\(\int \frac {(a+b x^2)^{5/2} (c+d x^2)^{3/2}}{(e+f x^2)^{3/2}} \, dx\) [520]

Optimal result

Integrand size = 34, antiderivative size = 925 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}}{\left (e+f x^2\right )^{3/2}} \, dx=\frac {\left (33 a^2 d^2 f^2-68 a b d f (d e-c f)+b^2 \left (35 d^2 e^2-38 c d e f+3 c^2 f^2\right )\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{48 d f^3 \sqrt {e+f x^2}}-\frac {b (7 b d e-7 b c f-13 a d f) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{24 f^2 \sqrt {e+f x^2}}+\frac {b^2 d x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{6 f \sqrt {e+f x^2}}+\frac {c \sqrt {-b e+a f} \left (2 a b d e f (95 d e-82 c f)-3 a^2 d f^2 (27 d e-16 c f)-b^2 e \left (105 d^2 e^2-100 c d e f+3 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{48 \sqrt {a} d e f^4 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}-\frac {\sqrt {-b e+a f} \left (3 a^2 d f^2 (5 d e-16 c f)-2 a b f \left (60 d^2 e^2-145 c d e f+72 c^2 f^2\right )+b^2 e \left (105 d^2 e^2-240 c d e f+127 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{48 \sqrt {a} f^5 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}+\frac {e \left (5 a^3 d^3 f^3-45 a^2 b d^2 f^2 (d e-c f)+15 a b^2 d f \left (5 d^2 e^2-6 c d e f+c^2 f^2\right )-b^3 \left (35 d^3 e^3-45 c d^2 e^2 f+9 c^2 d e f^2+c^3 f^3\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticPi}\left (-\frac {a f}{b e-a f},\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{16 \sqrt {a} d f^5 \sqrt {-b e+a f} \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:

1/48*(33*a^2*d^2*f^2-68*a*b*d*f*(-c*f+d*e)+b^2*(3*c^2*f^2-38*c*d*e*f+35*d^ 
2*e^2))*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/d/f^3/(f*x^2+e)^(1/2)-1/24*b*(-1 
3*a*d*f-7*b*c*f+7*b*d*e)*x^3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/f^2/(f*x^2+e) 
^(1/2)+1/6*b^2*d*x^5*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/f/(f*x^2+e)^(1/2)+1/4 
8*c*(a*f-b*e)^(1/2)*(2*a*b*d*e*f*(-82*c*f+95*d*e)-3*a^2*d*f^2*(-16*c*f+27* 
d*e)-b^2*e*(3*c^2*f^2-100*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)*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)/d/e/f^4/(d*x^2+c)^(1/2)/(e*(b*x 
^2+a)/a/(f*x^2+e))^(1/2)-1/48*(a*f-b*e)^(1/2)*(3*a^2*d*f^2*(-16*c*f+5*d*e) 
-2*a*b*f*(72*c^2*f^2-145*c*d*e*f+60*d^2*e^2)+b^2*e*(127*c^2*f^2-240*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)/f^5/(d*x^2+c)^(1/2)/(e*(b*x^2+a)/a/(f*x^2+e))^(1/2)+1/16*e*(5*a 
^3*d^3*f^3-45*a^2*b*d^2*f^2*(-c*f+d*e)+15*a*b^2*d*f*(c^2*f^2-6*c*d*e*f+5*d 
^2*e^2)-b^3*(c^3*f^3+9*c^2*d*e*f^2-45*c*d^2*e^2*f+35*d^3*e^3))*(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) 
/d/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]

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

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

Output:

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

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

\(\Big \downarrow \) 434

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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