\(\int \frac {(A+B x^2) \sqrt {d+e x^2}}{\sqrt {a-c x^4}} \, dx\) [108]

Optimal result

Integrand size = 31, antiderivative size = 469 \[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2}}{\sqrt {a-c x^4}} \, dx=-\frac {B \sqrt {d+e x^2} \sqrt {a-c x^4}}{2 c x}-\frac {B \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {(2 A c d+a B e) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 \sqrt {a} \sqrt {c} \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {(B d+2 A e) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

-1/2*B*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/c/x-1/2*B*(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)*El 
lipticE(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)+1/2*(2*A*c*d+B*a*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)*Elli 
pticF(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)/c^(1/2)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+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)*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]

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

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

Output:

Integrate[((A + B*x^2)*Sqrt[d + e*x^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.

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

\(\Big \downarrow \) 2261

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

Input:

Int[((A + B*x^2)*Sqrt[d + e*x^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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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