\(\int \frac {\sqrt {a+b x^2} \sqrt {e+f x^2}}{(c+d x^2)^{9/2}} \, dx\) [388]

Optimal result

Integrand size = 34, antiderivative size = 870 \[ \int \frac {\sqrt {a+b x^2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{9/2}} \, dx=\frac {x \sqrt {a+b x^2} \sqrt {e+f x^2}}{7 c \left (c+d x^2\right )^{7/2}}-\frac {(a d (6 d e-5 c f)-b c (5 d e-4 c f)) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{35 c^2 (b c-a d) (d e-c f) \left (c+d x^2\right )^{5/2}}+\frac {\left (b^2 c^2 \left (15 d^2 e^2-27 c d e f+8 c^2 f^2\right )+a^2 d^2 \left (24 d^2 e^2-43 c d e f+15 c^2 f^2\right )-a b c d \left (43 d^2 e^2-78 c d e f+27 c^2 f^2\right )\right ) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{105 c^3 (b c-a d)^2 (d e-c f)^2 \left (c+d x^2\right )^{3/2}}+\frac {\sqrt {e} \left (b^3 c^3 e \left (15 d^2 e^2-42 c d e f+35 c^2 f^2\right )-a b^2 c^2 \left (103 d^3 e^3-282 c d^2 e^2 f+238 c^2 d e f^2-35 c^3 f^3\right )+2 a^2 b c d \left (64 d^3 e^3-172 c d^2 e^2 f+141 c^2 d e f^2-21 c^3 f^3\right )-a^3 d^2 \left (48 d^3 e^3-128 c d^2 e^2 f+103 c^2 d e f^2-15 c^3 f^3\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{105 c^4 (b c-a d)^3 (d e-c f)^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}+\frac {\sqrt {e} (b e-a f) \left (a^2 d^2 \left (24 d^2 e^2-43 c d e f+15 c^2 f^2\right )+b^2 c^2 \left (45 d^2 e^2-84 c d e f+35 c^2 f^2\right )-a b c d \left (61 d^2 e^2-111 c d e f+42 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{105 c^3 (b c-a d)^3 (d e-c f)^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \] Output:

1/7*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/c/(d*x^2+c)^(7/2)-1/35*(a*d*(-5*c*f+ 
6*d*e)-b*c*(-4*c*f+5*d*e))*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/c^2/(-a*d+b*c 
)/(-c*f+d*e)/(d*x^2+c)^(5/2)+1/105*(b^2*c^2*(8*c^2*f^2-27*c*d*e*f+15*d^2*e 
^2)+a^2*d^2*(15*c^2*f^2-43*c*d*e*f+24*d^2*e^2)-a*b*c*d*(27*c^2*f^2-78*c*d* 
e*f+43*d^2*e^2))*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/c^3/(-a*d+b*c)^2/(-c*f+ 
d*e)^2/(d*x^2+c)^(3/2)+1/105*e^(1/2)*(b^3*c^3*e*(35*c^2*f^2-42*c*d*e*f+15* 
d^2*e^2)-a*b^2*c^2*(-35*c^3*f^3+238*c^2*d*e*f^2-282*c*d^2*e^2*f+103*d^3*e^ 
3)+2*a^2*b*c*d*(-21*c^3*f^3+141*c^2*d*e*f^2-172*c*d^2*e^2*f+64*d^3*e^3)-a^ 
3*d^2*(-15*c^3*f^3+103*c^2*d*e*f^2-128*c*d^2*e^2*f+48*d^3*e^3))*(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^4/(-a*d+b*c)^3/(-c* 
f+d*e)^(5/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(f*x^2+e)^(1/2)+1/105*e^(1/2) 
*(-a*f+b*e)*(a^2*d^2*(15*c^2*f^2-43*c*d*e*f+24*d^2*e^2)+b^2*c^2*(35*c^2*f^ 
2-84*c*d*e*f+45*d^2*e^2)-a*b*c*d*(42*c^2*f^2-111*c*d*e*f+61*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))/c^3/(-a*d+b*c)^ 
3/(-c*f+d*e)^(5/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(f*x^2+e)^(1/2)
 

Mathematica [F]

\[ \int \frac {\sqrt {a+b x^2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{9/2}} \, dx=\int \frac {\sqrt {a+b x^2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{9/2}} \, dx \] Input:

Integrate[(Sqrt[a + b*x^2]*Sqrt[e + f*x^2])/(c + d*x^2)^(9/2),x]
 

Output:

Integrate[(Sqrt[a + b*x^2]*Sqrt[e + f*x^2])/(c + d*x^2)^(9/2), 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.

\(\displaystyle \int \frac {\sqrt {a+b x^2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{9/2}} \, dx\)

\(\Big \downarrow \) 434

\(\displaystyle \int \frac {\sqrt {a+b x^2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{9/2}}dx\)

Input:

Int[(Sqrt[a + b*x^2]*Sqrt[e + f*x^2])/(c + d*x^2)^(9/2),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]

\[\int \frac {\sqrt {b \,x^{2}+a}\, \sqrt {f \,x^{2}+e}}{\left (x^{2} d +c \right )^{\frac {9}{2}}}d x\]

Input:

int((b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(9/2),x)
 

Output:

int((b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(9/2),x)
 

Fricas [F]

\[ \int \frac {\sqrt {a+b x^2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{9/2}} \, dx=\int { \frac {\sqrt {b x^{2} + a} \sqrt {f x^{2} + e}}{{\left (d x^{2} + c\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(9/2),x, algorithm="fr 
icas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{9/2}} \, dx=\text {Timed out} \] Input:

integrate((b*x**2+a)**(1/2)*(f*x**2+e)**(1/2)/(d*x**2+c)**(9/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\sqrt {a+b x^2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{9/2}} \, dx=\int { \frac {\sqrt {b x^{2} + a} \sqrt {f x^{2} + e}}{{\left (d x^{2} + c\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(9/2),x, algorithm="ma 
xima")
 

Output:

integrate(sqrt(b*x^2 + a)*sqrt(f*x^2 + e)/(d*x^2 + c)^(9/2), x)
 

Giac [F]

\[ \int \frac {\sqrt {a+b x^2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{9/2}} \, dx=\int { \frac {\sqrt {b x^{2} + a} \sqrt {f x^{2} + e}}{{\left (d x^{2} + c\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(9/2),x, algorithm="gi 
ac")
 

Output:

integrate(sqrt(b*x^2 + a)*sqrt(f*x^2 + e)/(d*x^2 + c)^(9/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{9/2}} \, dx=\int \frac {\sqrt {b\,x^2+a}\,\sqrt {f\,x^2+e}}{{\left (d\,x^2+c\right )}^{9/2}} \,d x \] Input:

int(((a + b*x^2)^(1/2)*(e + f*x^2)^(1/2))/(c + d*x^2)^(9/2),x)
 

Output:

int(((a + b*x^2)^(1/2)*(e + f*x^2)^(1/2))/(c + d*x^2)^(9/2), x)
 

Reduce [F]

\[ \int \frac {\sqrt {a+b x^2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{9/2}} \, dx=\int \frac {\sqrt {b \,x^{2}+a}\, \sqrt {f \,x^{2}+e}}{\left (d \,x^{2}+c \right )^{\frac {9}{2}}}d x \] Input:

int((b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(9/2),x)
 

Output:

int((b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(9/2),x)