\(\int \frac {x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}}{(e+f x^2)^{5/2}} \, dx\) [331]

Optimal result

Integrand size = 37, antiderivative size = 858 \[ \int \frac {x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}}{\left (e+f x^2\right )^{5/2}} \, dx=-\frac {e^2 x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 f^3 \left (e+f x^2\right )^{3/2}}-\frac {(9 b d e-b c f-a d f) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 b d f^3 \sqrt {e+f x^2}}+\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{4 f^2 \sqrt {e+f x^2}}-\frac {c \left (3 a^2 d f^2 (d e-c f)+b^2 e \left (105 d^2 e^2-100 c d e f+3 c^2 f^2\right )-a b f \left (100 d^2 e^2-95 c d e f+3 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{24 \sqrt {a} b d f^4 \sqrt {-b e+a f} (d e-c f) \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}+\frac {e \left (3 a^2 d f^2 (d e-c f)+a b f \left (30 d^2 e^2-101 c d e f+63 c^2 f^2\right )-b^2 e \left (105 d^2 e^2-240 c d e f+127 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{24 \sqrt {a} b f^5 \sqrt {-b e+a f} (d e-c f) \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}-\frac {e \left (a^2 d^2 f^2+2 a b d f (5 d e-c f)-b^2 \left (35 d^2 e^2-10 c d e f-c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticPi}\left (-\frac {a f}{b e-a f},\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{8 \sqrt {a} b d f^5 \sqrt {-b e+a f} \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:

-1/3*e^2*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/f^3/(f*x^2+e)^(3/2)-1/8*(-a*d*f 
-b*c*f+9*b*d*e)*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b/d/f^3/(f*x^2+e)^(1/2)+ 
1/4*x^3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/f^2/(f*x^2+e)^(1/2)-1/24*c*(3*a^2* 
d*f^2*(-c*f+d*e)+b^2*e*(3*c^2*f^2-100*c*d*e*f+105*d^2*e^2)-a*b*f*(3*c^2*f^ 
2-95*c*d*e*f+100*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^4/(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)+1/24*e*(3*a^2*d*f^2*(-c*f+d*e)+a*b*f*(63*c^ 
2*f^2-101*c*d*e*f+30*d^2*e^2)-b^2*e*(127*c^2*f^2-240*c*d*e*f+105*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^ 
5/(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)-1/8*e*(a^2*d^2*f^2+2*a*b*d*f*(-c*f+5*d*e)-b^2*(-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/d/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]

\[ \int \frac {x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}}{\left (e+f x^2\right )^{5/2}} \, dx=\int \frac {x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}}{\left (e+f x^2\right )^{5/2}} \, dx \] Input:

Integrate[(x^6*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(e + f*x^2)^(5/2),x]
 

Output:

Integrate[(x^6*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.

\(\displaystyle \int \frac {x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}}{\left (e+f x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 450

\(\displaystyle \int \frac {x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}}{\left (e+f x^2\right )^{5/2}}dx\)

Input:

Int[(x^6*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]

\[\int \frac {x^{6} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{\left (f \,x^{2}+e \right )^{\frac {5}{2}}}d x\]

Input:

int(x^6*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x)
 

Output:

int(x^6*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}}{\left (e+f x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(x^6*(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]

\[ \int \frac {x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}}{\left (e+f x^2\right )^{5/2}} \, dx=\int \frac {x^{6} \sqrt {a + b x^{2}} \sqrt {c + d x^{2}}}{\left (e + f x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(x**6*(b*x**2+a)**(1/2)*(d*x**2+c)**(1/2)/(f*x**2+e)**(5/2),x)
 

Output:

Integral(x**6*sqrt(a + b*x**2)*sqrt(c + d*x**2)/(e + f*x**2)**(5/2), x)
 

Maxima [F]

\[ \int \frac {x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}}{\left (e+f x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} x^{6}}{{\left (f x^{2} + e\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^6*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x, algorithm 
="maxima")
 

Output:

integrate(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*x^6/(f*x^2 + e)^(5/2), x)
 

Giac [F]

\[ \int \frac {x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}}{\left (e+f x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} x^{6}}{{\left (f x^{2} + e\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^6*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x, algorithm 
="giac")
 

Output:

integrate(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*x^6/(f*x^2 + e)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}}{\left (e+f x^2\right )^{5/2}} \, dx=\int \frac {x^6\,\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}}{{\left (f\,x^2+e\right )}^{5/2}} \,d x \] Input:

int((x^6*(a + b*x^2)^(1/2)*(c + d*x^2)^(1/2))/(e + f*x^2)^(5/2),x)
 

Output:

int((x^6*(a + b*x^2)^(1/2)*(c + d*x^2)^(1/2))/(e + f*x^2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}}{\left (e+f x^2\right )^{5/2}} \, dx=\int \frac {x^{6} \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{\left (f \,x^{2}+e \right )^{\frac {5}{2}}}d x \] Input:

int(x^6*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x)
 

Output:

int(x^6*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x)