Integrand size = 31, antiderivative size = 1007 \[ \int \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^{3/2} \, dx =\text {Too large to display} \] Output:
-1/15360*(12*A*e*(128*a^2*e^4-116*a*c*d^2*e^2+15*c^2*d^4)-B*(-2064*a^2*d*e ^4-512*a*c*d^3*e^2+105*c^2*d^5))*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/c/e^4/x+ 1/7680*(12*A*c*d*e*(236*a*e^2+5*c*d^2)-B*(240*a^2*e^4-168*a*c*d^2*e^2+35*c ^2*d^4))*x*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/c/e^3+1/1920*(384*A*a*e^3-12*A *c*d^2*e+436*B*a*d*e^2+7*B*c*d^3)*x^3*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/e^2 -1/960*(132*A*c*d*e-140*B*a*e^2+3*B*c*d^2)*x^5*(e*x^2+d)^(1/2)*(-c*x^4+a)^ (1/2)/e-1/120*c*(12*A*e+13*B*d)*x^7*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)-1/12* B*c*e*x^9*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)+1/15360*(c^(1/2)*d+a^(1/2)*e)*( -1536*A*a^2*e^5+1392*A*a*c*d^2*e^3-180*A*c^2*d^4*e-2064*B*a^2*d*e^4-512*B* a*c*d^3*e^2+105*B*c^2*d^5)*(1-a/c/x^4)^(1/2)*x^3*(a^(1/2)*(e*x^2+d)/(c^(1/ 2)*d+a^(1/2)*e)/x^2)^(1/2)*EllipticE(1/2*(1-a^(1/2)/c^(1/2)/x^2)^(1/2)*2^( 1/2),2^(1/2)*(d/(d+a^(1/2)*e/c^(1/2)))^(1/2))/c^(1/2)/e^4/(e*x^2+d)^(1/2)/ (-c*x^4+a)^(1/2)-1/15360*a^(1/2)*(-1536*A*a^2*e^5-8304*A*a*c*d^2*e^3-60*A* c^2*d^4*e-2544*B*a^2*d*e^4-176*B*a*c*d^3*e^2+35*B*c^2*d^5)*(1-a/c/x^4)^(1/ 2)*x^3*(a^(1/2)*(e*x^2+d)/(c^(1/2)*d+a^(1/2)*e)/x^2)^(1/2)*EllipticF(1/2*( 1-a^(1/2)/c^(1/2)/x^2)^(1/2)*2^(1/2),2^(1/2)*(d/(d+a^(1/2)*e/c^(1/2)))^(1/ 2))/c^(1/2)/e^3/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)-1/1024*(12*A*c*d*e*(-48*a ^2*e^4-8*a*c*d^2*e^2+c^2*d^4)-B*(64*a^3*e^6+144*a^2*c*d^2*e^4-36*a*c^2*d^4 *e^2+7*c^3*d^6))*(1-a/c/x^4)^(1/2)*x^3*(a^(1/2)*(e*x^2+d)/(c^(1/2)*d+a^(1/ 2)*e)/x^2)^(1/2)*EllipticPi(1/2*(1-a^(1/2)/c^(1/2)/x^2)^(1/2)*2^(1/2),2...
\[ \int \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^{3/2} \, dx=\int \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^{3/2} \, dx \] Input:
Integrate[(A + B*x^2)*(d + e*x^2)^(3/2)*(a - c*x^4)^(3/2),x]
Output:
Integrate[(A + B*x^2)*(d + e*x^2)^(3/2)*(a - c*x^4)^(3/2), x]
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.
