\(\int \frac {(A+B x^2) \sqrt {a-c x^4}}{x^2 \sqrt {d+e x^2}} \, dx\) [55]

Optimal result

Integrand size = 34, antiderivative size = 488 \[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^2 \sqrt {d+e x^2}} \, dx=\frac {B \sqrt {d+e x^2} \sqrt {a-c x^4}}{2 e x}+\frac {c (B d+2 A e) \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 d e \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\sqrt {a} \sqrt {c} (B d-2 A e) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 d \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {c (B d-2 A e) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 e \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

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

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 2251

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Giac [F]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(d + e*x**2)*sqrt(a - c*x**4)*a + 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*x - 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*d*x + 2*int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/(a*d*x**2 + a*e*x** 
4 - c*d*x**6 - c*e*x**8),x)*a**2*d*x + 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*x)/(d*x)