\(\int \frac {(a+b x^2)^{3/2} (e+f x^2)^{3/2}}{(c+d x^2)^{5/2}} \, dx\) [430]

Optimal result

Integrand size = 34, antiderivative size = 691 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{5/2}} \, dx=-\frac {(b c-a d) (d e-c f) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{3 c d^2 \left (c+d x^2\right )^{3/2}}+\frac {b f x \sqrt {a+b x^2} \sqrt {e+f x^2}}{2 d^2 \sqrt {c+d x^2}}+\frac {\sqrt {e} \sqrt {d e-c f} (b c (4 d e-15 c f)+4 a d (d e+c f)) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{6 c^2 d^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}-\frac {\sqrt {e} \left (3 b^2 c^2 f (3 d e-5 c f)-2 a^2 d^2 f (d e+2 c f)+2 a b d \left (d^2 e^2-7 c d e f+12 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{6 a c d^4 \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}+\frac {b c \sqrt {e} f (3 b d e-5 b c f+3 a d f) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticPi}\left (\frac {d e}{d e-c f},\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{2 a d^4 \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \] Output:

-1/3*(-a*d+b*c)*(-c*f+d*e)*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/c/d^2/(d*x^2+ 
c)^(3/2)+1/2*b*f*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/d^2/(d*x^2+c)^(1/2)+1/6 
*e^(1/2)*(-c*f+d*e)^(1/2)*(b*c*(-15*c*f+4*d*e)+4*a*d*(c*f+d*e))*(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))/c^2/d^3/(c*(b*x^2+a)/ 
a/(d*x^2+c))^(1/2)/(f*x^2+e)^(1/2)-1/6*e^(1/2)*(3*b^2*c^2*f*(-5*c*f+3*d*e) 
-2*a^2*d^2*f*(2*c*f+d*e)+2*a*b*d*(12*c^2*f^2-7*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)*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/d^4/(-c*f+d*e)^( 
1/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(f*x^2+e)^(1/2)+1/2*b*c*e^(1/2)*f*(3* 
a*d*f-5*b*c*f+3*b*d*e)*(b*x^2+a)^(1/2)*(c*(f*x^2+e)/e/(d*x^2+c))^(1/2)*Ell 
ipticPi((-c*f+d*e)^(1/2)*x/e^(1/2)/(d*x^2+c)^(1/2),d*e/(-c*f+d*e),(-(-a*d+ 
b*c)*e/a/(-c*f+d*e))^(1/2))/a/d^4/(-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]

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

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

Output:

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

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

\(\Big \downarrow \) 434

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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