\(\int \frac {1}{(a+b x^2)^{3/2} \sqrt {c+d x^2} (e+f x^2)^{5/2}} \, dx\) [555]

Optimal result

Integrand size = 34, antiderivative size = 651 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\frac {b^2 x \sqrt {c+d x^2}}{a (b c-a d) (b e-a f) \sqrt {a+b x^2} \left (e+f x^2\right )^{3/2}}-\frac {f \left (a b c f^2-a^2 d f^2-3 b^2 e (d e-c f)\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 a (b c-a d) e (b e-a f)^2 (d e-c f) \left (e+f x^2\right )^{3/2}}+\frac {c \left (a b^2 c e f^2 (9 d e-7 c f)-3 b^3 e^2 (d e-c f)^2+2 a^3 d f^3 (2 d e-c f)-a^2 b f^2 \left (9 d^2 e^2-3 c d e f-2 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{3 a^{3/2} (b c-a d) e^2 (-b e+a f)^{5/2} (d e-c f)^2 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}+\frac {\left (3 b^2 e (d e-c f)^2+a^2 d f^2 (3 d e-c f)-a b f \left (6 d^2 e^2-3 c d e f-c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{3 a^{3/2} e (-b e+a f)^{5/2} (d e-c f)^2 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:

b^2*x*(d*x^2+c)^(1/2)/a/(-a*d+b*c)/(-a*f+b*e)/(b*x^2+a)^(1/2)/(f*x^2+e)^(3 
/2)-1/3*f*(a*b*c*f^2-a^2*d*f^2-3*b^2*e*(-c*f+d*e))*x*(b*x^2+a)^(1/2)*(d*x^ 
2+c)^(1/2)/a/(-a*d+b*c)/e/(-a*f+b*e)^2/(-c*f+d*e)/(f*x^2+e)^(3/2)+1/3*c*(a 
*b^2*c*e*f^2*(-7*c*f+9*d*e)-3*b^3*e^2*(-c*f+d*e)^2+2*a^3*d*f^3*(-c*f+2*d*e 
)-a^2*b*f^2*(-2*c^2*f^2-3*c*d*e*f+9*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^(3/2)/(-a*d+b*c)/e^2/(a*f-b*e)^(5/2)/(- 
c*f+d*e)^2/(d*x^2+c)^(1/2)/(e*(b*x^2+a)/a/(f*x^2+e))^(1/2)+1/3*(3*b^2*e*(- 
c*f+d*e)^2+a^2*d*f^2*(-c*f+3*d*e)-a*b*f*(-c^2*f^2-3*c*d*e*f+6*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^(3/2)/e/(a*f- 
b*e)^(5/2)/(-c*f+d*e)^2/(d*x^2+c)^(1/2)/(e*(b*x^2+a)/a/(f*x^2+e))^(1/2)
 

Mathematica [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx \] Input:

Integrate[1/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]*(e + f*x^2)^(5/2)),x]
 

Output:

Integrate[1/((a + b*x^2)^(3/2)*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.

\(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 434

\(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}}dx\)

Input:

Int[1/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]*(e + f*x^2)^(5/2)),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 434

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2)^(r_.), x_Symbol] :> Unintegrable[(a + b*x^2)^p*(c + d*x^2)^q*(e + f* 
x^2)^r, x] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x]
 

Maple [F]

\[\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {3}{2}} \sqrt {x^{2} d +c}\, \left (f \,x^{2}+e \right )^{\frac {5}{2}}}d x\]

Input:

int(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x)
 

Output:

int(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x)
 

Fricas [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(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)/(b^2*d*f^3*x^12 + 
 (3*b^2*d*e*f^2 + (b^2*c + 2*a*b*d)*f^3)*x^10 + (3*b^2*d*e^2*f + 3*(b^2*c 
+ 2*a*b*d)*e*f^2 + (2*a*b*c + a^2*d)*f^3)*x^8 + (b^2*d*e^3 + a^2*c*f^3 + 3 
*(b^2*c + 2*a*b*d)*e^2*f + 3*(2*a*b*c + a^2*d)*e*f^2)*x^6 + a^2*c*e^3 + (3 
*a^2*c*e*f^2 + (b^2*c + 2*a*b*d)*e^3 + 3*(2*a*b*c + a^2*d)*e^2*f)*x^4 + (3 
*a^2*c*e^2*f + (2*a*b*c + a^2*d)*e^3)*x^2), x)
 

Sympy [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (a + b x^{2}\right )^{\frac {3}{2}} \sqrt {c + d x^{2}} \left (e + f x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(1/(b*x**2+a)**(3/2)/(d*x**2+c)**(1/2)/(f*x**2+e)**(5/2),x)
 

Output:

Integral(1/((a + b*x**2)**(3/2)*sqrt(c + d*x**2)*(e + f*x**2)**(5/2)), x)
 

Maxima [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x, algorithm=" 
maxima")
 

Output:

integrate(1/((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*(f*x^2 + e)^(5/2)), x)
 

Giac [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x, algorithm=" 
giac")
 

Output:

integrate(1/((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*(f*x^2 + e)^(5/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{3/2}\,\sqrt {d\,x^2+c}\,{\left (f\,x^2+e\right )}^{5/2}} \,d x \] Input:

int(1/((a + b*x^2)^(3/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^(5/2)),x)
 

Output:

int(1/((a + b*x^2)^(3/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {3}{2}} \sqrt {d \,x^{2}+c}\, \left (f \,x^{2}+e \right )^{\frac {5}{2}}}d x \] Input:

int(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x)
 

Output:

int(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(5/2),x)