B d e-B c f-2 A d f)-C \left (3 d^2 e^2-2 c d e f-c^2 f^2\right )\right )\right ) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticPi}\left (\frac {b e}{b e-a f},\arcsin \left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right ),\frac {(b c-a d) e}{c (b e-a f)}\right )}{8 b^2 c d f^2 \sqrt {b e-a f} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}} \] Output:

-a*A*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/e/x/(b*x^2+a)^(1/2)+1/8*(3*a*C*d*f-b* 
(-4*B*d*f-C*c*f+3*C*d*e))*x*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/d/f^2/(b*x^2+a 
)^(1/2)+1/4*b*C*x^3*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/f/(b*x^2+a)^(1/2)-1/8* 
(-a*f+b*e)^(1/2)*(a*C*d*e*f+b*(4*d*f*(2*A*f+B*e)-C*e*(-c*f+3*d*e)))*(d*x^2 
+c)^(1/2)*(a*(f*x^2+e)/e/(b*x^2+a))^(1/2)*EllipticE((-a*f+b*e)^(1/2)*x/e^( 
1/2)/(b*x^2+a)^(1/2),((-a*d+b*c)*e/c/(-a*f+b*e))^(1/2))/b/d/e^(1/2)/f^2/(a 
*(d*x^2+c)/c/(b*x^2+a))^(1/2)/(f*x^2+e)^(1/2)-1/8*(-a*f+b*e)^(1/2)*(a^2*C* 
d*e*f-8*A*b^2*c*f^2+a*b*e*(-4*B*d*f-3*C*c*f+3*C*d*e))*(d*x^2+c)^(1/2)*(a*( 
f*x^2+e)/e/(b*x^2+a))^(1/2)*EllipticF((-a*f+b*e)^(1/2)*x/e^(1/2)/(b*x^2+a) 
^(1/2),((-a*d+b*c)*e/c/(-a*f+b*e))^(1/2))/b^2/c/e^(1/2)/f^2/(a*(d*x^2+c)/c 
/(b*x^2+a))^(1/2)/(f*x^2+e)^(1/2)-1/8*a*e^(1/2)*(a^2*C*d^2*f^2+2*a*b*d*f*( 
-2*B*d*f-C*c*f+C*d*e)+b^2*(4*d*f*(-2*A*d*f-B*c*f+B*d*e)-C*(-c^2*f^2-2*c*d* 
e*f+3*d^2*e^2)))*(d*x^2+c)^(1/2)*(a*(f*x^2+e)/e/(b*x^2+a))^(1/2)*EllipticP 
i((-a*f+b*e)^(1/2)*x/e^(1/2)/(b*x^2+a)^(1/2),b*e/(-a*f+b*e),((-a*d+b*c)*e/ 
c/(-a*f+b*e))^(1/2))/b^2/c/d/f^2/(-a*f+b*e)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a) 
)^(1/2)/(f*x^2+e)^(1/2)
 

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle A \int \frac {\sqrt {b x^2+a} \sqrt {d x^2+c}}{x^2 \sqrt {f x^2+e}}dx+C \int \frac {x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}{\sqrt {f x^2+e}}dx+\frac {b B \sqrt {e} \sqrt {c+d x^2} (d e-c f) \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right ),\frac {(b c-a d) e}{c (b e-a f)}\right )}{2 d f \sqrt {e+f x^2} \sqrt {b e-a f} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {B \sqrt {e} \sqrt {a+b x^2} \sqrt {d e-c f} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {d x^2+c}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{2 f \sqrt {e+f x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {B c \sqrt {e} \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} (-a d f-b c f+b d e) \operatorname {EllipticPi}\left (\frac {d e}{d e-c f},\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {d x^2+c}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{2 a d f \sqrt {e+f x^2} \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {B d x \sqrt {a+b x^2} \sqrt {e+f x^2}}{2 f \sqrt {c+d x^2}}\)

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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