Integrand size = 34, antiderivative size = 569 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^4} \, dx=-\frac {A d \sqrt {d+e x^2} \sqrt {a-c x^4}}{3 x^3}+\frac {(5 B d+4 A e) \sqrt {d+e x^2} \sqrt {a-c x^4}}{8 x}+\frac {1}{4} B e x \sqrt {d+e x^2} \sqrt {a-c x^4}+\frac {c (39 B d+44 A e) \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{24 \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {\sqrt {c} \left (16 A c d^2-3 a B d e+20 a A e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{24 \sqrt {a} \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {\left (3 B c d^2+12 A c d e-4 a B e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{8 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:
-1/3*A*d*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/x^3+1/8*(4*A*e+5*B*d)*(e*x^2+d)^ (1/2)*(-c*x^4+a)^(1/2)/x+1/4*B*e*x*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)+1/24*c *(44*A*e+39*B*d)*(d+a^(1/2)*e/c^(1/2))*(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))/(e*x^2+d)^(1/2)/ (-c*x^4+a)^(1/2)-1/24*c^(1/2)*(20*A*a*e^2+16*A*c*d^2-3*B*a*d*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)*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))/a^(1/2)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)-1/8*(12*A*c*d*e-4*B*a*e^ 2+3*B*c*d^2)*(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))/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^4} \, dx=\int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^4} \, dx \] Input:
Integrate[((A + B*x^2)*(d + e*x^2)^(3/2)*Sqrt[a - c*x^4])/x^4,x]
Output:
Integrate[((A + B*x^2)*(d + e*x^2)^(3/2)*Sqrt[a - c*x^4])/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.
| \(\displaystyle \int \frac {\sqrt {a-c x^4} \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{x^4} \, dx\) |
| \(\Big \downarrow \) 2251 |
| \(\displaystyle \int \frac {\sqrt {a-c x^4} \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{x^4}dx\) |
Input:
Int[((A + B*x^2)*(d + e*x^2)^(3/2)*Sqrt[a - c*x^4])/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]
\[\int \frac {\left (B \,x^{2}+A \right ) \left (e \,x^{2}+d \right )^{\frac {3}{2}} \sqrt {-c \,x^{4}+a}}{x^{4}}d x\]
Input:
int((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2)/x^4,x)
Output:
int((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2)/x^4,x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^4} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}}}{x^{4}} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2)/x^4,x, algorithm="fri cas")
Output:
integral((B*e*x^4 + (B*d + A*e)*x^2 + A*d)*sqrt(-c*x^4 + a)*sqrt(e*x^2 + d )/x^4, x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^4} \, dx=\int \frac {\left (A + B x^{2}\right ) \sqrt {a - c x^{4}} \left (d + e x^{2}\right )^{\frac {3}{2}}}{x^{4}}\, dx \] Input:
integrate((B*x**2+A)*(e*x**2+d)**(3/2)*(-c*x**4+a)**(1/2)/x**4,x)
Output:
Integral((A + B*x**2)*sqrt(a - c*x**4)*(d + e*x**2)**(3/2)/x**4, x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^4} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}}}{x^{4}} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2)/x^4,x, algorithm="max ima")
Output:
integrate(sqrt(-c*x^4 + a)*(B*x^2 + A)*(e*x^2 + d)^(3/2)/x^4, x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^4} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}}}{x^{4}} \,d x } \] Input:
integrate((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2)/x^4,x, algorithm="gia c")
Output:
integrate(sqrt(-c*x^4 + a)*(B*x^2 + A)*(e*x^2 + d)^(3/2)/x^4, x)
Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^4} \, dx=\int \frac {\left (B\,x^2+A\right )\,\sqrt {a-c\,x^4}\,{\left (e\,x^2+d\right )}^{3/2}}{x^4} \,d x \] Input:
int(((A + B*x^2)*(a - c*x^4)^(1/2)*(d + e*x^2)^(3/2))/x^4,x)
Output:
int(((A + B*x^2)*(a - c*x^4)^(1/2)*(d + e*x^2)^(3/2))/x^4, x)
\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^4} \, dx=\frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, a b d \,e^{2}-2 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, a b \,e^{3} x^{2}-8 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, a c \,d^{2} e +8 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, a c d \,e^{2} x^{2}-4 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, b c \,d^{3}+4 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, b c \,d^{2} e \,x^{2}+\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, b c d \,e^{2} x^{4}-4 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a b c \,e^{4} x^{3}+12 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a \,c^{2} d \,e^{3} x^{3}+3 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) b \,c^{2} d^{2} e^{2} x^{3}+3 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{10}-c d \,x^{8}+a e \,x^{6}+a d \,x^{4}}d x \right ) a^{2} b \,d^{2} e^{2} x^{3}-20 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{10}-c d \,x^{8}+a e \,x^{6}+a d \,x^{4}}d x \right ) a^{2} c \,d^{3} e \,x^{3}-12 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{10}-c d \,x^{8}+a e \,x^{6}+a d \,x^{4}}d x \right ) a b c \,d^{4} x^{3}+4 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{2} c d \,e^{3} x^{3}+6 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a b c \,d^{2} e^{2} x^{3}+4 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a \,c^{2} d^{3} e \,x^{3}+4 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) b \,c^{2} d^{4} x^{3}}{4 c d e \,x^{3}} \] Input:
int((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2)/x^4,x)
Output:
(sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*b*d*e**2 - 2*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*b*e**3*x**2 - 8*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c*d**2*e + 8*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c*d*e**2*x**2 - 4*sqrt(d + e*x**2)* sqrt(a - c*x**4)*b*c*d**3 + 4*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c*d**2*e *x**2 + sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c*d*e**2*x**4 - 4*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*e**4*x**3 + 12*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*e**3*x**3 + 3*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**2*e**2*x**3 + 3*int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/(a*d*x**4 + a*e*x**6 - c*d*x**8 - c*e*x**10),x)*a**2*b*d**2*e**2*x**3 - 20*int((sqrt( d + e*x**2)*sqrt(a - c*x**4))/(a*d*x**4 + a*e*x**6 - c*d*x**8 - c*e*x**10) ,x)*a**2*c*d**3*e*x**3 - 12*int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/(a*d*x **4 + a*e*x**6 - c*d*x**8 - c*e*x**10),x)*a*b*c*d**4*x**3 + 4*int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a**2 *c*d*e**3*x**3 + 6*int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a*b*c*d**2*e**2*x**3 + 4*int((sqrt(d + e*x**2)* sqrt(a - c*x**4))/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a*c**2*d**3*e* x**3 + 4*int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/(a*d + a*e*x**2 - c*d*x** 4 - c*e*x**6),x)*b*c**2*d**4*x**3)/(4*c*d*e*x**3)