\(\int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (A+B x^2+C x^4)}{(e+f x^2)^{3/2}} \, dx\) [32]

Optimal result

Integrand size = 46, antiderivative size = 813 \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (A+B x^2+C x^4\right )}{\left (e+f x^2\right )^{3/2}} \, dx=\frac {(a C d f-b (5 C d e-c C f-4 B d f)) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 b d f^2 \sqrt {e+f x^2}}+\frac {C x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{4 f \sqrt {e+f x^2}}-\frac {c \sqrt {-b e+a f} (a C d e f+4 b d f (3 B e-2 A f)-b C e (15 d e-c f)) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{8 \sqrt {a} b d e f^3 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}+\frac {\left (a^2 C d e f^2+a b f (C e (6 d e-13 c f)-4 B f (d e-2 c f))-b^2 \left (3 C e^2 (5 d e-7 c f)-4 f (B e (3 d e-4 c f)-2 A f (d e-c f))\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{8 \sqrt {a} b f^4 \sqrt {-b e+a f} \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}-\frac {e \left (a^2 C d^2 f^2+2 a b d f (3 C d e-c C f-2 B d f)+b^2 \left (4 d f (3 B d e-B c f-2 A d f)-C \left (15 d^2 e^2-6 c d e f-c^2 f^2\right )\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticPi}\left (-\frac {a f}{b e-a f},\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{8 \sqrt {a} b d f^4 \sqrt {-b e+a f} \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:

1/8*(a*C*d*f-b*(-4*B*d*f-C*c*f+5*C*d*e))*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2) 
/b/d/f^2/(f*x^2+e)^(1/2)+1/4*C*x^3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/f/(f*x^ 
2+e)^(1/2)-1/8*c*(a*f-b*e)^(1/2)*(a*C*d*e*f+4*b*d*f*(-2*A*f+3*B*e)-b*C*e*( 
-c*f+15*d*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)/b/d/e/f^3/(d*x^2+c)^(1/2)/(e*(b*x^2+a)/a/(f*x^2+e))^(1/2)+1/8*(a^ 
2*C*d*e*f^2+a*b*f*(C*e*(-13*c*f+6*d*e)-4*B*f*(-2*c*f+d*e))-b^2*(3*C*e^2*(- 
7*c*f+5*d*e)-4*f*(B*e*(-4*c*f+3*d*e)-2*A*f*(-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)/b/f^4/(a*f-b*e)^(1/2)/ 
(d*x^2+c)^(1/2)/(e*(b*x^2+a)/a/(f*x^2+e))^(1/2)-1/8*e*(a^2*C*d^2*f^2+2*a*b 
*d*f*(-2*B*d*f-C*c*f+3*C*d*e)+b^2*(4*d*f*(-2*A*d*f-B*c*f+3*B*d*e)-C*(-c^2* 
f^2-6*c*d*e*f+15*d^2*e^2)))*(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)/b/d/f^4/(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]

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

Integrate[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(A + B*x^2 + C*x^4))/(e + f*x^2 
)^(3/2),x]
 

Output:

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

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

\(\Big \downarrow \) 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

\(\displaystyle B \int \frac {x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}{\left (f x^2+e\right )^{3/2}}dx+C \int \frac {x^4 \sqrt {b x^2+a} \sqrt {d x^2+c}}{\left (f x^2+e\right )^{3/2}}dx+\frac {A b c \sqrt {e} \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticPi}\left (\frac {d e}{d e-c f},\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {d x^2+c}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{a f \sqrt {e+f x^2} \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {A c^{3/2} \sqrt {a+b x^2} (b e-a f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d e-c f} x}{\sqrt {c} \sqrt {f x^2+e}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{a e f \sqrt {c+d x^2} \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {A \sqrt {c} \sqrt {a+b x^2} \sqrt {d e-c f} E\left (\arctan \left (\frac {\sqrt {d e-c f} x}{\sqrt {c} \sqrt {f x^2+e}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{e f \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {A x \sqrt {a+b x^2} (d e-c f)}{e f \sqrt {c+d x^2} \sqrt {e+f x^2}}\)

Input:

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

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

Input:

int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(C*x^4+B*x^2+A)/(f*x^2+e)^(3/2),x)
 

Output:

int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(C*x^4+B*x^2+A)/(f*x^2+e)^(3/2),x)
 

Fricas [F(-1)]

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

integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(C*x^4+B*x^2+A)/(f*x^2+e)^(3/2), 
x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

integrate((b*x**2+a)**(1/2)*(d*x**2+c)**(1/2)*(C*x**4+B*x**2+A)/(f*x**2+e) 
**(3/2),x)
 

Output:

Integral(sqrt(a + b*x**2)*sqrt(c + d*x**2)*(A + B*x**2 + C*x**4)/(e + f*x* 
*2)**(3/2), x)
 

Maxima [F]

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

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

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

integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(C*x^4+B*x^2+A)/(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. \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (A+B x^2+C x^4\right )}{\left (e+f x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}\,\left (C\,x^4+B\,x^2+A\right )}{{\left (f\,x^2+e\right )}^{3/2}} \,d x \] Input:

int(((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(A + B*x^2 + C*x^4))/(e + f*x^2)^ 
(3/2),x)
 

Output:

int(((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(A + B*x^2 + C*x^4))/(e + f*x^2)^ 
(3/2), x)
 

Reduce [F]

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

int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(C*x^4+B*x^2+A)/(f*x^2+e)^(3/2),x)
 

Output:

int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(C*x^4+B*x^2+A)/(f*x^2+e)^(3/2),x)