\(\int \frac {1}{(a+b x^2)^{3/2} (c+d x^2)^{5/2} \sqrt {e+f x^2}} \, dx\) [481]

Optimal result

Integrand size = 34, antiderivative size = 627 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx=\frac {b^2 x \sqrt {e+f x^2}}{a (b c-a d) (b e-a f) \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}+\frac {d \left (a b d^2 e-a^2 d^2 f+3 b^2 c (d e-c f)\right ) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{3 a c (b c-a d)^2 (b e-a f) (d e-c f) \left (c+d x^2\right )^{3/2}}+\frac {\sqrt {e} \left (a b^2 c d^2 e (7 d e-9 c f)+2 a^3 d^3 f (d e-2 c f)+3 b^3 c^2 (d e-c f)^2-a^2 b d^2 \left (2 d^2 e^2+3 c d e f-9 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{3 a c^2 (b c-a d)^3 (b e-a f) (d e-c f)^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}+\frac {d \sqrt {e} \left (a b d^2 e-a^2 d^2 f-9 b^2 c (d e-c f)\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{3 a c (b c-a d)^3 (d e-c f)^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \] Output:

b^2*x*(f*x^2+e)^(1/2)/a/(-a*d+b*c)/(-a*f+b*e)/(b*x^2+a)^(1/2)/(d*x^2+c)^(3 
/2)+1/3*d*(a*b*d^2*e-a^2*d^2*f+3*b^2*c*(-c*f+d*e))*x*(b*x^2+a)^(1/2)*(f*x^ 
2+e)^(1/2)/a/c/(-a*d+b*c)^2/(-a*f+b*e)/(-c*f+d*e)/(d*x^2+c)^(3/2)+1/3*e^(1 
/2)*(a*b^2*c*d^2*e*(-9*c*f+7*d*e)+2*a^3*d^3*f*(-2*c*f+d*e)+3*b^3*c^2*(-c*f 
+d*e)^2-a^2*b*d^2*(-9*c^2*f^2+3*c*d*e*f+2*d^2*e^2))*(b*x^2+a)^(1/2)*(c*(f* 
x^2+e)/e/(d*x^2+c))^(1/2)*EllipticE((-c*f+d*e)^(1/2)*x/e^(1/2)/(d*x^2+c)^( 
1/2),(-(-a*d+b*c)*e/a/(-c*f+d*e))^(1/2))/a/c^2/(-a*d+b*c)^3/(-a*f+b*e)/(-c 
*f+d*e)^(3/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(f*x^2+e)^(1/2)+1/3*d*e^(1/2 
)*(a*b*d^2*e-a^2*d^2*f-9*b^2*c*(-c*f+d*e))*(b*x^2+a)^(1/2)*(c*(f*x^2+e)/e/ 
(d*x^2+c))^(1/2)*EllipticF((-c*f+d*e)^(1/2)*x/e^(1/2)/(d*x^2+c)^(1/2),(-(- 
a*d+b*c)*e/a/(-c*f+d*e))^(1/2))/a/c/(-a*d+b*c)^3/(-c*f+d*e)^(3/2)/(c*(b*x^ 
2+a)/a/(d*x^2+c))^(1/2)/(f*x^2+e)^(1/2)
 

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 434

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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