\(\int \frac {1}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2} (e+f x^2)^{5/2}} \, dx\) [380]

Optimal result

Integrand size = 37, antiderivative size = 757 \[ \int \frac {1}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\frac {f^4 x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 e^3 (b e-a f) (d e-c f) \left (e+f x^2\right )^{3/2}}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{3 a c e^2 x^3 \sqrt {e+f x^2}}+\frac {(2 b c e+2 a d e+7 a c f) \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 a^2 c^2 e^3 x \sqrt {e+f x^2}}+\frac {2 \left (b^3 c e^3 (d e-c f)^2+a b^2 e^2 (d e-c f)^2 (d e+2 c f)+a^3 f^2 \left (d^3 e^3+2 c d^2 e^2 f-12 c^2 d e f^2+8 c^3 f^3\right )-a^2 b e f \left (2 d^3 e^3+3 c d^2 e^2 f-18 c^2 d e f^2+12 c^3 f^3\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{3 a^{5/2} c e^4 (-b e+a f)^{3/2} (d e-c f)^2 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}-\frac {\left (2 b^3 c e^2 (d e-c f)^2+a b^2 e (d e-c f)^2 (d e+5 c f)+a^3 d f^2 \left (d^2 e^2-11 c d e f+8 c^2 f^2\right )-2 a^2 b f \left (d^3 e^3-3 c d^2 e^2 f-3 c^2 d e f^2+4 c^3 f^3\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{3 a^{5/2} c e^3 (-b e+a f)^{3/2} (d e-c f)^2 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:

1/3*f^4*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/e^3/(-a*f+b*e)/(-c*f+d*e)/(f*x^2 
+e)^(3/2)-1/3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a/c/e^2/x^3/(f*x^2+e)^(1/2)+ 
1/3*(7*a*c*f+2*a*d*e+2*b*c*e)*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a^2/c^2/e^3/ 
x/(f*x^2+e)^(1/2)+2/3*(b^3*c*e^3*(-c*f+d*e)^2+a*b^2*e^2*(-c*f+d*e)^2*(2*c* 
f+d*e)+a^3*f^2*(8*c^3*f^3-12*c^2*d*e*f^2+2*c*d^2*e^2*f+d^3*e^3)-a^2*b*e*f* 
(12*c^3*f^3-18*c^2*d*e*f^2+3*c*d^2*e^2*f+2*d^3*e^3))*(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^(5/2)/c/e^4/(a*f-b*e)^(3/2)/(-c* 
f+d*e)^2/(d*x^2+c)^(1/2)/(e*(b*x^2+a)/a/(f*x^2+e))^(1/2)-1/3*(2*b^3*c*e^2* 
(-c*f+d*e)^2+a*b^2*e*(-c*f+d*e)^2*(5*c*f+d*e)+a^3*d*f^2*(8*c^2*f^2-11*c*d* 
e*f+d^2*e^2)-2*a^2*b*f*(4*c^3*f^3-3*c^2*d*e*f^2-3*c*d^2*e^2*f+d^3*e^3))*(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^(5/2)/c/e^3/( 
a*f-b*e)^(3/2)/(-c*f+d*e)^2/(d*x^2+c)^(1/2)/(e*(b*x^2+a)/a/(f*x^2+e))^(1/2 
)
 

Mathematica [F]

\[ \int \frac {1}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\int \frac {1}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx \] Input:

Integrate[1/(x^4*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^(5/2)),x]
 

Output:

Integrate[1/(x^4*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^(5/2)), 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.

\(\displaystyle \int \frac {1}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 450

\(\displaystyle \int \frac {1}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}}dx\)

Input:

Int[1/(x^4*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^(5/2)),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 450

Int[((g_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_)^2)^(p_.)*((c_.) + (d_.)*(x_)^2)^ 
(q_.)*((e_.) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Unintegrable[(g*x)^m*(a + 
b*x^2)^p*(c + d*x^2)^q*(e + f*x^2)^r, x] /; FreeQ[{a, b, c, d, e, f, g, m, 
p, q, r}, x]
 

Maple [F]

\[\int \frac {1}{x^{4} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \left (f \,x^{2}+e \right )^{\frac {5}{2}}}d x\]

Input:

int(1/x^4/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x)
 

Output:

int(1/x^4/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x)
 

Fricas [F]

\[ \int \frac {1}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\int { \frac {1}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{\frac {5}{2}} x^{4}} \,d x } \] Input:

integrate(1/x^4/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x, algorit 
hm="fricas")
 

Output:

integral(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)/(b*d*f^3*x^14 + ( 
3*b*d*e*f^2 + (b*c + a*d)*f^3)*x^12 + (3*b*d*e^2*f + a*c*f^3 + 3*(b*c + a* 
d)*e*f^2)*x^10 + a*c*e^3*x^4 + (b*d*e^3 + 3*a*c*e*f^2 + 3*(b*c + a*d)*e^2* 
f)*x^8 + (3*a*c*e^2*f + (b*c + a*d)*e^3)*x^6), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(1/x**4/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2)/(f*x**2+e)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\int { \frac {1}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{\frac {5}{2}} x^{4}} \,d x } \] Input:

integrate(1/x^4/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x, algorit 
hm="maxima")
 

Output:

integrate(1/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^(5/2)*x^4), x)
 

Giac [F]

\[ \int \frac {1}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\int { \frac {1}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{\frac {5}{2}} x^{4}} \,d x } \] Input:

integrate(1/x^4/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x, algorit 
hm="giac")
 

Output:

integrate(1/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^(5/2)*x^4), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\int \frac {1}{x^4\,\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}\,{\left (f\,x^2+e\right )}^{5/2}} \,d x \] Input:

int(1/(x^4*(a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^(5/2)),x)
 

Output:

int(1/(x^4*(a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {1}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\int \frac {1}{x^{4} \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (f \,x^{2}+e \right )^{\frac {5}{2}}}d x \] Input:

int(1/x^4/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x)
 

Output:

int(1/x^4/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x)