Integrand size = 34, antiderivative size = 874 \[ \int x^2 \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4} \, dx=-\frac {\left (30 A c d e \left (3 c d^2+28 a e^2\right )-B \left (45 c^2 d^4-108 a c d^2 e^2-256 a^2 e^4\right )\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{3840 c^2 e^3 x}-\frac {\left (15 B c d^3-30 A c d^2 e+148 a B d e^2+120 a A e^3\right ) x \sqrt {d+e x^2} \sqrt {a-c x^4}}{1920 c e^2}+\frac {\left (3 B c d^2+90 A c d e-16 a B e^2\right ) x^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}{480 c e}+\frac {1}{80} (11 B d+10 A e) x^5 \sqrt {d+e x^2} \sqrt {a-c x^4}+\frac {1}{10} B e x^7 \sqrt {d+e x^2} \sqrt {a-c x^4}-\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \left (30 A c d e \left (3 c d^2+28 a e^2\right )-B \left (45 c^2 d^4-108 a c d^2 e^2-256 a^2 e^4\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{3840 c^{3/2} e^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\sqrt {a} \left (30 A c d e \left (c d^2+36 a e^2\right )-B \left (15 c^2 d^4-404 a c d^2 e^2-256 a^2 e^4\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{3840 c^{3/2} e^2 \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\left (B \left (3 c^2 d^5-8 a c d^3 e^2+48 a^2 d e^4\right )-A \left (6 c^2 d^4 e-48 a c d^2 e^3-32 a^2 e^5\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{256 c e^3 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:
-1/3840*(30*A*c*d*e*(28*a*e^2+3*c*d^2)-B*(-256*a^2*e^4-108*a*c*d^2*e^2+45* c^2*d^4))*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/c^2/e^3/x-1/1920*(120*A*a*e^3-3 0*A*c*d^2*e+148*B*a*d*e^2+15*B*c*d^3)*x*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/c /e^2+1/480*(90*A*c*d*e-16*B*a*e^2+3*B*c*d^2)*x^3*(e*x^2+d)^(1/2)*(-c*x^4+a )^(1/2)/c/e+1/80*(10*A*e+11*B*d)*x^5*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)+1/10 *B*e*x^7*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)-1/3840*(c^(1/2)*d+a^(1/2)*e)*(30 *A*c*d*e*(28*a*e^2+3*c*d^2)-B*(-256*a^2*e^4-108*a*c*d^2*e^2+45*c^2*d^4))*( 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)*E llipticE(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^(3/2)/e^3/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+1/3840*a^(1 /2)*(30*A*c*d*e*(36*a*e^2+c*d^2)-B*(-256*a^2*e^4-404*a*c*d^2*e^2+15*c^2*d^ 4))*(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^(3/2)/e^2/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+1/256* (B*(48*a^2*d*e^4-8*a*c*d^3*e^2+3*c^2*d^5)-A*(-32*a^2*e^5-48*a*c*d^2*e^3+6* c^2*d^4*e))*(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),2,2^(1/2) *(d/(d+a^(1/2)*e/c^(1/2)))^(1/2))/c/e^3/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)
\[ \int x^2 \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4} \, dx=\int x^2 \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4} \, dx \] Input:
Integrate[x^2*(A + B*x^2)*(d + e*x^2)^(3/2)*Sqrt[a - c*x^4],x]
Output:
Integrate[x^2*(A + B*x^2)*(d + e*x^2)^(3/2)*Sqrt[a - c*x^4], 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.
