\(\int \frac {\sqrt {a+b x^2}}{\sqrt {1-x^4}} \, dx\) [473]

Optimal result

Integrand size = 23, antiderivative size = 172 \[ \int \frac {\sqrt {a+b x^2}}{\sqrt {1-x^4}} \, dx=\frac {a \sqrt {1-\frac {1}{x^4}} x^3 \sqrt {\frac {a+b x^2}{(a+b) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {1}{x^2}}}{\sqrt {2}}\right ),\frac {2 a}{a+b}\right )}{\sqrt {a+b x^2} \sqrt {1-x^4}}+\frac {b \sqrt {1-\frac {1}{x^4}} x^3 \sqrt {\frac {a+b x^2}{(a+b) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {1}{x^2}}}{\sqrt {2}}\right ),\frac {2 a}{a+b}\right )}{\sqrt {a+b x^2} \sqrt {1-x^4}} \] Output:

a*(1-1/x^4)^(1/2)*x^3*((b*x^2+a)/(a+b)/x^2)^(1/2)*EllipticF(1/2*(1-1/x^2)^ 
(1/2)*2^(1/2),2^(1/2)*(a/(a+b))^(1/2))/(b*x^2+a)^(1/2)/(-x^4+1)^(1/2)+b*(1 
-1/x^4)^(1/2)*x^3*((b*x^2+a)/(a+b)/x^2)^(1/2)*EllipticPi(1/2*(1-1/x^2)^(1/ 
2)*2^(1/2),2,2^(1/2)*(a/(a+b))^(1/2))/(b*x^2+a)^(1/2)/(-x^4+1)^(1/2)
 

Mathematica [F]

\[ \int \frac {\sqrt {a+b x^2}}{\sqrt {1-x^4}} \, dx=\int \frac {\sqrt {a+b x^2}}{\sqrt {1-x^4}} \, dx \] Input:

Integrate[Sqrt[a + b*x^2]/Sqrt[1 - x^4],x]
 

Output:

Integrate[Sqrt[a + b*x^2]/Sqrt[1 - 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.

\(\displaystyle \int \frac {\sqrt {a+b x^2}}{\sqrt {1-x^4}} \, dx\)

\(\Big \downarrow \) 1571

\(\displaystyle \int \frac {\sqrt {a+b x^2}}{\sqrt {1-x^4}}dx\)

Input:

Int[Sqrt[a + b*x^2]/Sqrt[1 - 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]

\[\int \frac {\sqrt {b \,x^{2}+a}}{\sqrt {-x^{4}+1}}d x\]

Input:

int((b*x^2+a)^(1/2)/(-x^4+1)^(1/2),x)
 

Output:

int((b*x^2+a)^(1/2)/(-x^4+1)^(1/2),x)
 

Fricas [F]

\[ \int \frac {\sqrt {a+b x^2}}{\sqrt {1-x^4}} \, dx=\int { \frac {\sqrt {b x^{2} + a}}{\sqrt {-x^{4} + 1}} \,d x } \] Input:

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

Output:

integral(-sqrt(-x^4 + 1)*sqrt(b*x^2 + a)/(x^4 - 1), x)
 

Sympy [F]

\[ \int \frac {\sqrt {a+b x^2}}{\sqrt {1-x^4}} \, dx=\int \frac {\sqrt {a + b x^{2}}}{\sqrt {- \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}}\, dx \] Input:

integrate((b*x**2+a)**(1/2)/(-x**4+1)**(1/2),x)
 

Output:

Integral(sqrt(a + b*x**2)/sqrt(-(x - 1)*(x + 1)*(x**2 + 1)), x)
 

Maxima [F]

\[ \int \frac {\sqrt {a+b x^2}}{\sqrt {1-x^4}} \, dx=\int { \frac {\sqrt {b x^{2} + a}}{\sqrt {-x^{4} + 1}} \,d x } \] Input:

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

Output:

integrate(sqrt(b*x^2 + a)/sqrt(-x^4 + 1), x)
 

Giac [F]

\[ \int \frac {\sqrt {a+b x^2}}{\sqrt {1-x^4}} \, dx=\int { \frac {\sqrt {b x^{2} + a}}{\sqrt {-x^{4} + 1}} \,d x } \] Input:

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

Output:

integrate(sqrt(b*x^2 + a)/sqrt(-x^4 + 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2}}{\sqrt {1-x^4}} \, dx=\int \frac {\sqrt {b\,x^2+a}}{\sqrt {1-x^4}} \,d x \] Input:

int((a + b*x^2)^(1/2)/(1 - x^4)^(1/2),x)
 

Output:

int((a + b*x^2)^(1/2)/(1 - x^4)^(1/2), x)
 

Reduce [F]

\[ \int \frac {\sqrt {a+b x^2}}{\sqrt {1-x^4}} \, dx=-\left (\int \frac {\sqrt {b \,x^{2}+a}\, \sqrt {-x^{4}+1}}{x^{4}-1}d x \right ) \] Input:

int((b*x^2+a)^(1/2)/(-x^4+1)^(1/2),x)
 

Output:

 - int((sqrt(a + b*x**2)*sqrt( - x**4 + 1))/(x**4 - 1),x)