\(\int \frac {1}{x^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx\) [2028]

Optimal result

Integrand size = 23, antiderivative size = 144 \[ \int \frac {1}{x^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx=-\frac {1}{2 x \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{4 b x}-\frac {\sqrt {a} \arctan \left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2} \sqrt {b}}\right )}{2 \sqrt {2} b^{3/2}} \]

[Out]

-1/2/x/(b+(a*x^2+b^2)^(1/2))^(1/2)-1/4*(b+(a*x^2+b^2)^(1/2))^(1/2)/b/x-1/4*2^(1/2)*a^(1/2)*arctan(1/2*a^(1/2)*
x*2^(1/2)/b^(1/2)/(b+(a*x^2+b^2)^(1/2))^(1/2)-1/2*(b+(a*x^2+b^2)^(1/2))^(1/2)*2^(1/2)/b^(1/2))/b^(3/2)

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.72 \[ \int \frac {1}{x^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\frac {1}{8} \left (-\frac {2 \left (3 b+\sqrt {b^2+a x^2}\right )}{b x \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {2} \sqrt {a} \arctan \left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{b^{3/2}}\right ) \]

[In]

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

[Out]

((-2*(3*b + Sqrt[b^2 + a*x^2]))/(b*x*Sqrt[b + Sqrt[b^2 + a*x^2]]) - (Sqrt[2]*Sqrt[a]*ArcTan[(Sqrt[a]*x)/(Sqrt[
2]*Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/b^(3/2))/8

Maple [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3.

Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.22

method	result	size
meijerg	\(-\frac {\sqrt {2}\, \operatorname {hypergeom}\left (\left [-\frac {1}{2}, \frac {1}{4}, \frac {3}{4}\right ], \left [\frac {1}{2}, \frac {3}{2}\right ], -\frac {x^{2} a}{b^{2}}\right )}{2 \left (b^{2}\right )^{\frac {1}{4}} x}\)	\(31\)

[In]

int(1/x^2/(b+(a*x^2+b^2)^(1/2))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/2/(b^2)^(1/4)*2^(1/2)/x*hypergeom([-1/2,1/4,3/4],[1/2,3/2],-x^2*a/b^2)

Fricas [A] (verification not implemented)

none

Time = 25.65 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.96 \[ \int \frac {1}{x^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\left [\frac {\sqrt {\frac {1}{2}} a x^{3} \sqrt {-\frac {a}{b}} \log \left (-\frac {a^{2} x^{3} + 4 \, a b^{2} x - 4 \, \sqrt {a x^{2} + b^{2}} a b x + 4 \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {a x^{2} + b^{2}} b^{2} \sqrt {-\frac {a}{b}} - \sqrt {\frac {1}{2}} {\left (a b x^{2} + 2 \, b^{3}\right )} \sqrt {-\frac {a}{b}}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{x^{3}}\right ) - 2 \, {\left (a x^{2} - 2 \, b^{2} + 2 \, \sqrt {a x^{2} + b^{2}} b\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{8 \, a b x^{3}}, \frac {\sqrt {\frac {1}{2}} a x^{3} \sqrt {\frac {a}{b}} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {b + \sqrt {a x^{2} + b^{2}}} b \sqrt {\frac {a}{b}}}{a x}\right ) - {\left (a x^{2} - 2 \, b^{2} + 2 \, \sqrt {a x^{2} + b^{2}} b\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{4 \, a b x^{3}}\right ] \]

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.75 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.32 \[ \int \frac {1}{x^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx=- \frac {\Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right ) {{}_{3}F_{2}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4}, \frac {3}{4} \\ \frac {1}{2}, \frac {3}{2} \end {matrix}\middle | {\frac {a x^{2} e^{i \pi }}{b^{2}}} \right )}}{2 \pi \sqrt {b} x} \]

[In]

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

[Out]

-gamma(1/4)*gamma(3/4)*hyper((-1/2, 1/4, 3/4), (1/2, 3/2), a*x**2*exp_polar(I*pi)/b**2)/(2*pi*sqrt(b)*x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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