\(\int x^4 \sqrt {a+b x^2} \sqrt {c+d x^2} (e+f x^2)^{3/2} \, dx\) [310]

Optimal result

Integrand size = 37, antiderivative size = 1544 \[ \int x^4 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2} \, dx =\text {Too large to display} \] Output:

-1/3840*(105*a^4*d^4*f^4-10*a^3*b*d^3*f^3*(4*c*f+19*d*e)+2*a^2*b^2*d^2*f^2 
*(-17*c^2*f^2+47*c*d*e*f+18*d^2*e^2)+2*a*b^3*d*f*(-20*c^3*f^3+47*c^2*d*e*f 
^2-18*c*d^2*e^2*f+15*d^3*e^3)-b^4*(-105*c^4*f^4+190*c^3*d*e*f^3-36*c^2*d^2 
*e^2*f^2-30*c*d^3*e^3*f+45*d^4*e^4))*x*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/b^3 
/d^4/f^3/(b*x^2+a)^(1/2)+1/1920*(35*a^3*d^3*f^3-a^2*b*d^2*f^2*(11*c*f+61*d 
*e)+a*b^2*d*f*(-11*c^2*f^2+26*c*d*e*f+9*d^2*e^2)-b^3*(-35*c^3*f^3+61*c^2*d 
*e*f^2-9*c*d^2*e^2*f+15*d^3*e^3))*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2 
+e)^(1/2)/b^3/d^3/f^2-1/480*(7*a^2*d*f^2/b-2*a*f*(c*f+6*d*e)-b*(3*d*e^2+12 
*c*e*f-7*c^2*f^2/d))*x^3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/b 
/d/f+1/80*(a*d*f+b*c*f+11*b*d*e)*x^5*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^ 
2+e)^(1/2)/b/d+1/10*f*x^7*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)+ 
1/3840*(-a*d+b*c)^(1/2)*e*(105*a^4*d^4*f^4-10*a^3*b*d^3*f^3*(4*c*f+19*d*e) 
+2*a^2*b^2*d^2*f^2*(-17*c^2*f^2+47*c*d*e*f+18*d^2*e^2)+2*a*b^3*d*f*(-20*c^ 
3*f^3+47*c^2*d*e*f^2-18*c*d^2*e^2*f+15*d^3*e^3)-b^4*(-105*c^4*f^4+190*c^3* 
d*e*f^3-36*c^2*d^2*e^2*f^2-30*c*d^3*e^3*f+45*d^4*e^4))*(d*x^2+c)^(1/2)*(a* 
(f*x^2+e)/e/(b*x^2+a))^(1/2)*EllipticE((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a 
)^(1/2),(c*(-a*f+b*e)/(-a*d+b*c)/e)^(1/2))/b^4/c^(1/2)/d^4/f^3/(a*(d*x^2+c 
)/c/(b*x^2+a))^(1/2)/(f*x^2+e)^(1/2)+1/3840*a*(-a*d+b*c)^(1/2)*(105*a^4*d^ 
4*f^4+40*a^2*b^2*d^3*e*f^2*(-c*f+7*d*e)-30*a^3*b*d^3*f^3*(-c*f+11*d*e)-2*a 
*b^3*d*f*(15*c^3*f^3-31*c^2*d*e*f^2+13*c*d^2*e^2*f+3*d^3*e^3)+b^4*(-105...
 

Mathematica [F]

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

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

\(\Big \downarrow \) 450

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

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]

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

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

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

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

Giac [F]

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

Mupad [F(-1)]

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

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