\(\int \frac {(-b^2+a x^2)^2 \sqrt {b+\sqrt {b^2+a x^2}}}{(b^2+a x^2)^2} \, dx\) [2012]

Optimal result

Integrand size = 44, antiderivative size = 142 \[ \int \frac {\left (-b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^2} \, dx=\frac {x \left (11 b^2+2 a x^2\right )}{3 \sqrt {b^2+a x^2} \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {2 x \left (5 b^3+2 a b x^2\right )}{3 \left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {5 b^{3/2} \arctan \left (\frac {\sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}} \] Output:

1/3*x*(2*a*x^2+11*b^2)/(a*x^2+b^2)^(1/2)/(b+(a*x^2+b^2)^(1/2))^(1/2)+2/3*x 
*(2*a*b*x^2+5*b^3)/(a*x^2+b^2)/(b+(a*x^2+b^2)^(1/2))^(1/2)-5*b^(3/2)*arcta 
n(a^(1/2)*x/b^(1/2)/(b+(a*x^2+b^2)^(1/2))^(1/2))/a^(1/2)
 

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^2} \, dx=\frac {x \left (11 b^2+2 a x^2\right )}{3 \sqrt {b^2+a x^2} \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {2 x \left (5 b^3+2 a b x^2\right )}{3 \left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {5 b^{3/2} \arctan \left (\frac {\sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}} \] Input:

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

Output:

(x*(11*b^2 + 2*a*x^2))/(3*Sqrt[b^2 + a*x^2]*Sqrt[b + Sqrt[b^2 + a*x^2]]) + 
 (2*x*(5*b^3 + 2*a*b*x^2))/(3*(b^2 + a*x^2)*Sqrt[b + Sqrt[b^2 + a*x^2]]) - 
 (5*b^(3/2)*ArcTan[(Sqrt[a]*x)/(Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/Sqr 
t[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.

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {4 b^2 \sqrt {\sqrt {a x^2+b^2}+b}}{a x^2+b^2}+\sqrt {\sqrt {a x^2+b^2}+b}+\frac {4 b^4 \sqrt {\sqrt {a x^2+b^2}+b}}{\left (a x^2+b^2\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

$Aborted
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (-b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^2} \, dx=\left (\int \frac {\sqrt {\sqrt {a \,x^{2}+b^{2}}+b}}{a^{2} x^{4}+2 a \,b^{2} x^{2}+b^{4}}d x \right ) b^{4}+\left (\int \frac {\sqrt {\sqrt {a \,x^{2}+b^{2}}+b}\, x^{4}}{a^{2} x^{4}+2 a \,b^{2} x^{2}+b^{4}}d x \right ) a^{2}-2 \left (\int \frac {\sqrt {\sqrt {a \,x^{2}+b^{2}}+b}\, x^{2}}{a^{2} x^{4}+2 a \,b^{2} x^{2}+b^{4}}d x \right ) a \,b^{2} \] Input:

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

Output:

int(sqrt(sqrt(a*x**2 + b**2) + b)/(a**2*x**4 + 2*a*b**2*x**2 + b**4),x)*b* 
*4 + int((sqrt(sqrt(a*x**2 + b**2) + b)*x**4)/(a**2*x**4 + 2*a*b**2*x**2 + 
 b**4),x)*a**2 - 2*int((sqrt(sqrt(a*x**2 + b**2) + b)*x**2)/(a**2*x**4 + 2 
*a*b**2*x**2 + b**4),x)*a*b**2