\(\int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} (e+f x^2)^{5/2}} \, dx\) [374]

Optimal result

Integrand size = 37, antiderivative size = 819 \[ \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\frac {e (b e (5 d e-3 c f)-3 a f (d e-c f)) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{6 b d f^2 (b e-a f) (d e-c f) \left (e+f x^2\right )^{3/2}}+\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 b d f \left (e+f x^2\right )^{3/2}}-\frac {c \left (3 a^2 f^2 (d e-c f)^2+b^2 e^2 \left (15 d^2 e^2-22 c d e f+3 c^2 f^2\right )-2 a b e f \left (11 d^2 e^2-16 c d e f+3 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{6 \sqrt {a} b d f^3 (-b e+a f)^{3/2} (d e-c f)^2 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}+\frac {e \left (3 a^2 f^2 (d e-c f)^2+2 a b e f \left (6 d^2 e^2-19 c d e f+15 c^2 f^2\right )-b^2 e^2 \left (15 d^2 e^2-42 c d e f+31 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{6 \sqrt {a} b f^4 (-b e+a f)^{3/2} (d e-c f)^2 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}-\frac {e (5 b d e+b c f+a d f) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticPi}\left (-\frac {a f}{b e-a f},\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{2 \sqrt {a} b d f^4 \sqrt {-b e+a f} \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:

1/6*e*(b*e*(-3*c*f+5*d*e)-3*a*f*(-c*f+d*e))*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1 
/2)/b/d/f^2/(-a*f+b*e)/(-c*f+d*e)/(f*x^2+e)^(3/2)+1/2*x^3*(b*x^2+a)^(1/2)* 
(d*x^2+c)^(1/2)/b/d/f/(f*x^2+e)^(3/2)-1/6*c*(3*a^2*f^2*(-c*f+d*e)^2+b^2*e^ 
2*(3*c^2*f^2-22*c*d*e*f+15*d^2*e^2)-2*a*b*e*f*(3*c^2*f^2-16*c*d*e*f+11*d^2 
*e^2))*(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/f^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)+1/6*e*(3*a^2*f^2*(-c*f+d*e)^2+2*a*b*e*f*(15*c^2*f^2-19*c*d*e* 
f+6*d^2*e^2)-b^2*e^2*(31*c^2*f^2-42*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)*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)^(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)-1/2*e*(a*d*f+ 
b*c*f+5*b*d*e)*(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]

\[ \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx \] Input:

Integrate[x^8/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^(5/2)),x]
 

Output:

Integrate[x^8/(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.

\(\displaystyle \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 450

\(\displaystyle \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}}dx\)

Input:

Int[x^8/(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]

\[\int \frac {x^{8}}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \left (f \,x^{2}+e \right )^{\frac {5}{2}}}d x\]

Input:

int(x^8/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x)
 

Output:

int(x^8/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(x^8/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x, algorithm 
="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\int \frac {x^{8}}{\sqrt {a + b x^{2}} \sqrt {c + d x^{2}} \left (e + f x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(x**8/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2)/(f*x**2+e)**(5/2),x)
 

Output:

Integral(x**8/(sqrt(a + b*x**2)*sqrt(c + d*x**2)*(e + f*x**2)**(5/2)), x)
 

Maxima [F]

\[ \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\int { \frac {x^{8}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^8/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x, algorithm 
="maxima")
 

Output:

integrate(x^8/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^(5/2)), x)
 

Giac [F]

\[ \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\int { \frac {x^{8}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^8/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x, algorithm 
="giac")
 

Output:

integrate(x^8/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^(5/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\int \frac {x^8}{\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}\,{\left (f\,x^2+e\right )}^{5/2}} \,d x \] Input:

int(x^8/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^(5/2)),x)
 

Output:

int(x^8/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {x^8}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\int \frac {x^{8}}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (f \,x^{2}+e \right )^{\frac {5}{2}}}d x \] Input:

int(x^8/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x)
 

Output:

int(x^8/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x)