\(\int \frac {(A+B x^2) (a+b x^2+c x^4)^{3/2}}{(d+e x^2)^{11/2}} \, dx\) [237]

Optimal result

Integrand size = 35, antiderivative size = 1593 \[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx =\text {Too large to display} \] Output:

-1/9*(-A*e+B*d)*(a*e^2-b*d*e+c*d^2)*x*(c*x^4+b*x^2+a)^(1/2)/d/e^3/(e*x^2+d 
)^(9/2)+1/63*(B*d*(21*c*d^2-e*(-a*e+11*b*d))-2*A*e*(6*c*d^2-e*(4*a*e+b*d)) 
)*x*(c*x^4+b*x^2+a)^(1/2)/d^2/e^3/(e*x^2+d)^(7/2)+1/315*(A*e*(15*c^2*d^4-c 
*d^2*e*(-51*a*e+2*b*d)-e^2*(-48*a^2*e^2+35*a*b*d*e+10*b^2*d^2))-B*d*(105*c 
^2*d^4-c*d^2*e*(-87*a*e+110*b*d)+e^2*(-6*a^2*e^2+a*b*d*e+8*b^2*d^2)))*x*(c 
*x^4+b*x^2+a)^(1/2)/d^3/e^3/(a*e^2-b*d*e+c*d^2)/(e*x^2+d)^(5/2)+1/315*(B*d 
*(35*c^3*d^6-c^2*d^4*e*(-82*a*e+55*b*d)+c*d^2*e^2*(23*a^2*e^2-52*a*b*d*e+8 
*b^2*d^2)+e^3*(8*a^3*e^3-9*a^2*b*d*e^2-3*a*b^2*d^2*e+8*b^3*d^3))+2*A*e*(5* 
c^3*d^6-2*c^2*d^4*e*(-11*a*e+2*b*d)-c*d^2*e^2*(-65*a^2*e^2+46*a*b*d*e+4*b^ 
2*d^2)+e^3*(32*a^3*e^3-54*a^2*b*d*e^2+15*a*b^2*d^2*e+5*b^3*d^3)))*x*(c*x^4 
+b*x^2+a)^(1/2)/d^4/e^3/(a*e^2-b*d*e+c*d^2)^2/(e*x^2+d)^(3/2)+1/315*(2*b^4 
*d^4*(5*A*e+4*B*d)-b^3*d^3*(-25*A*a*e^2+18*A*c*d^2+7*B*a*d*e)-3*a*b^2*d^2* 
(81*A*a*e^3+41*A*c*d^2*e+3*B*a*d*e^2+19*B*c*d^3)+4*a*b*d*(a*B*d*e*(8*a*e^2 
+15*c*d^2)+A*(82*a^2*e^4+147*a*c*d^2*e^2+36*c^2*d^4))-4*a^2*(A*e*(32*a^2*e 
^4+93*a*c*d^2*e^2+93*c^2*d^4)-B*(-4*a^2*d*e^4-15*a*c*d^3*e^2+21*c^2*d^5))) 
*(c*x^4+b*x^2+a)^(1/2)/d^4/(a*e^2-b*d*e+c*d^2)^3/x/(e*x^2+d)^(1/2)-1/630*( 
-4*a*c+b^2)^(1/2)*(2*b^4*d^4*(5*A*e+4*B*d)-b^3*d^3*(-25*A*a*e^2+18*A*c*d^2 
+7*B*a*d*e)-3*a*b^2*d^2*(81*A*a*e^3+41*A*c*d^2*e+3*B*a*d*e^2+19*B*c*d^3)+4 
*a*b*d*(a*B*d*e*(8*a*e^2+15*c*d^2)+A*(82*a^2*e^4+147*a*c*d^2*e^2+36*c^2*d^ 
4))-4*a^2*(A*e*(32*a^2*e^4+93*a*c*d^2*e^2+93*c^2*d^4)-B*(-4*a^2*d*e^4-1...
 

Mathematica [F]

\[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx=\int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx \] Input:

Integrate[((A + B*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(d + e*x^2)^(11/2),x]
 

Output:

Integrate[((A + B*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(d + e*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.

\(\displaystyle \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx\)

\(\Big \downarrow \) 2260

\(\displaystyle \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}}dx\)

Input:

Int[((A + B*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(d + e*x^2)^(11/2),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 2260

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^ 
(p_.), x_Symbol] :> Unintegrable[Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x] 
 /; FreeQ[{a, b, c, d, e, p, q}, x] && PolyQ[Px, x]
 

Maple [F]

\[\int \frac {\left (B \,x^{2}+A \right ) \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{\left (e \,x^{2}+d \right )^{\frac {11}{2}}}d x\]

Input:

int((B*x^2+A)*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d)^(11/2),x)
 

Output:

int((B*x^2+A)*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d)^(11/2),x)
 

Fricas [F]

\[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx=\int { \frac {{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {11}{2}}} \,d x } \] Input:

integrate((B*x^2+A)*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d)^(11/2),x, algorithm="f 
ricas")
 

Output:

integral((B*c*x^6 + (B*b + A*c)*x^4 + (B*a + A*b)*x^2 + A*a)*sqrt(c*x^4 + 
b*x^2 + a)*sqrt(e*x^2 + d)/(e^6*x^12 + 6*d*e^5*x^10 + 15*d^2*e^4*x^8 + 20* 
d^3*e^3*x^6 + 15*d^4*e^2*x^4 + 6*d^5*e*x^2 + d^6), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx=\text {Timed out} \] Input:

integrate((B*x**2+A)*(c*x**4+b*x**2+a)**(3/2)/(e*x**2+d)**(11/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx=\int { \frac {{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {11}{2}}} \,d x } \] Input:

integrate((B*x^2+A)*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d)^(11/2),x, algorithm="m 
axima")
 

Output:

integrate((c*x^4 + b*x^2 + a)^(3/2)*(B*x^2 + A)/(e*x^2 + d)^(11/2), x)
 

Giac [F]

\[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx=\int { \frac {{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {11}{2}}} \,d x } \] Input:

integrate((B*x^2+A)*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d)^(11/2),x, algorithm="g 
iac")
 

Output:

integrate((c*x^4 + b*x^2 + a)^(3/2)*(B*x^2 + A)/(e*x^2 + d)^(11/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2}}{{\left (e\,x^2+d\right )}^{11/2}} \,d x \] Input:

int(((A + B*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(d + e*x^2)^(11/2),x)
 

Output:

int(((A + B*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(d + e*x^2)^(11/2), x)
 

Reduce [F]

\[ \int \frac {\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\left (d+e x^2\right )^{11/2}} \, dx=\int \frac {\left (B \,x^{2}+A \right ) \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{\left (e \,x^{2}+d \right )^{\frac {11}{2}}}d x \] Input:

int((B*x^2+A)*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d)^(11/2),x)
 

Output:

int((B*x^2+A)*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d)^(11/2),x)