Integrand size = 34, antiderivative size = 874 \[ \int x^4 \left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4} \, dx=\frac {\left (50 A c d e \left (3 c d^2-4 a e^2\right )-B \left (105 c^2 d^4-92 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^4 x}+\frac {\left (35 B c d^3-50 A c d^2 e-28 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^3}-\frac {\left (7 B c d^2-10 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^2}+\frac {(B d+10 A e) x^5 \sqrt {d+e x^2} \sqrt {a-c x^4}}{80 e}+\frac {1}{10} B x^7 \sqrt {d+e x^2} \sqrt {a-c x^4}+\frac {\left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (50 A c d e \left (3 c d^2-4 a e^2\right )-B \left (105 c^2 d^4-92 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 e^4 \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {\sqrt {a} \left (10 A c d e \left (5 c d^2-44 a e^2\right )-B \left (35 c^2 d^4-36 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^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\left (2 A e \left (5 c^2 d^4-8 a c d^2 e^2+16 a^2 e^4\right )-B \left (7 c^2 d^5-8 a c d^3 e^2-16 a^2 d 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 {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^4 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:
1/3840*(50*A*c*d*e*(-4*a*e^2+3*c*d^2)-B*(256*a^2*e^4-92*a*c*d^2*e^2+105*c^ 2*d^4))*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/c^2/e^4/x+1/1920*(-120*A*a*e^3-50 *A*c*d^2*e-28*B*a*d*e^2+35*B*c*d^3)*x*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/c/e ^3-1/480*(-10*A*c*d*e+16*B*a*e^2+7*B*c*d^2)*x^3*(e*x^2+d)^(1/2)*(-c*x^4+a) ^(1/2)/c/e^2+1/80*(10*A*e+B*d)*x^5*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/e+1/10 *B*x^7*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)+1/3840*(d+a^(1/2)*e/c^(1/2))*(50*A *c*d*e*(-4*a*e^2+3*c*d^2)-B*(256*a^2*e^4-92*a*c*d^2*e^2+105*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)*Elli pticE(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/e^4/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)-1/3840*a^(1/2)*(10*A *c*d*e*(-44*a*e^2+5*c*d^2)-B*(256*a^2*e^4-36*a*c*d^2*e^2+35*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)*Elli pticF(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)+1/256*(2*A*e*( 16*a^2*e^4-8*a*c*d^2*e^2+5*c^2*d^4)-B*(-16*a^2*d*e^4-8*a*c*d^3*e^2+7*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)*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^4/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)
\[ \int x^4 \left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4} \, dx=\int x^4 \left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4} \, dx \] Input:
Integrate[x^4*(A + B*x^2)*Sqrt[d + e*x^2]*Sqrt[a - c*x^4],x]
Output:
Integrate[x^4*(A + B*x^2)*Sqrt[d + e*x^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.
\(\displaystyle \int x^4 \sqrt {a-c x^4} \left (A+B x^2\right ) \sqrt {d+e x^2} \, dx\) |
\(\Big \downarrow \) 2251 |
\(\displaystyle \int x^4 \sqrt {a-c x^4} \left (A+B x^2\right ) \sqrt {d+e x^2}dx\) |
Input:
Int[x^4*(A + B*x^2)*Sqrt[d + e*x^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]
\[\int x^{4} \left (B \,x^{2}+A \right ) \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}d x\]
Input:
int(x^4*(B*x^2+A)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2),x)
Output:
int(x^4*(B*x^2+A)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2),x)
\[ \int x^4 \left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4} \, dx=\int { \sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )} \sqrt {e x^{2} + d} x^{4} \,d x } \] Input:
integrate(x^4*(B*x^2+A)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2),x, algorithm="fri cas")
Output:
integral((B*x^6 + A*x^4)*sqrt(-c*x^4 + a)*sqrt(e*x^2 + d), x)
\[ \int x^4 \left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4} \, dx=\int x^{4} \left (A + B x^{2}\right ) \sqrt {a - c x^{4}} \sqrt {d + e x^{2}}\, dx \] Input:
integrate(x**4*(B*x**2+A)*(e*x**2+d)**(1/2)*(-c*x**4+a)**(1/2),x)
Output:
Integral(x**4*(A + B*x**2)*sqrt(a - c*x**4)*sqrt(d + e*x**2), x)
\[ \int x^4 \left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4} \, dx=\int { \sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )} \sqrt {e x^{2} + d} x^{4} \,d x } \] Input:
integrate(x^4*(B*x^2+A)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2),x, algorithm="max ima")
Output:
integrate(sqrt(-c*x^4 + a)*(B*x^2 + A)*sqrt(e*x^2 + d)*x^4, x)
\[ \int x^4 \left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4} \, dx=\int { \sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )} \sqrt {e x^{2} + d} x^{4} \,d x } \] Input:
integrate(x^4*(B*x^2+A)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2),x, algorithm="gia c")
Output:
integrate(sqrt(-c*x^4 + a)*(B*x^2 + A)*sqrt(e*x^2 + d)*x^4, x)
Timed out. \[ \int x^4 \left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4} \, dx=\int x^4\,\left (B\,x^2+A\right )\,\sqrt {a-c\,x^4}\,\sqrt {e\,x^2+d} \,d x \] Input:
int(x^4*(A + B*x^2)*(a - c*x^4)^(1/2)*(d + e*x^2)^(1/2),x)
Output:
int(x^4*(A + B*x^2)*(a - c*x^4)^(1/2)*(d + e*x^2)^(1/2), x)
\[ \int x^4 \left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4} \, dx =\text {Too large to display} \] Input:
int(x^4*(B*x^2+A)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2),x)
Output:
( - 120*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*e**3*x - 28*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 - 50*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c*d**2*e*x + 40*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 + 35*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c*d**3*x - 28*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c*d**2*e*x**3 + 24*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 + 200*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 - 92*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 - 150*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**2*d**3*e + 105*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 + 248*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*b*d*e**3 - 20*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 + 14*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)*...