\(\int \frac {(a+b x^2)^{5/2} \sqrt {e+f x^2}}{(c+d x^2)^{11/2}} \, dx\) [401]

Optimal result

Integrand size = 34, antiderivative size = 1221 \[ \int \frac {\left (a+b x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{11/2}} \, dx =\text {Too large to display} \] Output:

1/9*(-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)^(9/2)-1 
/63*(-a*d+b*c)*(b*c*(-12*c*f+11*d*e)+a*d*(-7*c*f+8*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)^(7/2)+1/315*(b^2*c^2*(15*c^2 
*f^2-29*c*d*e*f+8*d^2*e^2)+a*b*c*d*(25*c^2*f^2-32*c*d*e*f+19*d^2*e^2)+a^2* 
d^2*(35*c^2*f^2-89*c*d*e*f+48*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)^(5/2)-1/315*(a^3*d^3*(-35*c^3*f^3+159*c^2*d 
*e*f^2-180*c*d^2*e^2*f+64*d^3*e^3)-a*b^2*c^2*d*(-10*c^3*f^3+14*c^2*d*e*f^2 
-43*c*d^2*e^2*f+15*d^3*e^3)-2*b^3*c^3*(-5*c^3*f^3+13*c^2*d*e*f^2-8*c*d^2*e 
^2*f+4*d^3*e^3)-2*a^2*b*c*d^2*(-5*c^3*f^3+52*c^2*d*e*f^2-53*c*d^2*e^2*f+18 
*d^3*e^3))*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/c^4/d^2/(-a*d+b*c)/(-c*f+d*e) 
^3/(d*x^2+c)^(3/2)+1/315*e^(1/2)*(8*b^4*c^4*e^3*(-3*c*f+d*e)+a*b^3*c^3*e^2 
*(99*c^2*f^2-46*c*d*e*f+11*d^2*e^2)+3*a^2*b^2*c^2*e*(-58*c^3*f^3+53*c^2*d* 
e*f^2-36*c*d^2*e^2*f+9*d^3*e^3)+a^4*d*(35*c^4*f^4-334*c^3*d*e*f^3+627*c^2* 
d^2*e^2*f^2-472*c*d^3*e^3*f+128*d^4*e^4)-a^3*b*c*(45*c^4*f^4-548*c^3*d*e*f 
^3+945*c^2*d^2*e^2*f^2-690*c*d^3*e^3*f+184*d^4*e^4))*(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^5/(-a*d+b*c)^2/(-c*f+d*e)^(7/2 
)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(f*x^2+e)^(1/2)-1/315*e^(1/2)*(-a*f+b*e) 
*(4*b^3*c^3*e^2*(-3*c*f+d*e)+3*a*b^2*c^2*e*(13*c^2*f^2-7*c*d*e*f+2*d^2*e^2 
)+a^3*d*(-35*c^3*f^3+159*c^2*d*e*f^2-180*c*d^2*e^2*f+64*d^3*e^3)-3*a^2*...
 

Mathematica [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{11/2}} \, dx=\int \frac {\left (a+b x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{11/2}} \, dx \] Input:

Integrate[((a + b*x^2)^(5/2)*Sqrt[e + f*x^2])/(c + d*x^2)^(11/2),x]
 

Output:

Integrate[((a + b*x^2)^(5/2)*Sqrt[e + f*x^2])/(c + d*x^2)^(11/2), 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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{11/2}} \, dx\)

\(\Big \downarrow \) 434

\(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{11/2}}dx\)

Input:

Int[((a + b*x^2)^(5/2)*Sqrt[e + f*x^2])/(c + d*x^2)^(11/2),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]

\[\int \frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} \sqrt {f \,x^{2}+e}}{\left (x^{2} d +c \right )^{\frac {11}{2}}}d x\]

Input:

int((b*x^2+a)^(5/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(11/2),x)
 

Output:

int((b*x^2+a)^(5/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(11/2),x)
 

Fricas [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{11/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {f x^{2} + e}}{{\left (d x^{2} + c\right )}^{\frac {11}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(5/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(11/2),x, algorithm="f 
ricas")
 

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^6*x^12 + 6*c*d^5*x^10 + 15*c^2*d^4*x^8 + 20*c^3*d^3*x^6 + 15 
*c^4*d^2*x^4 + 6*c^5*d*x^2 + c^6), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{11/2}} \, dx=\text {Timed out} \] Input:

integrate((b*x**2+a)**(5/2)*(f*x**2+e)**(1/2)/(d*x**2+c)**(11/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{11/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {f x^{2} + e}}{{\left (d x^{2} + c\right )}^{\frac {11}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(5/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(11/2),x, algorithm="m 
axima")
 

Output:

integrate((b*x^2 + a)^(5/2)*sqrt(f*x^2 + e)/(d*x^2 + c)^(11/2), x)
 

Giac [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{11/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {f x^{2} + e}}{{\left (d x^{2} + c\right )}^{\frac {11}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(5/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(11/2),x, algorithm="g 
iac")
 

Output:

integrate((b*x^2 + a)^(5/2)*sqrt(f*x^2 + e)/(d*x^2 + c)^(11/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{11/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}\,\sqrt {f\,x^2+e}}{{\left (d\,x^2+c\right )}^{11/2}} \,d x \] Input:

int(((a + b*x^2)^(5/2)*(e + f*x^2)^(1/2))/(c + d*x^2)^(11/2),x)
 

Output:

int(((a + b*x^2)^(5/2)*(e + f*x^2)^(1/2))/(c + d*x^2)^(11/2), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \sqrt {e+f x^2}}{\left (c+d x^2\right )^{11/2}} \, dx=\int \frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} \sqrt {f \,x^{2}+e}}{\left (d \,x^{2}+c \right )^{\frac {11}{2}}}d x \] Input:

int((b*x^2+a)^(5/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(11/2),x)
 

Output:

int((b*x^2+a)^(5/2)*(f*x^2+e)^(1/2)/(d*x^2+c)^(11/2),x)