\(\int \frac {1}{(a+b x^2)^{5/2} (c+d x^2)^{3/2} \sqrt {e+f x^2}} \, dx\) [483]

Optimal result

Integrand size = 34, antiderivative size = 636 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}} \, dx=\frac {b^2 x \sqrt {e+f x^2}}{3 a (b c-a d) (b e-a f) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}+\frac {2 b^2 \left (b^2 c e-3 a b d e-2 a b c f+4 a^2 d f\right ) x \sqrt {e+f x^2}}{3 a^2 (b c-a d)^2 (b e-a f)^2 \sqrt {a+b x^2} \sqrt {c+d x^2}}+\frac {\sqrt {e} \left (6 a^3 b d^3 e f-3 a^4 d^3 f^2+2 b^4 c^2 e (d e-c f)-a b^3 c \left (7 d^2 e^2-3 c d e f-4 c^2 f^2\right )-3 a^2 b^2 d \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{3 a^2 c (b c-a d)^3 (b e-a f)^2 \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}+\frac {b \sqrt {e} \left (9 a b d^2 e-9 a^2 d^2 f-b^2 c (d e-c f)\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{3 a^2 (b c-a d)^3 (b e-a f) \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \] Output:

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

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 434

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

integrate(1/(b*x^2+a)^(5/2)/(d*x^2+c)^(3/2)/(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^3*d^2*f*x^12 + 
 (b^3*d^2*e + (2*b^3*c*d + 3*a*b^2*d^2)*f)*x^10 + ((2*b^3*c*d + 3*a*b^2*d^ 
2)*e + (b^3*c^2 + 6*a*b^2*c*d + 3*a^2*b*d^2)*f)*x^8 + ((b^3*c^2 + 6*a*b^2* 
c*d + 3*a^2*b*d^2)*e + (3*a*b^2*c^2 + 6*a^2*b*c*d + a^3*d^2)*f)*x^6 + a^3* 
c^2*e + ((3*a*b^2*c^2 + 6*a^2*b*c*d + a^3*d^2)*e + (3*a^2*b*c^2 + 2*a^3*c* 
d)*f)*x^4 + (a^3*c^2*f + (3*a^2*b*c^2 + 2*a^3*c*d)*e)*x^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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