\(\int \frac {(A+B x^2) \sqrt {a+b x^2+c x^4}}{(d+e x^2)^{9/2}} \, dx\) [230]

Optimal result

Integrand size = 35, antiderivative size = 1240 \[ \int \frac {\left (A+B x^2\right ) \sqrt {a+b x^2+c x^4}}{\left (d+e x^2\right )^{9/2}} \, dx =\text {Too large to display} \] Output:

-1/7*(-A*e+B*d)*x*(c*x^4+b*x^2+a)^(1/2)/d/e/(e*x^2+d)^(7/2)+1/35*(A*e*(4*c 
*d^2-e*(-6*a*e+5*b*d))+B*d*(3*c*d^2-e*(-a*e+2*b*d)))*x*(c*x^4+b*x^2+a)^(1/ 
2)/d^2/e/(a*e^2-b*d*e+c*d^2)/(e*x^2+d)^(5/2)+1/105*(2*B*d*(3*c^2*d^4-c*d^2 
*e*(3*a*e+4*b*d)+e^2*(2*a^2*e^2-3*a*b*d*e+3*b^2*d^2))+A*e*(8*c^2*d^4-3*c*d 
^2*e*(-16*a*e+9*b*d)+e^2*(24*a^2*e^2-43*a*b*d*e+15*b^2*d^2)))*x*(c*x^4+b*x 
^2+a)^(1/2)/d^3/e/(a*e^2-b*d*e+c*d^2)^2/(e*x^2+d)^(3/2)+1/105*(3*b^3*d^3*e 
*(5*A*e+2*B*d)-b^2*d^2*(103*A*a*e^3+42*A*c*d^2*e+9*B*a*d*e^2+14*B*c*d^3)+b 
*d*(a*B*d*e*(19*a*e^2+c*d^2)+A*(128*a^2*e^4+237*a*c*d^2*e^2+35*c^2*d^4))-2 
*a*(A*e*(24*a^2*e^4+69*a*c*d^2*e^2+77*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^3/(a*e^2-b*d*e+c*d^2)^3/x/(e*x^2 
+d)^(1/2)-1/210*(-4*a*c+b^2)^(1/2)*(3*b^3*d^3*e*(5*A*e+2*B*d)-b^2*d^2*(103 
*A*a*e^3+42*A*c*d^2*e+9*B*a*d*e^2+14*B*c*d^3)+b*d*(a*B*d*e*(19*a*e^2+c*d^2 
)+A*(128*a^2*e^4+237*a*c*d^2*e^2+35*c^2*d^4))-2*a*(A*e*(24*a^2*e^4+69*a*c* 
d^2*e^2+77*c^2*d^4)-B*(-4*a^2*d*e^4-15*a*c*d^3*e^2+21*c^2*d^5)))*(-a*(c+a/ 
x^4+b/x^2)/(-4*a*c+b^2))^(1/2)*x*(e*x^2+d)^(1/2)*EllipticE(1/2*(1+(b+2*a/x 
^2)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2)*((-4*a*c+b^2)^(1/2)*d/(b*d+( 
-4*a*c+b^2)^(1/2)*d-2*a*e))^(1/2))*2^(1/2)/d^4/(a*e^2-b*d*e+c*d^2)^3/(-a*( 
e+d/x^2)/((b+(-4*a*c+b^2)^(1/2))*d-2*a*e))^(1/2)/(c*x^4+b*x^2+a)^(1/2)-1/1 
05*2^(1/2)*(-4*a*c+b^2)^(1/2)*(3*b^2*d^2*e*(20*A*e+B*d)+8*a*B*d*e*(a*e^2+3 
*c*d^2)-b*d*(104*A*a*e^3+126*A*c*d^2*e+15*B*a*d*e^2+7*B*c*d^3)+2*A*(24*...
 

Mathematica [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {a+b x^2+c x^4}}{\left (d+e x^2\right )^{9/2}} \, dx=\int \frac {\left (A+B x^2\right ) \sqrt {a+b x^2+c x^4}}{\left (d+e x^2\right )^{9/2}} \, dx \] Input:

Integrate[((A + B*x^2)*Sqrt[a + b*x^2 + c*x^4])/(d + e*x^2)^(9/2),x]
 

Output:

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

\(\displaystyle \int \frac {\left (A+B x^2\right ) \sqrt {a+b x^2+c x^4}}{\left (d+e x^2\right )^{9/2}} \, dx\)

\(\Big \downarrow \) 2260

\(\displaystyle \int \frac {\left (A+B x^2\right ) \sqrt {a+b x^2+c x^4}}{\left (d+e x^2\right )^{9/2}}dx\)

Input:

Int[((A + B*x^2)*Sqrt[a + b*x^2 + c*x^4])/(d + e*x^2)^(9/2),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]

\[\int \frac {\left (B \,x^{2}+A \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{\left (e \,x^{2}+d \right )^{\frac {9}{2}}}d x\]

Input:

int((B*x^2+A)*(c*x^4+b*x^2+a)^(1/2)/(e*x^2+d)^(9/2),x)
 

Output:

int((B*x^2+A)*(c*x^4+b*x^2+a)^(1/2)/(e*x^2+d)^(9/2),x)
 

Fricas [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {a+b x^2+c x^4}}{\left (d+e x^2\right )^{9/2}} \, dx=\int { \frac {\sqrt {c x^{4} + b x^{2} + a} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate((B*x^2+A)*(c*x^4+b*x^2+a)^(1/2)/(e*x^2+d)^(9/2),x, algorithm="fr 
icas")
 

Output:

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

Sympy [F(-1)]

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

integrate((B*x**2+A)*(c*x**4+b*x**2+a)**(1/2)/(e*x**2+d)**(9/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {a+b x^2+c x^4}}{\left (d+e x^2\right )^{9/2}} \, dx=\int { \frac {\sqrt {c x^{4} + b x^{2} + a} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate((B*x^2+A)*(c*x^4+b*x^2+a)^(1/2)/(e*x^2+d)^(9/2),x, algorithm="ma 
xima")
 

Output:

integrate(sqrt(c*x^4 + b*x^2 + a)*(B*x^2 + A)/(e*x^2 + d)^(9/2), x)
 

Giac [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {a+b x^2+c x^4}}{\left (d+e x^2\right )^{9/2}} \, dx=\int { \frac {\sqrt {c x^{4} + b x^{2} + a} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate((B*x^2+A)*(c*x^4+b*x^2+a)^(1/2)/(e*x^2+d)^(9/2),x, algorithm="gi 
ac")
 

Output:

integrate(sqrt(c*x^4 + b*x^2 + a)*(B*x^2 + A)/(e*x^2 + d)^(9/2), x)
 

Mupad [F(-1)]

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

int(((A + B*x^2)*(a + b*x^2 + c*x^4)^(1/2))/(d + e*x^2)^(9/2),x)
 

Output:

int(((A + B*x^2)*(a + b*x^2 + c*x^4)^(1/2))/(d + e*x^2)^(9/2), x)
 

Reduce [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {a+b x^2+c x^4}}{\left (d+e x^2\right )^{9/2}} \, dx=\int \frac {\left (B \,x^{2}+A \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{\left (e \,x^{2}+d \right )^{\frac {9}{2}}}d x \] Input:

int((B*x^2+A)*(c*x^4+b*x^2+a)^(1/2)/(e*x^2+d)^(9/2),x)
 

Output:

int((B*x^2+A)*(c*x^4+b*x^2+a)^(1/2)/(e*x^2+d)^(9/2),x)