\(\int \frac {(b^2+a x^2) \sqrt {b+\sqrt {b^2+a x^2}}}{-b^2+a x^2} \, dx\) [3090]

Optimal result

Integrand size = 42, antiderivative size = 530 \[ \int \frac {\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}{-b^2+a x^2} \, dx=\frac {4 b x}{3 \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {2 x \sqrt {b^2+a x^2}}{3 \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {2 i \left (\sqrt {-1+\sqrt {2}} b^{3/2}+\sqrt {2 \left (-1+\sqrt {2}\right )} b^{3/2}\right ) \arctan \left (\frac {\frac {i a x}{\sqrt {b+\sqrt {b^2+a x^2}}}-i \sqrt {a} \sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {a} \sqrt {b}}\right )}{\sqrt {a}}+\frac {2 \sqrt {-1+\sqrt {2}} b^{3/2} \arctan \left (\frac {\sqrt {a} x}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {b}}\right )}{\sqrt {a}}-\frac {2 i \left (-\sqrt {1+\sqrt {2}} b^{3/2}+\sqrt {2 \left (1+\sqrt {2}\right )} b^{3/2}\right ) \text {arctanh}\left (\frac {\frac {i a x}{\sqrt {b+\sqrt {b^2+a x^2}}}-i \sqrt {a} \sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {a} \sqrt {b}}\right )}{\sqrt {a}}+\frac {2 \sqrt {1+\sqrt {2}} b^{3/2} \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {b}}\right )}{\sqrt {a}} \] Output:

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)
 

Mathematica [A] (verified)

Time = 0.76 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.33 \[ \int \frac {\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}{-b^2+a x^2} \, dx=\frac {2 x \left (2 b+\sqrt {b^2+a x^2}\right )}{3 \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {2 b^{3/2} \arctan \left (\frac {\sqrt {-1+\sqrt {2}} \sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {1+\sqrt {2}} \sqrt {a}}-\frac {2 \sqrt {1+\sqrt {2}} b^{3/2} \text {arctanh}\left (\frac {\sqrt {1+\sqrt {2}} \sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}} \] Input:

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

Output:

(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]
 

Rubi [F]

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

\(\displaystyle -b \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{b-\sqrt {a} x}dx-b \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{b+\sqrt {a} x}dx+\frac {2 b x}{\sqrt {\sqrt {a x^2+b^2}+b}}+\frac {2 a x^3}{3 \left (\sqrt {a x^2+b^2}+b\right )^{3/2}}\)

Input:

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

Output:

$Aborted
 

Maple [F]

\[\int \frac {\left (a \,x^{2}+b^{2}\right ) \sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}{a \,x^{2}-b^{2}}d x\]

Input:

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

Output:

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

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}{-b^2+a x^2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}{-b^2+a x^2} \, dx=\int \frac {\sqrt {b + \sqrt {a x^{2} + b^{2}}} \left (a x^{2} + b^{2}\right )}{a x^{2} - b^{2}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}{-b^2+a x^2} \, dx=\int { \frac {{\left (a x^{2} + b^{2}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{a x^{2} - b^{2}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}{-b^2+a x^2} \, dx=\int { \frac {{\left (a x^{2} + b^{2}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{a x^{2} - b^{2}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}{-b^2+a x^2} \, dx=\int \frac {\left (b^2+a\,x^2\right )\,\sqrt {b+\sqrt {b^2+a\,x^2}}}{a\,x^2-b^2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}{-b^2+a x^2} \, dx=\int \frac {\left (a \,x^{2}+b^{2}\right ) \sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}{a \,x^{2}-b^{2}}d x \] Input:

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

Output:

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