Integrand size = 33, antiderivative size = 149 \[ \int \frac {\sqrt {b^2+a x^2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx=-\frac {2 b x}{3 \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {2 x \sqrt {b^2+a x^2}}{3 \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {2 \sqrt {2} b^{3/2} \arctan \left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2} \sqrt {b}}\right )}{\sqrt {a}} \]
-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)*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))/a ^(1/2)
Time = 0.17 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.54 \[ \int \frac {\sqrt {b^2+a x^2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\frac {2 a x^3}{3 \left (b+\sqrt {b^2+a x^2}\right )^{3/2}}+\frac {\sqrt {2} b^{3/2} \arctan \left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}} \]
(2*a*x^3)/(3*(b + Sqrt[b^2 + a*x^2])^(3/2)) + (Sqrt[2]*b^(3/2)*ArcTan[(Sqr t[a]*x)/(Sqrt[2]*Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/Sqrt[a]
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.
\(\displaystyle \int \frac {\sqrt {a x^2+b^2}}{\sqrt {\sqrt {a x^2+b^2}+b}} \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \frac {\sqrt {a x^2+b^2}}{\sqrt {\sqrt {a x^2+b^2}+b}}dx\) |
3.21.68.3.1 Defintions of rubi rules used
\[\int \frac {\sqrt {a \,x^{2}+b^{2}}}{\sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}d x\]
Time = 145.28 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.77 \[ \int \frac {\sqrt {b^2+a x^2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\left [\frac {3 \, \sqrt {2} a b x \sqrt {-\frac {b}{a}} \log \left (-\frac {a x^{3} + 4 \, b^{2} x - 4 \, \sqrt {a x^{2} + b^{2}} b x - 2 \, {\left (2 \, \sqrt {2} \sqrt {a x^{2} + b^{2}} b \sqrt {-\frac {b}{a}} - \sqrt {2} {\left (a x^{2} + 2 \, b^{2}\right )} \sqrt {-\frac {b}{a}}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{x^{3}}\right ) + 4 \, {\left (a x^{2} + 2 \, b^{2} - 2 \, \sqrt {a x^{2} + b^{2}} b\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{6 \, a x}, -\frac {3 \, \sqrt {2} a b x \sqrt {\frac {b}{a}} \arctan \left (\frac {\sqrt {2} \sqrt {b + \sqrt {a x^{2} + b^{2}}} \sqrt {\frac {b}{a}}}{x}\right ) - 2 \, {\left (a x^{2} + 2 \, b^{2} - 2 \, \sqrt {a x^{2} + b^{2}} b\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{3 \, a x}\right ] \]
[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 )]
\[ \int \frac {\sqrt {b^2+a x^2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\int \frac {\sqrt {a x^{2} + b^{2}}}{\sqrt {b + \sqrt {a x^{2} + b^{2}}}}\, dx \]
\[ \int \frac {\sqrt {b^2+a x^2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\int { \frac {\sqrt {a x^{2} + b^{2}}}{\sqrt {b + \sqrt {a x^{2} + b^{2}}}} \,d x } \]
\[ \int \frac {\sqrt {b^2+a x^2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\int { \frac {\sqrt {a x^{2} + b^{2}}}{\sqrt {b + \sqrt {a x^{2} + b^{2}}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {b^2+a x^2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\int \frac {\sqrt {b^2+a\,x^2}}{\sqrt {b+\sqrt {b^2+a\,x^2}}} \,d x \]