\(\int \frac {x^{10}}{\sqrt {a+b x^2} \sqrt {c+d x^2} (e+f x^2)^{5/2}} \, dx\) [373]

Optimal result

Integrand size = 37, antiderivative size = 1042 \[ \int \frac {x^{10}}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=-\frac {e^2 (b e (7 d e-3 c f)-3 a f (d e-c f)) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{12 b d f^3 (b e-a f) (d e-c f) \left (e+f x^2\right )^{3/2}}+\frac {x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{4 b d f \left (e+f x^2\right )^{3/2}}-\frac {(7 b d e+3 b c f+3 a d f) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 b^2 d^2 f^3 \sqrt {e+f x^2}}+\frac {c \left (9 a^3 d f^3 (d e-c f)^2+3 a^2 b f^2 (d e-c f)^2 (5 d e+3 c f)+b^3 e^2 \left (105 d^3 e^3-145 c d^2 e^2 f+15 c^2 d e f^2+9 c^3 f^3\right )-a b^2 e f \left (145 d^3 e^3-200 c d^2 e^2 f+21 c^2 d e f^2+18 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 )}{24 \sqrt {a} b^2 d^2 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 \left (9 a^3 d f^3 (d e-c f)^2+3 a^2 b f^2 (d e-c f)^2 (7 d e+c f)+a b^2 e f \left (75 d^3 e^3-236 c d^2 e^2 f+183 c^2 d e f^2-6 c^3 f^3\right )-b^3 e^2 \left (105 d^3 e^3-285 c d^2 e^2 f+199 c^2 d e f^2-3 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 )}{24 \sqrt {a} b^2 d f^5 (-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 d^2 f^2+2 a b d f (5 d e+c f)+b^2 \left (35 d^2 e^2+10 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 )}} \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^2 d^2 f^5 \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:

-1/12*e^2*(b*e*(-3*c*f+7*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^3/(-a*f+b*e)/(-c*f+d*e)/(f*x^2+e)^(3/2)+1/4*x^5*(b*x^2+a)^(1 
/2)*(d*x^2+c)^(1/2)/b/d/f/(f*x^2+e)^(3/2)-1/8*(3*a*d*f+3*b*c*f+7*b*d*e)*x* 
(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b^2/d^2/f^3/(f*x^2+e)^(1/2)+1/24*c*(9*a^3* 
d*f^3*(-c*f+d*e)^2+3*a^2*b*f^2*(-c*f+d*e)^2*(3*c*f+5*d*e)+b^3*e^2*(9*c^3*f 
^3+15*c^2*d*e*f^2-145*c*d^2*e^2*f+105*d^3*e^3)-a*b^2*e*f*(18*c^3*f^3+21*c^ 
2*d*e*f^2-200*c*d^2*e^2*f+145*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^(1/2)/b^2/d^2/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)-1/24*e*(9*a^3*d*f^3*(-c* 
f+d*e)^2+3*a^2*b*f^2*(-c*f+d*e)^2*(c*f+7*d*e)+a*b^2*e*f*(-6*c^3*f^3+183*c^ 
2*d*e*f^2-236*c*d^2*e^2*f+75*d^3*e^3)-b^3*e^2*(-3*c^3*f^3+199*c^2*d*e*f^2- 
285*c*d^2*e^2*f+105*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^(1/2)/b^2/d/f^5/(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)+1/8*e*(3*a^2*d^2*f^2+2*a*b*d*f*(c*f+ 
5*d*e)+b^2*(3*c^2*f^2+10*c*d*e*f+35*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^2/d^2/f^5/(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]

\[ \int \frac {x^{10}}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\int \frac {x^{10}}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx \] Input:

Integrate[x^10/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^(5/2)),x]
 

Output:

Integrate[x^10/(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.

\(\displaystyle \int \frac {x^{10}}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 450

\(\displaystyle \int \frac {x^{10}}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}}dx\)

Input:

Int[x^10/(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]

\[\int \frac {x^{10}}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \left (f \,x^{2}+e \right )^{\frac {5}{2}}}d x\]

Input:

int(x^10/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x)
 

Output:

int(x^10/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x)
 

Fricas [F]

\[ \int \frac {x^{10}}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\int { \frac {x^{10}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^10/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x, algorith 
m="fricas")
 

Output:

integral(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)*x^10/(b*d*f^3*x^1 
0 + (3*b*d*e*f^2 + (b*c + a*d)*f^3)*x^8 + (3*b*d*e^2*f + a*c*f^3 + 3*(b*c 
+ a*d)*e*f^2)*x^6 + a*c*e^3 + (b*d*e^3 + 3*a*c*e*f^2 + 3*(b*c + a*d)*e^2*f 
)*x^4 + (3*a*c*e^2*f + (b*c + a*d)*e^3)*x^2), x)
 

Sympy [F]

\[ \int \frac {x^{10}}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\int \frac {x^{10}}{\sqrt {a + b x^{2}} \sqrt {c + d x^{2}} \left (e + f x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(x**10/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2)/(f*x**2+e)**(5/2),x)
 

Output:

Integral(x**10/(sqrt(a + b*x**2)*sqrt(c + d*x**2)*(e + f*x**2)**(5/2)), x)
 

Maxima [F]

\[ \int \frac {x^{10}}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\int { \frac {x^{10}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^10/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x, algorith 
m="maxima")
 

Output:

integrate(x^10/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^(5/2)), x)
 

Giac [F]

\[ \int \frac {x^{10}}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\int { \frac {x^{10}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^10/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x, algorith 
m="giac")
 

Output:

integrate(x^10/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^(5/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{10}}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\int \frac {x^{10}}{\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}\,{\left (f\,x^2+e\right )}^{5/2}} \,d x \] Input:

int(x^10/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^(5/2)),x)
 

Output:

int(x^10/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {x^{10}}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\int \frac {x^{10}}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (f \,x^{2}+e \right )^{\frac {5}{2}}}d x \] Input:

int(x^10/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x)
 

Output:

int(x^10/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x)