\(\int \frac {(A+B x^2) \sqrt {d+e x^2} \sqrt {a-c x^4}}{x^4} \, dx\) [38]

Optimal result

Integrand size = 34, antiderivative size = 524 \[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{x^4} \, dx=-\frac {A \sqrt {d+e x^2} \sqrt {a-c x^4}}{3 x^3}+\frac {B \sqrt {d+e x^2} \sqrt {a-c x^4}}{2 x}+\frac {c (9 B d+2 A e) \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{6 d \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {\sqrt {c} \left (4 A c d^2+3 a B d e+2 a A e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{6 \sqrt {a} d \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {c (B d+2 A e) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

-1/3*A*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/x^3+1/2*B*(e*x^2+d)^(1/2)*(-c*x^4+ 
a)^(1/2)/x+1/6*c*(2*A*e+9*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))/d/ 
(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)-1/6*c^(1/2)*(2*A*a*e^2+4*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)/d/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)-1/2*c*(2* 
A*e+B*d)*(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)
 

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 2251

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 2251

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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

integrate((B*x^2+A)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/x^4,x, algorithm="fri 
cas")
 

Output:

integral(sqrt(-c*x^4 + a)*(B*x^2 + A)*sqrt(e*x^2 + d)/x^4, x)
 

Sympy [F]

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

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

Output:

Integral((A + B*x**2)*sqrt(a - c*x**4)*sqrt(d + e*x**2)/x**4, x)
 

Maxima [F]

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

integrate((B*x^2+A)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/x^4,x, algorithm="max 
ima")
 

Output:

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

Giac [F]

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

integrate((B*x^2+A)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/x^4,x, algorithm="gia 
c")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{x^4} \, dx=\frac {-\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, a d e +\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, a \,e^{2} x^{2}-\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, b \,d^{2}+\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, b d e \,x^{2}+2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a c \,e^{3} x^{3}+\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) b c d \,e^{2} x^{3}-2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{10}-c d \,x^{8}+a e \,x^{6}+a d \,x^{4}}d x \right ) a^{2} d^{2} e \,x^{3}-3 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{10}-c d \,x^{8}+a e \,x^{6}+a d \,x^{4}}d x \right ) a b \,d^{3} x^{3}+\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a b d \,e^{2} x^{3}+\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) b c \,d^{3} x^{3}}{d e \,x^{3}} \] Input:

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

Output:

( - sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*d*e + sqrt(d + e*x**2)*sqrt(a - c* 
x**4)*a*e**2*x**2 - sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*d**2 + sqrt(d + e* 
x**2)*sqrt(a - c*x**4)*b*d*e*x**2 + 2*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*e**3*x**3 + int((sq 
rt(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*d*e**2*x**3 - 2*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*d**2*e*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*b*d**3*x**3 + 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*d*e**2*x**3 + 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*d**3*x**3)/(d*e*x** 
3)