\(\int \frac {\sqrt {b^2+a x^2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx\) [2068]

Optimal result

Integrand size = 33, antiderivative size = 149 \[ \int \frac {\sqrt {b^2+a x^2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx=-\frac {2 b x}{3 \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {2 x \sqrt {b^2+a x^2}}{3 \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {2 \sqrt {2} b^{3/2} \arctan \left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2} \sqrt {b}}\right )}{\sqrt {a}} \]

[Out]

-2/3*b*x/(b+(a*x^2+b^2)^(1/2))^(1/2)+2/3*x*(a*x^2+b^2)^(1/2)/(b+(a*x^2+b^2)^(1/2))^(1/2)+2*2^(1/2)*b^(3/2)*arc
tan(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))
/a^(1/2)

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.54 \[ \int \frac {\sqrt {b^2+a x^2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\frac {2 a x^3}{3 \left (b+\sqrt {b^2+a x^2}\right )^{3/2}}+\frac {\sqrt {2} b^{3/2} \arctan \left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 145.28 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.77 \[ \int \frac {\sqrt {b^2+a x^2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\left [\frac {3 \, \sqrt {2} a b x \sqrt {-\frac {b}{a}} \log \left (-\frac {a x^{3} + 4 \, b^{2} x - 4 \, \sqrt {a x^{2} + b^{2}} b x - 2 \, {\left (2 \, \sqrt {2} \sqrt {a x^{2} + b^{2}} b \sqrt {-\frac {b}{a}} - \sqrt {2} {\left (a x^{2} + 2 \, b^{2}\right )} \sqrt {-\frac {b}{a}}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{x^{3}}\right ) + 4 \, {\left (a x^{2} + 2 \, b^{2} - 2 \, \sqrt {a x^{2} + b^{2}} b\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{6 \, a x}, -\frac {3 \, \sqrt {2} a b x \sqrt {\frac {b}{a}} \arctan \left (\frac {\sqrt {2} \sqrt {b + \sqrt {a x^{2} + b^{2}}} \sqrt {\frac {b}{a}}}{x}\right ) - 2 \, {\left (a x^{2} + 2 \, b^{2} - 2 \, \sqrt {a x^{2} + b^{2}} b\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{3 \, a x}\right ] \]

[In]

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

[Out]

[1/6*(3*sqrt(2)*a*b*x*sqrt(-b/a)*log(-(a*x^3 + 4*b^2*x - 4*sqrt(a*x^2 + b^2)*b*x - 2*(2*sqrt(2)*sqrt(a*x^2 + b
^2)*b*sqrt(-b/a) - sqrt(2)*(a*x^2 + 2*b^2)*sqrt(-b/a))*sqrt(b + sqrt(a*x^2 + b^2)))/x^3) + 4*(a*x^2 + 2*b^2 -
2*sqrt(a*x^2 + b^2)*b)*sqrt(b + sqrt(a*x^2 + b^2)))/(a*x), -1/3*(3*sqrt(2)*a*b*x*sqrt(b/a)*arctan(sqrt(2)*sqrt
(b + sqrt(a*x^2 + b^2))*sqrt(b/a)/x) - 2*(a*x^2 + 2*b^2 - 2*sqrt(a*x^2 + b^2)*b)*sqrt(b + sqrt(a*x^2 + b^2)))/
(a*x)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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