\(\int \frac {(a+b x^2)^{5/2} (e+f x^2)^{3/2}}{(c+d x^2)^{5/2}} \, dx\) [436]

Optimal result

Integrand size = 34, antiderivative size = 862 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {(b c-a d)^2 (d e-c f) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{3 c d^3 \left (c+d x^2\right )^{3/2}}+\frac {b (5 b d e-9 b c f+9 a d f) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{8 d^3 \sqrt {c+d x^2}}+\frac {b^2 f x^3 \sqrt {a+b x^2} \sqrt {e+f x^2}}{4 d^2 \sqrt {c+d x^2}}+\frac {\sqrt {e} \sqrt {d e-c f} \left (a b c d (24 d e-115 c f)-5 b^2 c^2 (11 d e-21 c f)+16 a^2 d^2 (d e+c f)\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{24 c^2 d^4 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}+\frac {\sqrt {e} \left (8 a^3 d^3 f (d e+2 c f)-8 a^2 b d^2 \left (d^2 e^2-12 c d e f+20 c^2 f^2\right )-3 b^3 c^2 \left (3 d^2 e^2-30 c d e f+35 c^2 f^2\right )+a b^2 c d \left (32 d^2 e^2-215 c d e f+255 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{24 a c d^5 \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}+\frac {b c \sqrt {e} \left (15 a^2 d^2 f^2+10 a b d f (3 d e-5 c f)+b^2 \left (3 d^2 e^2-30 c d e f+35 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticPi}\left (\frac {d e}{d e-c f},\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{8 a d^5 \sqrt {d e-c f} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \] Output:

1/3*(-a*d+b*c)^2*(-c*f+d*e)*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/c/d^3/(d*x^2 
+c)^(3/2)+1/8*b*(9*a*d*f-9*b*c*f+5*b*d*e)*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2 
)/d^3/(d*x^2+c)^(1/2)+1/4*b^2*f*x^3*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/d^2/(d 
*x^2+c)^(1/2)+1/24*e^(1/2)*(-c*f+d*e)^(1/2)*(a*b*c*d*(-115*c*f+24*d*e)-5*b 
^2*c^2*(-21*c*f+11*d*e)+16*a^2*d^2*(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^4/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2) 
/(f*x^2+e)^(1/2)+1/24*e^(1/2)*(8*a^3*d^3*f*(2*c*f+d*e)-8*a^2*b*d^2*(20*c^2 
*f^2-12*c*d*e*f+d^2*e^2)-3*b^3*c^2*(35*c^2*f^2-30*c*d*e*f+3*d^2*e^2)+a*b^2 
*c*d*(255*c^2*f^2-215*c*d*e*f+32*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^5/(-c*f+d*e)^(1/2)/(c*(b*x^2+a)/a/(d 
*x^2+c))^(1/2)/(f*x^2+e)^(1/2)+1/8*b*c*e^(1/2)*(15*a^2*d^2*f^2+10*a*b*d*f* 
(-5*c*f+3*d*e)+b^2*(35*c^2*f^2-30*c*d*e*f+3*d^2*e^2))*(b*x^2+a)^(1/2)*(c*( 
f*x^2+e)/e/(d*x^2+c))^(1/2)*EllipticPi((-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^5/(-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]

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

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

Output:

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

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

\(\Big \downarrow \) 434

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((b^2*f*x^6 + (b^2*e + 2*a*b*f)*x^4 + a^2*e + (2*a*b*e + a^2*f)*x^ 
2)*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. \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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