\(\int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{x^2 \sqrt {b^2+a x^2}} \, dx\) [1705]

Optimal result

Integrand size = 36, antiderivative size = 114 \[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{x^2 \sqrt {b^2+a x^2}} \, dx=-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{b x}-\frac {\sqrt {2} \sqrt {a} \arctan \left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2} \sqrt {b}}\right )}{b^{3/2}} \]

[Out]

-(b+(a*x^2+b^2)^(1/2))^(1/2)/b/x-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]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{x^2 \sqrt {b^2+a x^2}} \, dx=-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{b x}-\frac {\sqrt {a} \arctan \left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {2} b^{3/2}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 23.02 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.92 \[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{x^2 \sqrt {b^2+a x^2}} \, dx=\left [\frac {\sqrt {2} x \sqrt {-\frac {a}{b}} \log \left (-\frac {a^{2} x^{3} + 4 \, a b^{2} x - 4 \, \sqrt {a x^{2} + b^{2}} a b x + 2 \, {\left (2 \, \sqrt {2} \sqrt {a x^{2} + b^{2}} b^{2} \sqrt {-\frac {a}{b}} - \sqrt {2} {\left (a b x^{2} + 2 \, b^{3}\right )} \sqrt {-\frac {a}{b}}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{x^{3}}\right ) - 4 \, \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{4 \, b x}, \frac {\sqrt {2} x \sqrt {\frac {a}{b}} \arctan \left (\frac {\sqrt {2} \sqrt {b + \sqrt {a x^{2} + b^{2}}} b \sqrt {\frac {a}{b}}}{a x}\right ) - 2 \, \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{2 \, b x}\right ] \]

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.63 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.39 \[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{x^2 \sqrt {b^2+a x^2}} \, dx=- \frac {\Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right ) {{}_{3}F_{2}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4}, \frac {3}{4} \\ \frac {1}{2}, \frac {1}{2} \end {matrix}\middle | {\frac {a x^{2} e^{i \pi }}{b^{2}}} \right )}}{\pi \sqrt {b} x} \]

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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