\(\int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^6 (e+f x^2)^{5/2}} \, dx\) [337]

Optimal result

Integrand size = 37, antiderivative size = 845 \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{x^6 \left (e+f x^2\right )^{5/2}} \, dx=-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{5 e x^5 \left (e+f x^2\right )^{3/2}}-\frac {\left (\frac {b}{a}+\frac {d}{c}-\frac {8 f}{e}\right ) \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 e x^3 \left (e+f x^2\right )^{3/2}}+\frac {\left (\frac {2 b^2 c e}{a}-2 b d e+\frac {2 a d^2 e}{c}+11 b c f+11 a d f-\frac {48 a c f^2}{e}\right ) \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 a c e^2 x \left (e+f x^2\right )^{3/2}}+\frac {2 f \left (b^2 c^2 e^2-a b c e (d e-6 c f)+a^2 \left (d^2 e^2+6 c d e f-32 c^2 f^2\right )\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 a^2 c^2 e^4 \left (e+f x^2\right )^{3/2}}-\frac {\left (2 b^3 c^2 e^3 (d e-c f)-a b^2 c e^2 \left (2 d^2 e^2-13 c d e f+11 c^2 f^2\right )-a^3 f \left (2 d^3 e^3+11 c d^2 e^2 f-136 c^2 d e f^2+128 c^3 f^3\right )+a^2 b e \left (2 d^3 e^3+13 c d^2 e^2 f-146 c^2 d e f^2+136 c^3 f^3\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{15 a^{5/2} c e^5 \sqrt {-b e+a f} (d e-c f) \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}+\frac {(b c-a d) \left (2 b^2 c e^2 (d e-c f)-a b e \left (d^2 e^2-13 c d e f+12 c^2 f^2\right )+a^2 f \left (d^2 e^2-60 c d e f+64 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{15 a^{5/2} c e^4 \sqrt {-b e+a f} (d e-c f) \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:

-1/5*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/e/x^5/(f*x^2+e)^(3/2)-1/15*(b/a+d/c-8 
*f/e)*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/e/x^3/(f*x^2+e)^(3/2)+1/15*(2*b^2*c* 
e/a-2*b*d*e+2*a*d^2*e/c+11*b*c*f+11*a*d*f-48*a*c*f^2/e)*(b*x^2+a)^(1/2)*(d 
*x^2+c)^(1/2)/a/c/e^2/x/(f*x^2+e)^(3/2)+2/15*f*(b^2*c^2*e^2-a*b*c*e*(-6*c* 
f+d*e)+a^2*(-32*c^2*f^2+6*c*d*e*f+d^2*e^2))*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1 
/2)/a^2/c^2/e^4/(f*x^2+e)^(3/2)-1/15*(2*b^3*c^2*e^3*(-c*f+d*e)-a*b^2*c*e^2 
*(11*c^2*f^2-13*c*d*e*f+2*d^2*e^2)-a^3*f*(128*c^3*f^3-136*c^2*d*e*f^2+11*c 
*d^2*e^2*f+2*d^3*e^3)+a^2*b*e*(136*c^3*f^3-146*c^2*d*e*f^2+13*c*d^2*e^2*f+ 
2*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 
^(5/2)/c/e^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)+1/15*(-a*d+b*c)*(2*b^2*c*e^2*(-c*f+d*e)-a*b*e*(12*c^2*f^2-13 
*c*d*e*f+d^2*e^2)+a^2*f*(64*c^2*f^2-60*c*d*e*f+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^(5/2)/c/e^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)
 

Mathematica [F]

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

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

Output:

Integrate[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(x^6*(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.

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

\(\Big \downarrow \) 450

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

Input:

Int[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(x^6*(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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)/(f^3*x^12 + 3*e*f 
^2*x^10 + 3*e^2*f*x^8 + e^3*x^6), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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