Integrand size = 42, antiderivative size = 530 \[ \int \frac {\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}{-b^2+a x^2} \, dx=\frac {4 b x}{3 \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {2 x \sqrt {b^2+a x^2}}{3 \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {2 i \left (\sqrt {-1+\sqrt {2}} b^{3/2}+\sqrt {2 \left (-1+\sqrt {2}\right )} b^{3/2}\right ) \arctan \left (\frac {\frac {i a x}{\sqrt {b+\sqrt {b^2+a x^2}}}-i \sqrt {a} \sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {a} \sqrt {b}}\right )}{\sqrt {a}}+\frac {2 \sqrt {-1+\sqrt {2}} b^{3/2} \arctan \left (\frac {\sqrt {a} x}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {b}}\right )}{\sqrt {a}}-\frac {2 i \left (-\sqrt {1+\sqrt {2}} b^{3/2}+\sqrt {2 \left (1+\sqrt {2}\right )} b^{3/2}\right ) \text {arctanh}\left (\frac {\frac {i a x}{\sqrt {b+\sqrt {b^2+a x^2}}}-i \sqrt {a} \sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {a} \sqrt {b}}\right )}{\sqrt {a}}+\frac {2 \sqrt {1+\sqrt {2}} b^{3/2} \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {b}}\right )}{\sqrt {a}} \]
4/3*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*I*((2^(1/2)-1)^(1/2)*b^(3/2)+(-2+2*2^(1/2))^(1/2)*b^(3/2)) *arctan((I*a*x/(b+(a*x^2+b^2)^(1/2))^(1/2)-I*a^(1/2)*(b+(a*x^2+b^2)^(1/2)) ^(1/2))/(2+2*2^(1/2))^(1/2)/a^(1/2)/b^(1/2))/a^(1/2)+2*(2^(1/2)-1)^(1/2)*b ^(3/2)*arctan(a^(1/2)*x/(2+2*2^(1/2))^(1/2)/b^(1/2)/(b+(a*x^2+b^2)^(1/2))^ (1/2)-(b+(a*x^2+b^2)^(1/2))^(1/2)/(2+2*2^(1/2))^(1/2)/b^(1/2))/a^(1/2)-2*I *(-(1+2^(1/2))^(1/2)*b^(3/2)+(2+2*2^(1/2))^(1/2)*b^(3/2))*arctanh((I*a*x/( b+(a*x^2+b^2)^(1/2))^(1/2)-I*a^(1/2)*(b+(a*x^2+b^2)^(1/2))^(1/2))/(-2+2*2^ (1/2))^(1/2)/a^(1/2)/b^(1/2))/a^(1/2)+2*(1+2^(1/2))^(1/2)*b^(3/2)*arctanh( a^(1/2)*x/(-2+2*2^(1/2))^(1/2)/b^(1/2)/(b+(a*x^2+b^2)^(1/2))^(1/2)-(b+(a*x ^2+b^2)^(1/2))^(1/2)/(-2+2*2^(1/2))^(1/2)/b^(1/2))/a^(1/2)
Time = 0.71 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.33 \[ \int \frac {\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}{-b^2+a x^2} \, dx=\frac {2 x \left (2 b+\sqrt {b^2+a x^2}\right )}{3 \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {2 b^{3/2} \arctan \left (\frac {\sqrt {-1+\sqrt {2}} \sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {1+\sqrt {2}} \sqrt {a}}-\frac {2 \sqrt {1+\sqrt {2}} b^{3/2} \text {arctanh}\left (\frac {\sqrt {1+\sqrt {2}} \sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}} \]
(2*x*(2*b + Sqrt[b^2 + a*x^2]))/(3*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (2*b^(3/ 2)*ArcTan[(Sqrt[-1 + Sqrt[2]]*Sqrt[a]*x)/(Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^ 2]])])/(Sqrt[1 + Sqrt[2]]*Sqrt[a]) - (2*Sqrt[1 + Sqrt[2]]*b^(3/2)*ArcTanh[ (Sqrt[1 + Sqrt[2]]*Sqrt[a]*x)/(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.
\(\displaystyle \int \frac {\left (a x^2+b^2\right ) \sqrt {\sqrt {a x^2+b^2}+b}}{a x^2-b^2} \, dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (\frac {2 b^2 \sqrt {\sqrt {a x^2+b^2}+b}}{a x^2-b^2}+\sqrt {\sqrt {a x^2+b^2}+b}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -b \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{b-\sqrt {a} x}dx-b \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{b+\sqrt {a} x}dx+\frac {2 b x}{\sqrt {\sqrt {a x^2+b^2}+b}}+\frac {2 a x^3}{3 \left (\sqrt {a x^2+b^2}+b\right )^{3/2}}\) |
3.31.90.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
\[\int \frac {\left (a \,x^{2}+b^{2}\right ) \sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}{a \,x^{2}-b^{2}}d x\]
Timed out. \[ \int \frac {\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}{-b^2+a x^2} \, dx=\text {Timed out} \]
\[ \int \frac {\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}{-b^2+a x^2} \, dx=\int \frac {\sqrt {b + \sqrt {a x^{2} + b^{2}}} \left (a x^{2} + b^{2}\right )}{a x^{2} - b^{2}}\, dx \]
\[ \int \frac {\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}{-b^2+a x^2} \, dx=\int { \frac {{\left (a x^{2} + b^{2}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{a x^{2} - b^{2}} \,d x } \]
\[ \int \frac {\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}{-b^2+a x^2} \, dx=\int { \frac {{\left (a x^{2} + b^{2}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{a x^{2} - b^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}{-b^2+a x^2} \, dx=\int \frac {\left (b^2+a\,x^2\right )\,\sqrt {b+\sqrt {b^2+a\,x^2}}}{a\,x^2-b^2} \,d x \]