| \(\displaystyle \int \left (a-c x^4\right )^{3/2} \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 2261 |
| \(\displaystyle \int \left (a-c x^4\right )^{3/2} \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}dx\) |
Input:
Int[(A + B*x^2)*(d + e*x^2)^(3/2)*(a - c*x^4)^(3/2),x]
Output:
$Aborted
Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol ] :> Unintegrable[Px*(d + e*x^2)^q*(a + c*x^4)^p, x] /; FreeQ[{a, c, d, e, p, q}, x] && PolyQ[Px, x]
\[\int \left (B \,x^{2}+A \right ) \left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (-c \,x^{4}+a \right )^{\frac {3}{2}}d x\]
Input:
int((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(3/2),x)
Output:
int((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(3/2),x)
\[ \int \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^{3/2} \, dx=\int { {\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(3/2),x, algorithm="fricas" )
Output:
integral(-(B*c*e*x^8 + (B*c*d + A*c*e)*x^6 + (A*c*d - B*a*e)*x^4 - A*a*d - (B*a*d + A*a*e)*x^2)*sqrt(-c*x^4 + a)*sqrt(e*x^2 + d), x)
\[ \int \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^{3/2} \, dx=\int \left (A + B x^{2}\right ) \left (a - c x^{4}\right )^{\frac {3}{2}} \left (d + e x^{2}\right )^{\frac {3}{2}}\, dx \] Input:
integrate((B*x**2+A)*(e*x**2+d)**(3/2)*(-c*x**4+a)**(3/2),x)
Output:
Integral((A + B*x**2)*(a - c*x**4)**(3/2)*(d + e*x**2)**(3/2), x)
\[ \int \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^{3/2} \, dx=\int { {\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(3/2),x, algorithm="maxima" )
Output:
integrate((-c*x^4 + a)^(3/2)*(B*x^2 + A)*(e*x^2 + d)^(3/2), x)
\[ \int \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^{3/2} \, dx=\int { {\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate((-c*x^4 + a)^(3/2)*(B*x^2 + A)*(e*x^2 + d)^(3/2), x)
Timed out. \[ \int \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^{3/2} \, dx=\int \left (B\,x^2+A\right )\,{\left (a-c\,x^4\right )}^{3/2}\,{\left (e\,x^2+d\right )}^{3/2} \,d x \] Input:
int((A + B*x^2)*(a - c*x^4)^(3/2)*(d + e*x^2)^(3/2),x)
Output:
int((A + B*x^2)*(a - c*x^4)^(3/2)*(d + e*x^2)^(3/2), x)
\[ \int \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^{3/2} \, dx =\text {Too large to display} \] Input:
int((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(3/2),x)
Output:
( - 240*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*b*e**4*x + 2832*sqrt(d + e* x**2)*sqrt(a - c*x**4)*a**2*c*d*e**3*x + 1536*sqrt(d + e*x**2)*sqrt(a - c* x**4)*a**2*c*e**4*x**3 + 168*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*b*c*d**2* e**2*x + 1744*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*b*c*d*e**3*x**3 + 1120*s qrt(d + e*x**2)*sqrt(a - c*x**4)*a*b*c*e**4*x**5 + 60*sqrt(d + e*x**2)*sqr t(a - c*x**4)*a*c**2*d**3*e*x - 48*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c** 2*d**2*e**2*x**3 - 1056*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c**2*d*e**3*x* *5 - 768*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c**2*e**4*x**7 - 35*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c**2*d**4*x + 28*sqrt(d + e*x**2)*sqrt(a - c*x* *4)*b*c**2*d**3*e*x**3 - 24*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c**2*d**2* e**2*x**5 - 832*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c**2*d*e**3*x**7 - 640 *sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c**2*e**4*x**9 + 1536*int((sqrt(d + e *x**2)*sqrt(a - c*x**4)*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a* *3*c*e**5 + 2064*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**4)/(a*d + a*e*x **2 - c*d*x**4 - c*e*x**6),x)*a**2*b*c*d*e**4 - 1392*int((sqrt(d + e*x**2) *sqrt(a - c*x**4)*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a**2*c** 2*d**2*e**3 + 512*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**4)/(a*d + a*e* x**2 - c*d*x**4 - c*e*x**6),x)*a*b*c**2*d**3*e**2 + 180*int((sqrt(d + e*x* *2)*sqrt(a - c*x**4)*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a*c** 3*d**4*e - 105*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**4)/(a*d + a*e*...