\(\int \frac {1}{\sqrt {a+b x^2} (c+d x^2)^{3/2} (e+f x^2)^{5/2}} \, dx\) [553]

Optimal result

Integrand size = 34, antiderivative size = 643 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{5/2}} \, dx=-\frac {d^2 x \sqrt {a+b x^2}}{c (b c-a d) (d e-c f) \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}+\frac {f \left (a d f (3 d e+c f)-b \left (3 d^2 e^2+c^2 f^2\right )\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c (b c-a d) e (b e-a f) (d e-c f)^2 \left (e+f x^2\right )^{3/2}}-\frac {\left (a^2 d f^2 \left (3 d^2 e^2+7 c d e f-2 c^2 f^2\right )+b^2 e \left (3 d^3 e^3+9 c^2 d e f^2-4 c^3 f^3\right )-a b f \left (6 d^3 e^3+9 c d^2 e^2 f+3 c^2 d e f^2-2 c^3 f^3\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{3 \sqrt {a} (b c-a d) e^2 (-b e+a f)^{3/2} (d e-c f)^3 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}-\frac {f \left (a d f (9 d e-c f)-b \left (9 d^2 e^2-c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{3 \sqrt {a} e (-b e+a f)^{3/2} (d e-c f)^3 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:

-d^2*x*(b*x^2+a)^(1/2)/c/(-a*d+b*c)/(-c*f+d*e)/(d*x^2+c)^(1/2)/(f*x^2+e)^( 
3/2)+1/3*f*(a*d*f*(c*f+3*d*e)-b*(c^2*f^2+3*d^2*e^2))*x*(b*x^2+a)^(1/2)*(d* 
x^2+c)^(1/2)/c/(-a*d+b*c)/e/(-a*f+b*e)/(-c*f+d*e)^2/(f*x^2+e)^(3/2)-1/3*(a 
^2*d*f^2*(-2*c^2*f^2+7*c*d*e*f+3*d^2*e^2)+b^2*e*(-4*c^3*f^3+9*c^2*d*e*f^2+ 
3*d^3*e^3)-a*b*f*(-2*c^3*f^3+3*c^2*d*e*f^2+9*c*d^2*e^2*f+6*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)/(-a*d+b*c) 
/e^2/(a*f-b*e)^(3/2)/(-c*f+d*e)^3/(d*x^2+c)^(1/2)/(e*(b*x^2+a)/a/(f*x^2+e) 
)^(1/2)-1/3*f*(a*d*f*(-c*f+9*d*e)-b*(-c^2*f^2+9*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)/e/(a*f-b*e)^(3/2)/(-c 
*f+d*e)^3/(d*x^2+c)^(1/2)/(e*(b*x^2+a)/a/(f*x^2+e))^(1/2)
 

Mathematica [F]

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

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

Output:

Integrate[1/(Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)*(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.

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

\(\Big \downarrow \) 434

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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