\(\int \frac {\sqrt {a+b x^2} (e+f x^2)^{3/2}}{(c+d x^2)^{11/2}} \, dx\) [426]

Optimal result

Integrand size = 34, antiderivative size = 1296 \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{11/2}} \, dx =\text {Too large to display} \] Output:

1/9*(-c*f+d*e)*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/c/d/(d*x^2+c)^(9/2)-1/63* 
(2*a*d*(c*f+4*d*e)-b*c*(3*c*f+7*d*e))*x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/c^ 
2/d/(-a*d+b*c)/(d*x^2+c)^(7/2)-1/315*(a*b*c*d*(-16*c^2*f^2-67*c*d*e*f+89*d 
^2*e^2)-b^2*c^2*(-12*c^2*f^2-20*c*d*e*f+35*d^2*e^2)-a^2*d^2*(-10*c^2*f^2-3 
5*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/(-a*d+b*c)^ 
2/(-c*f+d*e)/(d*x^2+c)^(5/2)-1/315*(2*a^3*d^3*(5*c^3*f^3+15*c^2*d*e*f^2-54 
*c*d^2*e^2*f+32*d^3*e^3)-b^3*c^3*(8*c^3*f^3+8*c^2*d*e*f^2-55*c*d^2*e^2*f+3 
5*d^3*e^3)+a*b^2*c^2*d*(16*c^3*f^3+91*c^2*d*e*f^2-278*c*d^2*e^2*f+159*d^3* 
e^3)-a^2*b*c*d^2*(26*c^3*f^3+89*c^2*d*e*f^2-307*c*d^2*e^2*f+180*d^3*e^3))* 
x*(b*x^2+a)^(1/2)*(f*x^2+e)^(1/2)/c^4/d/(-a*d+b*c)^3/(-c*f+d*e)^2/(d*x^2+c 
)^(3/2)+1/315*e^(1/2)*(b^4*c^4*e^2*(63*c^2*f^2-90*c*d*e*f+35*d^2*e^2)-a*b^ 
3*c^3*e*(-147*c^3*f^3+747*c^2*d*e*f^2-902*c*d^2*e^2*f+334*d^3*e^3)-a^3*b*c 
*d*(-36*c^4*f^4-101*c^3*d*e*f^3+915*c^2*d^2*e^2*f^2-1218*c*d^3*e^3*f+472*d 
^4*e^4)+3*a^2*b^2*c^2*(-14*c^4*f^4-53*c^3*d*e*f^3+420*c^2*d^2*e^2*f^2-546* 
c*d^3*e^3*f+209*d^4*e^4)+a^4*d^2*(-10*c^4*f^4-25*c^3*d*e*f^3+243*c^2*d^2*e 
^2*f^2-328*c*d^3*e^3*f+128*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)^4/(-c*f+d*e)^(5/2)/(c*(b*x^2+a)/a 
/(d*x^2+c))^(1/2)/(f*x^2+e)^(1/2)+2/315*e^(1/2)*(-a*f+b*e)*(b^3*c^3*e*(63* 
c^2*f^2-135*c*d*e*f+70*d^2*e^2)-a^3*d^2*(5*c^3*f^3+15*c^2*d*e*f^2-54*c*...
 

Mathematica [F]

\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{11/2}} \, dx=\int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{11/2}} \, dx \] Input:

Integrate[(Sqrt[a + b*x^2]*(e + f*x^2)^(3/2))/(c + d*x^2)^(11/2),x]
 

Output:

Integrate[(Sqrt[a + b*x^2]*(e + f*x^2)^(3/2))/(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.

\(\displaystyle \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{11/2}} \, dx\)

\(\Big \downarrow \) 434

\(\displaystyle \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{11/2}}dx\)

Input:

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

\[\int \frac {\sqrt {b \,x^{2}+a}\, \left (f \,x^{2}+e \right )^{\frac {3}{2}}}{\left (x^{2} d +c \right )^{\frac {11}{2}}}d x\]

Input:

int((b*x^2+a)^(1/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(11/2),x)
 

Output:

int((b*x^2+a)^(1/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(11/2),x)
 

Fricas [F]

\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{11/2}} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (f x^{2} + e\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{\frac {11}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(1/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(11/2),x, algorithm="f 
ricas")
 

Output:

integral(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^(3/2)/(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. \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{11/2}} \, dx=\text {Timed out} \] Input:

integrate((b*x**2+a)**(1/2)*(f*x**2+e)**(3/2)/(d*x**2+c)**(11/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{11/2}} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (f x^{2} + e\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{\frac {11}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(1/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(11/2),x, algorithm="m 
axima")
 

Output:

integrate(sqrt(b*x^2 + a)*(f*x^2 + e)^(3/2)/(d*x^2 + c)^(11/2), x)
 

Giac [F]

\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{11/2}} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (f x^{2} + e\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{\frac {11}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(1/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(11/2),x, algorithm="g 
iac")
 

Output:

integrate(sqrt(b*x^2 + a)*(f*x^2 + e)^(3/2)/(d*x^2 + c)^(11/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{11/2}} \, dx=\int \frac {\sqrt {b\,x^2+a}\,{\left (f\,x^2+e\right )}^{3/2}}{{\left (d\,x^2+c\right )}^{11/2}} \,d x \] Input:

int(((a + b*x^2)^(1/2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(11/2),x)
 

Output:

int(((a + b*x^2)^(1/2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(11/2), x)
 

Reduce [F]

\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{11/2}} \, dx=\int \frac {\sqrt {b \,x^{2}+a}\, \left (f \,x^{2}+e \right )^{\frac {3}{2}}}{\left (d \,x^{2}+c \right )^{\frac {11}{2}}}d x \] Input:

int((b*x^2+a)^(1/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(11/2),x)
 

Output:

int((b*x^2+a)^(1/2)*(f*x^2+e)^(3/2)/(d*x^2+c)^(11/2),x)