\(\int \frac {A+B x^2}{(d+e x^2)^{3/2} \sqrt {a-c x^4}} \, dx\) [110]

Optimal result

Integrand size = 31, antiderivative size = 354 \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}} \, dx=-\frac {(B d-A e) \sqrt {a-c x^4}}{\left (c d^2-a e^2\right ) x \sqrt {d+e x^2}}-\frac {\sqrt {c} (B d-A e) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{d \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {A \sqrt {c} \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{\sqrt {a} d \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

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

Mathematica [F]

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

Integrate[(A + B*x^2)/((d + e*x^2)^(3/2)*Sqrt[a - c*x^4]),x]
 

Output:

Integrate[(A + B*x^2)/((d + e*x^2)^(3/2)*Sqrt[a - c*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.

\(\displaystyle \int \frac {A+B x^2}{\sqrt {a-c x^4} \left (d+e x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 2261

\(\displaystyle \int \frac {A+B x^2}{\sqrt {a-c x^4} \left (d+e x^2\right )^{3/2}}dx\)

Input:

Int[(A + B*x^2)/((d + e*x^2)^(3/2)*Sqrt[a - c*x^4]),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 2261

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol 
] :> Unintegrable[Px*(d + e*x^2)^q*(a + c*x^4)^p, x] /; FreeQ[{a, c, d, e, 
p, q}, x] && PolyQ[Px, x]
 

Maple [F]

\[\int \frac {B \,x^{2}+A}{\left (e \,x^{2}+d \right )^{\frac {3}{2}} \sqrt {-c \,x^{4}+a}}d x\]

Input:

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

Output:

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

Fricas [F]

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

integrate((B*x^2+A)/(e*x^2+d)^(3/2)/(-c*x^4+a)^(1/2),x, algorithm="fricas" 
)
 

Output:

integral(-sqrt(-c*x^4 + a)*(B*x^2 + A)*sqrt(e*x^2 + d)/(c*e^2*x^8 + 2*c*d* 
e*x^6 - 2*a*d*e*x^2 + (c*d^2 - a*e^2)*x^4 - a*d^2), x)
 

Sympy [F]

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

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

Output:

Integral((A + B*x**2)/(sqrt(a - c*x**4)*(d + e*x**2)**(3/2)), x)
 

Maxima [F]

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

integrate((B*x^2+A)/(e*x^2+d)^(3/2)/(-c*x^4+a)^(1/2),x, algorithm="maxima" 
)
 

Output:

integrate((B*x^2 + A)/(sqrt(-c*x^4 + a)*(e*x^2 + d)^(3/2)), x)
 

Giac [F]

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

integrate((B*x^2+A)/(e*x^2+d)^(3/2)/(-c*x^4+a)^(1/2),x, algorithm="giac")
 

Output:

integrate((B*x^2 + A)/(sqrt(-c*x^4 + a)*(e*x^2 + d)^(3/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}} \, dx=\int \frac {B\,x^2+A}{\sqrt {a-c\,x^4}\,{\left (e\,x^2+d\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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