\(\int \frac {x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2} (e+f x^2)^{3/2}} \, dx\) [368]

Optimal result

Integrand size = 37, antiderivative size = 493 \[ \int \frac {x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx=-\frac {c e \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{\sqrt {a} f \sqrt {-b e+a f} (d e-c f) \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}-\frac {e (d e-2 c f) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{\sqrt {a} f^2 \sqrt {-b e+a f} (d e-c f) \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}+\frac {e \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticPi}\left (-\frac {a f}{b e-a f},\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{\sqrt {a} f^2 \sqrt {-b e+a f} \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:

-c*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)/ 
f/(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)-e*(-2*c*f+d*e)*(b*x^2+a)^(1/2)*(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)*Elliptic 
F((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)/f^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)+e*(b*x^2+a)^(1/2)*(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)*Ellipt 
icPi((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)/f^2/(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]

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

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

Output:

Integrate[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.

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

\(\Big \downarrow \) 450

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

Input:

Int[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 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]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

integrate(x^4/(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]

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

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

Output:

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

Maxima [F]

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

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

Output:

integrate(x^4/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^(3/2)), x)
 

Giac [F]

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

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

Output:

integrate(x^4/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^(3/2)), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int(x^4/(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)