\(\int \frac {(A+B x^2) \sqrt {a-c x^4}}{x^4 \sqrt {d+e x^2}} \, dx\) [56]

Optimal result

Integrand size = 34, antiderivative size = 486 \[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^4 \sqrt {d+e x^2}} \, dx=-\frac {A \sqrt {d+e x^2} \sqrt {a-c x^4}}{3 d x^3}+\frac {c (3 B d-2 A e) \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{3 d^2 \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {\sqrt {c} \left (2 A c d^2+a e (3 B d-2 A e)\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{3 \sqrt {a} d^2 \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {B c \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{\sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

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

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

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

Output:

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

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

\(\Big \downarrow \) 2251

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(sqrt(-c*x^4 + a)*(B*x^2 + A)*sqrt(e*x^2 + d)/(e*x^6 + d*x^4), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Giac [F]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - sqrt(d + e*x**2)*sqrt(a - c*x**4)*b - 2*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)*b*c*e*x**3 + 2*in 
t((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*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*x**3 - 2*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*e*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*x**3)/(2*e*x**3)