\(\int \frac {(a+b x^2)^{5/2} \sqrt {e+f x^2}}{(c+d x^2)^{9/2}} \, dx\) [400]

Optimal result

Integrand size = 34, antiderivative size = 777 \[ \int \frac {\left (a+b x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{9/2}} \, dx=\frac {(b c-a d)^2 x \sqrt {a+b x^2} \sqrt {e+f x^2}}{7 c d^2 \left (c+d x^2\right )^{7/2}}-\frac {(b c-a d) (b c (9 d e-10 c f)+a d (6 d e-5 c f)) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{35 c^2 d^2 (d e-c f) \left (c+d x^2\right )^{5/2}}+\frac {\left (a^2 d^2 \left (24 d^2 e^2-43 c d e f+15 c^2 f^2\right )+b^2 c^2 \left (8 d^2 e^2-27 c d e f+15 c^2 f^2\right )+a b c d \left (13 d^2 e^2-20 c d e f+15 c^2 f^2\right )\right ) x \sqrt {a+b x^2} \sqrt {e+f x^2}}{105 c^3 d^2 (d e-c f)^2 \left (c+d x^2\right )^{3/2}}+\frac {\sqrt {e} \left (8 b^3 c^3 e^3+3 a b^2 c^2 e^2 (3 d e-11 c f)+2 a^2 b c e \left (8 d^2 e^2-25 c d e f+29 c^2 f^2\right )-a^3 \left (48 d^3 e^3-128 c d^2 e^2 f+103 c^2 d e f^2-15 c^3 f^3\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )|-\frac {(b c-a d) e}{a (d e-c f)}\right )}{105 c^4 (b c-a d) (d e-c f)^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}}-\frac {\sqrt {e} (b e-a f) \left (4 b^2 c^2 e^2+a b c e (5 d e-13 c f)-a^2 \left (24 d^2 e^2-43 c d e f+15 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right ),-\frac {(b c-a d) e}{a (d e-c f)}\right )}{105 c^3 (b c-a d) (d e-c f)^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {e+f x^2}} \] Output:

1/7*(-a*d+b*c)^2*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/c/d^2/(d*x^2+c)^(7/2)-1 
/35*(-a*d+b*c)*(b*c*(-10*c*f+9*d*e)+a*d*(-5*c*f+6*d*e))*x*(b*x^2+a)^(1/2)* 
(f*x^2+e)^(1/2)/c^2/d^2/(-c*f+d*e)/(d*x^2+c)^(5/2)+1/105*(a^2*d^2*(15*c^2* 
f^2-43*c*d*e*f+24*d^2*e^2)+b^2*c^2*(15*c^2*f^2-27*c*d*e*f+8*d^2*e^2)+a*b*c 
*d*(15*c^2*f^2-20*c*d*e*f+13*d^2*e^2))*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/c 
^3/d^2/(-c*f+d*e)^2/(d*x^2+c)^(3/2)+1/105*e^(1/2)*(8*b^3*c^3*e^3+3*a*b^2*c 
^2*e^2*(-11*c*f+3*d*e)+2*a^2*b*c*e*(29*c^2*f^2-25*c*d*e*f+8*d^2*e^2)-a^3*( 
-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)/(-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)*(4*b^2*c^2*e^2+a*b*c*e*(-13*c*f+5*d*e)-a^2*(15*c^2*f^2-43*c*d*e*f+24*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)/(-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]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{9/2}} \, dx=\int \frac {\left (a+b x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{9/2}} \, dx \] Input:

Integrate[((a + b*x^2)^(5/2)*Sqrt[e + f*x^2])/(c + d*x^2)^(9/2),x]
 

Output:

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

\(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{9/2}} \, dx\)

\(\Big \downarrow \) 434

\(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{9/2}}dx\)

Input:

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

\[\int \frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} \sqrt {f \,x^{2}+e}}{\left (x^{2} d +c \right )^{\frac {9}{2}}}d x\]

Input:

int((b*x^2+a)^(5/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(9/2),x)
 

Output:

int((b*x^2+a)^(5/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(9/2),x)
 

Fricas [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{9/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {f x^{2} + e}}{{\left (d x^{2} + c\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(5/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(9/2),x, algorithm="fr 
icas")
 

Output:

integral((b^2*x^4 + 2*a*b*x^2 + a^2)*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. \[ \int \frac {\left (a+b x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{9/2}} \, dx=\text {Timed out} \] Input:

integrate((b*x**2+a)**(5/2)*(f*x**2+e)**(1/2)/(d*x**2+c)**(9/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{9/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {f x^{2} + e}}{{\left (d x^{2} + c\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(5/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(9/2),x, algorithm="ma 
xima")
 

Output:

integrate((b*x^2 + a)^(5/2)*sqrt(f*x^2 + e)/(d*x^2 + c)^(9/2), x)
 

Giac [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{9/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {f x^{2} + e}}{{\left (d x^{2} + c\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(5/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(9/2),x, algorithm="gi 
ac")
 

Output:

integrate((b*x^2 + a)^(5/2)*sqrt(f*x^2 + e)/(d*x^2 + c)^(9/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{9/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}\,\sqrt {f\,x^2+e}}{{\left (d\,x^2+c\right )}^{9/2}} \,d x \] Input:

int(((a + b*x^2)^(5/2)*(e + f*x^2)^(1/2))/(c + d*x^2)^(9/2),x)
 

Output:

int(((a + b*x^2)^(5/2)*(e + f*x^2)^(1/2))/(c + d*x^2)^(9/2), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{9/2}} \, dx=\int \frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} \sqrt {f \,x^{2}+e}}{\left (d \,x^{2}+c \right )^{\frac {9}{2}}}d x \] Input:

int((b*x^2+a)^(5/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(9/2),x)
 

Output:

int((b*x^2+a)^(5/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(9/2),x)