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

Optimal result

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

[Out]

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)

Rubi [F]

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

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt {a x^2+\sqrt {b+a^2 x^4}}} \, dx \\ \end{align*}

Mathematica [A] (verified)

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

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 1.11 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.34 \[ \int \frac {1}{\sqrt {a x^2+\sqrt {b+a^2 x^4}}} \, dx=\left [\frac {\frac {\sqrt {2} b \log \left (4 \, a^{2} x^{4} + 4 \, \sqrt {a^{2} x^{4} + b} a x^{2} + 2 \, {\left (\sqrt {2} a^{\frac {3}{2}} x^{3} + \sqrt {2} \sqrt {a^{2} x^{4} + b} \sqrt {a} x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} + b\right )}{\sqrt {a}} - 4 \, {\left (a x^{3} - \sqrt {a^{2} x^{4} + b} x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{8 \, b}, -\frac {\sqrt {2} b \sqrt {-\frac {1}{a}} \arctan \left (\frac {\sqrt {2} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} \sqrt {-\frac {1}{a}}}{2 \, x}\right ) + 2 \, {\left (a x^{3} - \sqrt {a^{2} x^{4} + b} x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{4 \, b}\right ] \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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