\(\int \frac {1}{\sqrt {a x^2+\sqrt {b+a^2 x^4}}} \, dx\) [1368]

Optimal result

Integrand size = 23, antiderivative size = 98 \[ \int \frac {1}{\sqrt {a x^2+\sqrt {b+a^2 x^4}}} \, dx=\frac {x}{2 \sqrt {a x^2+\sqrt {b+a^2 x^4}}}+\frac {\log \left (a x^2+\sqrt {b+a^2 x^4}+\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )}{2 \sqrt {2} \sqrt {a}} \] Output:

1/2*x/(a*x^2+(a^2*x^4+b)^(1/2))^(1/2)+1/4*ln(a*x^2+(a^2*x^4+b)^(1/2)+2^(1/ 
2)*a^(1/2)*x*(a*x^2+(a^2*x^4+b)^(1/2))^(1/2))*2^(1/2)/a^(1/2)
 

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {a x^2+\sqrt {b+a^2 x^4}}} \, dx=\frac {x}{2 \sqrt {a x^2+\sqrt {b+a^2 x^4}}}+\frac {\log \left (a x^2+\sqrt {b+a^2 x^4}+\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )}{2 \sqrt {2} \sqrt {a}} \] Input:

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

Output:

x/(2*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]) + Log[a*x^2 + Sqrt[b + a^2*x^4] + Sq 
rt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]]/(2*Sqrt[2]*Sqrt[a])
 

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.

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

\(\Big \downarrow \) 7299

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

Input:

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

Output:

$Aborted
 

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 1.03 (sec) , antiderivative size = 262, normalized size of antiderivative = 2.67 \[ \int \frac {1}{\sqrt {a x^2+\sqrt {b+a^2 x^4}}} \, dx=\left [\frac {\frac {\sqrt {2} b \log \left (4 \, a^{2} x^{4} + 4 \, \sqrt {a^{2} x^{4} + b} a x^{2} + 2 \, {\left (\sqrt {2} a^{\frac {3}{2}} x^{3} + \sqrt {2} \sqrt {a^{2} x^{4} + b} \sqrt {a} x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} + b\right )}{\sqrt {a}} - 4 \, {\left (a x^{3} - \sqrt {a^{2} x^{4} + b} x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{8 \, b}, -\frac {\sqrt {2} b \sqrt {-\frac {1}{a}} \arctan \left (-\frac {{\left (\sqrt {2} a^{2} x^{3} \sqrt {-\frac {1}{a}} - \sqrt {2} \sqrt {a^{2} x^{4} + b} a x \sqrt {-\frac {1}{a}}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{b}\right ) + 2 \, {\left (a x^{3} - \sqrt {a^{2} x^{4} + b} x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{4 \, b}\right ] \] Input:

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

Output:

[1/8*(sqrt(2)*b*log(4*a^2*x^4 + 4*sqrt(a^2*x^4 + b)*a*x^2 + 2*(sqrt(2)*a^( 
3/2)*x^3 + sqrt(2)*sqrt(a^2*x^4 + b)*sqrt(a)*x)*sqrt(a*x^2 + sqrt(a^2*x^4 
+ b)) + b)/sqrt(a) - 4*(a*x^3 - sqrt(a^2*x^4 + b)*x)*sqrt(a*x^2 + sqrt(a^2 
*x^4 + b)))/b, -1/4*(sqrt(2)*b*sqrt(-1/a)*arctan(-(sqrt(2)*a^2*x^3*sqrt(-1 
/a) - sqrt(2)*sqrt(a^2*x^4 + b)*a*x*sqrt(-1/a))*sqrt(a*x^2 + sqrt(a^2*x^4 
+ b))/b) + 2*(a*x^3 - sqrt(a^2*x^4 + b)*x)*sqrt(a*x^2 + sqrt(a^2*x^4 + b)) 
)/b]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\sqrt {a x^2+\sqrt {b+a^2 x^4}}} \, dx=\frac {8 \sqrt {a^{2} x^{4}+b}\, \sqrt {\sqrt {a^{2} x^{4}+b}+a \,x^{2}}\, a x -3 \sqrt {a}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {a^{2} x^{4}+b}+a \,x^{2}}-\sqrt {a}\, \sqrt {2}\, x \right ) b +3 \sqrt {a}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {a^{2} x^{4}+b}+a \,x^{2}}+\sqrt {a}\, \sqrt {2}\, x \right ) b -32 \left (\int \frac {\sqrt {\sqrt {a^{2} x^{4}+b}+a \,x^{2}}\, x^{6}}{a^{2} x^{4}+b}d x \right ) a^{4}-32 \left (\int \frac {\sqrt {\sqrt {a^{2} x^{4}+b}+a \,x^{2}}\, x^{2}}{a^{2} x^{4}+b}d x \right ) a^{2} b +4 \left (\int \frac {\sqrt {a^{2} x^{4}+b}\, \sqrt {\sqrt {a^{2} x^{4}+b}+a \,x^{2}}}{a^{2} x^{4}+b}d x \right ) a b}{24 a b} \] Input:

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

Output:

(8*sqrt(a**2*x**4 + b)*sqrt(sqrt(a**2*x**4 + b) + a*x**2)*a*x - 3*sqrt(a)* 
sqrt(2)*log(sqrt(sqrt(a**2*x**4 + b) + a*x**2) - sqrt(a)*sqrt(2)*x)*b + 3* 
sqrt(a)*sqrt(2)*log(sqrt(sqrt(a**2*x**4 + b) + a*x**2) + sqrt(a)*sqrt(2)*x 
)*b - 32*int((sqrt(sqrt(a**2*x**4 + b) + a*x**2)*x**6)/(a**2*x**4 + b),x)* 
a**4 - 32*int((sqrt(sqrt(a**2*x**4 + b) + a*x**2)*x**2)/(a**2*x**4 + b),x) 
*a**2*b + 4*int((sqrt(a**2*x**4 + b)*sqrt(sqrt(a**2*x**4 + b) + a*x**2))/( 
a**2*x**4 + b),x)*a*b)/(24*a*b)