\(\int \frac {(a+b x^2)^{3/2} \sqrt {e+f x^2}}{(c+d x^2)^{9/2}} \, dx\) [394]

Optimal result

Integrand size = 34, antiderivative size = 830 \[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{9/2}} \, dx=\frac {\left (\frac {a}{c}-\frac {b}{d}\right ) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{7 \left (c+d x^2\right )^{7/2}}+\frac {(a d (6 d e-5 c f)+b c (2 d e-3 c f)) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{35 c^2 d (d e-c f) \left (c+d x^2\right )^{5/2}}+\frac {\left (2 b^2 c^2 \left (3 d^2 e^2-4 c d e f+3 c^2 f^2\right )+a b c d \left (15 d^2 e^2-29 c d e f+6 c^2 f^2\right )-a^2 d^2 \left (24 d^2 e^2-43 c d e f+15 c^2 f^2\right )\right ) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{105 c^3 d (b c-a d) (d e-c f)^2 \left (c+d x^2\right )^{3/2}}+\frac {\sqrt {e} \left (2 b^3 c^3 e^2 (3 d e-7 c f)+a b^2 c^2 e \left (12 d^2 e^2-37 c d e f+49 c^2 f^2\right )-a^2 b c \left (72 d^3 e^3-197 c d^2 e^2 f+170 c^2 d e f^2-21 c^3 f^3\right )+a^3 d \left (48 d^3 e^3-128 c d^2 e^2 f+103 c^2 d e f^2-15 c^3 f^3\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{105 c^4 (b c-a d)^2 (d e-c f)^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}-\frac {\sqrt {e} (b e-a f) \left (b^2 c^2 e (3 d e-7 c f)+a^2 d \left (24 d^2 e^2-43 c d e f+15 c^2 f^2\right )-a b c \left (33 d^2 e^2-62 c d e f+21 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{105 c^3 (b c-a d)^2 (d e-c f)^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \] Output:

1/7*(a/c-b/d)*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(7/2)+1/35*(a*d* 
(-5*c*f+6*d*e)+b*c*(-3*c*f+2*d*e))*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/c^2/d 
/(-c*f+d*e)/(d*x^2+c)^(5/2)+1/105*(2*b^2*c^2*(3*c^2*f^2-4*c*d*e*f+3*d^2*e^ 
2)+a*b*c*d*(6*c^2*f^2-29*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))*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/c^3/d/(-a*d+b*c)/(-c*f+d* 
e)^2/(d*x^2+c)^(3/2)+1/105*e^(1/2)*(2*b^3*c^3*e^2*(-7*c*f+3*d*e)+a*b^2*c^2 
*e*(49*c^2*f^2-37*c*d*e*f+12*d^2*e^2)-a^2*b*c*(-21*c^3*f^3+170*c^2*d*e*f^2 
-197*c*d^2*e^2*f+72*d^3*e^3)+a^3*d*(-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)*Ellipti 
cE((-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)^2/(-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)*(b^2*c^2*e*(-7*c*f+3*d*e)+a^2*d*( 
15*c^2*f^2-43*c*d*e*f+24*d^2*e^2)-a*b*c*(21*c^2*f^2-62*c*d*e*f+33*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)^2/(-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]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{9/2}} \, dx=\int \frac {\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{9/2}} \, dx \] Input:

Integrate[((a + b*x^2)^(3/2)*Sqrt[e + f*x^2])/(c + d*x^2)^(9/2),x]
 

Output:

Integrate[((a + b*x^2)^(3/2)*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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{9/2}} \, dx\)

\(\Big \downarrow \) 434

\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{9/2}}dx\)

Input:

Int[((a + b*x^2)^(3/2)*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]

\[\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} \sqrt {f \,x^{2}+e}}{\left (x^{2} d +c \right )^{\frac {9}{2}}}d x\]

Input:

int((b*x^2+a)^(3/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(9/2),x)
 

Output:

int((b*x^2+a)^(3/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(9/2),x)
 

Fricas [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{9/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {f x^{2} + e}}{{\left (d x^{2} + c\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(9/2),x, algorithm="fr 
icas")
 

Output:

integral((b*x^2 + a)^(3/2)*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. \[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{9/2}} \, dx=\text {Timed out} \] Input:

integrate((b*x**2+a)**(3/2)*(f*x**2+e)**(1/2)/(d*x**2+c)**(9/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{9/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {f x^{2} + e}}{{\left (d x^{2} + c\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(9/2),x, algorithm="ma 
xima")
 

Output:

integrate((b*x^2 + a)^(3/2)*sqrt(f*x^2 + e)/(d*x^2 + c)^(9/2), x)
 

Giac [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{9/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {f x^{2} + e}}{{\left (d x^{2} + c\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(9/2),x, algorithm="gi 
ac")
 

Output:

integrate((b*x^2 + a)^(3/2)*sqrt(f*x^2 + e)/(d*x^2 + c)^(9/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{9/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,\sqrt {f\,x^2+e}}{{\left (d\,x^2+c\right )}^{9/2}} \,d x \] Input:

int(((a + b*x^2)^(3/2)*(e + f*x^2)^(1/2))/(c + d*x^2)^(9/2),x)
 

Output:

int(((a + b*x^2)^(3/2)*(e + f*x^2)^(1/2))/(c + d*x^2)^(9/2), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{9/2}} \, dx=\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} \sqrt {f \,x^{2}+e}}{\left (d \,x^{2}+c \right )^{\frac {9}{2}}}d x \] Input:

int((b*x^2+a)^(3/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(9/2),x)
 

Output:

int((b*x^2+a)^(3/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(9/2),x)