| \(\displaystyle \int x^2 \sqrt {a-c x^4} \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 2251 |
| \(\displaystyle \int x^2 \sqrt {a-c x^4} \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}dx\) |
Input:
Int[x^2*(A + B*x^2)*(d + e*x^2)^(3/2)*Sqrt[a - c*x^4],x]
Output:
$Aborted
Int[(Px_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_) ^4)^(p_.), x_Symbol] :> Unintegrable[Px*(f*x)^m*(d + e*x^2)^q*(a + c*x^4)^p , x] /; FreeQ[{a, c, d, e, f, m, p, q}, x] && PolyQ[Px, x]
\[\int x^{2} \left (B \,x^{2}+A \right ) \left (e \,x^{2}+d \right )^{\frac {3}{2}} \sqrt {-c \,x^{4}+a}d x\]
Input:
int(x^2*(B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2),x)
Output:
int(x^2*(B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2),x)
\[ \int x^2 \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4} \, dx=\int { \sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{2} \,d x } \] Input:
integrate(x^2*(B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2),x, algorithm="fri cas")
Output:
integral((B*e*x^6 + (B*d + A*e)*x^4 + A*d*x^2)*sqrt(-c*x^4 + a)*sqrt(e*x^2 + d), x)
\[ \int x^2 \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4} \, dx=\int x^{2} \left (A + B x^{2}\right ) \sqrt {a - c x^{4}} \left (d + e x^{2}\right )^{\frac {3}{2}}\, dx \] Input:
integrate(x**2*(B*x**2+A)*(e*x**2+d)**(3/2)*(-c*x**4+a)**(1/2),x)
Output:
Integral(x**2*(A + B*x**2)*sqrt(a - c*x**4)*(d + e*x**2)**(3/2), x)
\[ \int x^2 \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4} \, dx=\int { \sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{2} \,d x } \] Input:
integrate(x^2*(B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2),x, algorithm="max ima")
Output:
integrate(sqrt(-c*x^4 + a)*(B*x^2 + A)*(e*x^2 + d)^(3/2)*x^2, x)
\[ \int x^2 \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4} \, dx=\int { \sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{2} \,d x } \] Input:
integrate(x^2*(B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2),x, algorithm="gia c")
Output:
integrate(sqrt(-c*x^4 + a)*(B*x^2 + A)*(e*x^2 + d)^(3/2)*x^2, x)
Timed out. \[ \int x^2 \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4} \, dx=\int x^2\,\left (B\,x^2+A\right )\,\sqrt {a-c\,x^4}\,{\left (e\,x^2+d\right )}^{3/2} \,d x \] Input:
int(x^2*(A + B*x^2)*(a - c*x^4)^(1/2)*(d + e*x^2)^(3/2),x)
Output:
int(x^2*(A + B*x^2)*(a - c*x^4)^(1/2)*(d + e*x^2)^(3/2), x)
\[ \int x^2 \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4} \, dx =\text {Too large to display} \] Input:
int(x^2*(B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2),x)
Output:
( - 120*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*e**3*x - 148*sqrt(d + e*x** 2)*sqrt(a - c*x**4)*a*b*d*e**2*x - 64*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a* b*e**3*x**3 + 30*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c*d**2*e*x + 360*sqrt (d + e*x**2)*sqrt(a - c*x**4)*a*c*d*e**2*x**3 + 240*sqrt(d + e*x**2)*sqrt( a - c*x**4)*a*c*e**3*x**5 - 15*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c*d**3* x + 12*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c*d**2*e*x**3 + 264*sqrt(d + e* x**2)*sqrt(a - c*x**4)*b*c*d*e**2*x**5 + 192*sqrt(d + e*x**2)*sqrt(a - c*x **4)*b*c*e**3*x**7 + 256*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*e**4 + 840*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*d*e**3 + 108*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*d**2*e**2 + 90*int((sqrt(d + e*x**2)*sqr t(a - c*x**4)*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a*c**2*d**3* e - 45*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)*b*c**2*d**4 + 240*int((sqrt(d + e*x**2)*sqrt(a - c*x** 4)*x**2)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a**3*e**4 + 488*int((sq rt(d + e*x**2)*sqrt(a - c*x**4)*x**2)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x** 6),x)*a**2*b*d*e**3 + 780*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**2)/(a* d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a**2*c*d**2*e**2 - 6*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**2)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x...