\(\int \frac {(a+b x^2)^{5/2} (c+d x^2)^{3/2}}{(e+f x^2)^{5/2}} \, dx\) [546]

Optimal result

Integrand size = 34, antiderivative size = 812 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}}{\left (e+f x^2\right )^{5/2}} \, dx=-\frac {(b e-a f)^2 (d e-c f) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 e f^3 \left (e+f x^2\right )^{3/2}}-\frac {b (9 b d e-5 b c f-9 a d f) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 f^3 \sqrt {e+f x^2}}+\frac {b^2 d x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{4 f^2 \sqrt {e+f x^2}}-\frac {c \sqrt {-b e+a f} \left (a b e f (115 d e-24 c f)-5 b^2 e^2 (21 d e-11 c f)-16 a^2 f^2 (d e+c f)\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{24 \sqrt {a} e^2 f^4 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}-\frac {\sqrt {-b e+a f} \left (8 a^2 c d f^3+a b f \left (45 d^2 e^2-80 c d e f-8 c^2 f^2\right )-b^2 e \left (105 d^2 e^2-195 c d e f+64 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{24 \sqrt {a} e f^5 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}+\frac {b e \left (15 a^2 d^2 f^2-10 a b d f (5 d e-3 c f)+b^2 \left (35 d^2 e^2-30 c d e f+3 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticPi}\left (-\frac {a f}{b e-a f},\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{8 \sqrt {a} f^5 \sqrt {-b e+a f} \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:

-1/3*(-a*f+b*e)^2*(-c*f+d*e)*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/e/f^3/(f*x^ 
2+e)^(3/2)-1/8*b*(-9*a*d*f-5*b*c*f+9*b*d*e)*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1 
/2)/f^3/(f*x^2+e)^(1/2)+1/4*b^2*d*x^3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/f^2/ 
(f*x^2+e)^(1/2)-1/24*c*(a*f-b*e)^(1/2)*(a*b*e*f*(-24*c*f+115*d*e)-5*b^2*e^ 
2*(-11*c*f+21*d*e)-16*a^2*f^2*(c*f+d*e))*(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)/e^2/f^4/(d*x^2+c)^(1/2)/(e*(b*x^2+a)/a 
/(f*x^2+e))^(1/2)-1/24*(a*f-b*e)^(1/2)*(8*a^2*c*d*f^3+a*b*f*(-8*c^2*f^2-80 
*c*d*e*f+45*d^2*e^2)-b^2*e*(64*c^2*f^2-195*c*d*e*f+105*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/f^5/(d*x^2+c 
)^(1/2)/(e*(b*x^2+a)/a/(f*x^2+e))^(1/2)+1/8*b*e*(15*a^2*d^2*f^2-10*a*b*d*f 
*(-3*c*f+5*d*e)+b^2*(3*c^2*f^2-30*c*d*e*f+35*d^2*e^2))*(b*x^2+a)^(1/2)*(e* 
(d*x^2+c)/c/(f*x^2+e))^(1/2)*EllipticPi((a*f-b*e)^(1/2)*x/a^(1/2)/(f*x^2+e 
)^(1/2),-a*f/(-a*f+b*e),(a*(-c*f+d*e)/c/(-a*f+b*e))^(1/2))/a^(1/2)/f^5/(a* 
f-b*e)^(1/2)/(d*x^2+c)^(1/2)/(e*(b*x^2+a)/a/(f*x^2+e))^(1/2)
 

Mathematica [F]

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

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

Output:

Integrate[((a + b*x^2)^(5/2)*(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.

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

\(\Big \downarrow \) 434

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

Input:

Int[((a + b*x^2)^(5/2)*(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]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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