\(\int \frac {\sqrt {d+e x^2}}{\sqrt {a-c x^4}} \, dx\) [453]

Optimal result

Integrand size = 24, antiderivative size = 268 \[ \int \frac {\sqrt {d+e x^2}}{\sqrt {a-c x^4}} \, dx=\frac {\sqrt {c} d \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{\sqrt {a} \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {e \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{\sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

c^(1/2)*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)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+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)*El 
lipticPi(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]

\[ \int \frac {\sqrt {d+e x^2}}{\sqrt {a-c x^4}} \, dx=\int \frac {\sqrt {d+e x^2}}{\sqrt {a-c x^4}} \, dx \] Input:

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

Output:

Integrate[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.

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

\(\Big \downarrow \) 1571

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 1571

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

Maple [F]

\[\int \frac {\sqrt {e \,x^{2}+d}}{\sqrt {-c \,x^{4}+a}}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {\sqrt {d+e x^2}}{\sqrt {a-c x^4}} \, dx=\int { \frac {\sqrt {e x^{2} + d}}{\sqrt {-c x^{4} + a}} \,d x } \] Input:

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

Output:

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

Sympy [F]

\[ \int \frac {\sqrt {d+e x^2}}{\sqrt {a-c x^4}} \, dx=\int \frac {\sqrt {d + e x^{2}}}{\sqrt {a - c x^{4}}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {\sqrt {d+e x^2}}{\sqrt {a-c x^4}} \, dx=\int { \frac {\sqrt {e x^{2} + d}}{\sqrt {-c x^{4} + a}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {\sqrt {d+e x^2}}{\sqrt {a-c x^4}} \, dx=\int { \frac {\sqrt {e x^{2} + d}}{\sqrt {-c x^{4} + a}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {d+e x^2}}{\sqrt {a-c x^4}} \, dx=\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c \,x^{4}+a}d x \] Input:

int((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))/(a - c*x**4),x)