\(\int \frac {(b^2+a x^2)^{3/2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx\) [2280]

Optimal result

Integrand size = 33, antiderivative size = 173 \[ \int \frac {\left (b^2+a x^2\right )^{3/2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\frac {2 x \sqrt {b^2+a x^2} \left (46 b^2+15 a x^2\right )}{105 \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {2 x \left (46 b^3+3 a b x^2\right )}{105 \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {2 \sqrt {2} b^{7/2} \arctan \left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2} \sqrt {b}}\right )}{\sqrt {a}} \] Output:

2/105*x*(a*x^2+b^2)^(1/2)*(15*a*x^2+46*b^2)/(b+(a*x^2+b^2)^(1/2))^(1/2)-2/ 
105*x*(3*a*b*x^2+46*b^3)/(b+(a*x^2+b^2)^(1/2))^(1/2)+2*2^(1/2)*b^(7/2)*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)
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.74 \[ \int \frac {\left (b^2+a x^2\right )^{3/2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\frac {2 x \left (-46 b^3-3 a b x^2+46 b^2 \sqrt {b^2+a x^2}+15 a x^2 \sqrt {b^2+a x^2}\right )}{105 \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {\sqrt {2} b^{7/2} \arctan \left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}} \] Input:

Integrate[(b^2 + a*x^2)^(3/2)/Sqrt[b + Sqrt[b^2 + a*x^2]],x]
 

Output:

(2*x*(-46*b^3 - 3*a*b*x^2 + 46*b^2*Sqrt[b^2 + a*x^2] + 15*a*x^2*Sqrt[b^2 + 
 a*x^2]))/(105*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (Sqrt[2]*b^(7/2)*ArcTan[(Sqr 
t[a]*x)/(Sqrt[2]*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.

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

\(\Big \downarrow \) 7299

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

Input:

Int[(b^2 + a*x^2)^(3/2)/Sqrt[b + Sqrt[b^2 + a*x^2]],x]
 

Output:

$Aborted
 

Maple [F]

\[\int \frac {\left (a \,x^{2}+b^{2}\right )^{\frac {3}{2}}}{\sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}d x\]

Input:

int((a*x^2+b^2)^(3/2)/(b+(a*x^2+b^2)^(1/2))^(1/2),x)
 

Output:

int((a*x^2+b^2)^(3/2)/(b+(a*x^2+b^2)^(1/2))^(1/2),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (b^2+a x^2\right )^{3/2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\text {Timed out} \] Input:

integrate((a*x^2+b^2)^(3/2)/(b+(a*x^2+b^2)^(1/2))^(1/2),x, algorithm="fric 
as")
                                                                                    
                                                                                    
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\left (b^2+a x^2\right )^{3/2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\int \frac {\left (a x^{2} + b^{2}\right )^{\frac {3}{2}}}{\sqrt {b + \sqrt {a x^{2} + b^{2}}}}\, dx \] Input:

integrate((a*x**2+b**2)**(3/2)/(b+(a*x**2+b**2)**(1/2))**(1/2),x)
 

Output:

Integral((a*x**2 + b**2)**(3/2)/sqrt(b + sqrt(a*x**2 + b**2)), x)
 

Maxima [F]

\[ \int \frac {\left (b^2+a x^2\right )^{3/2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\int { \frac {{\left (a x^{2} + b^{2}\right )}^{\frac {3}{2}}}{\sqrt {b + \sqrt {a x^{2} + b^{2}}}} \,d x } \] Input:

integrate((a*x^2+b^2)^(3/2)/(b+(a*x^2+b^2)^(1/2))^(1/2),x, algorithm="maxi 
ma")
 

Output:

integrate((a*x^2 + b^2)^(3/2)/sqrt(b + sqrt(a*x^2 + b^2)), x)
 

Giac [F]

\[ \int \frac {\left (b^2+a x^2\right )^{3/2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\int { \frac {{\left (a x^{2} + b^{2}\right )}^{\frac {3}{2}}}{\sqrt {b + \sqrt {a x^{2} + b^{2}}}} \,d x } \] Input:

integrate((a*x^2+b^2)^(3/2)/(b+(a*x^2+b^2)^(1/2))^(1/2),x, algorithm="giac 
")
 

Output:

integrate((a*x^2 + b^2)^(3/2)/sqrt(b + sqrt(a*x^2 + b^2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (b^2+a x^2\right )^{3/2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\int \frac {{\left (b^2+a\,x^2\right )}^{3/2}}{\sqrt {b+\sqrt {b^2+a\,x^2}}} \,d x \] Input:

int((a*x^2 + b^2)^(3/2)/(b + (a*x^2 + b^2)^(1/2))^(1/2),x)
 

Output:

int((a*x^2 + b^2)^(3/2)/(b + (a*x^2 + b^2)^(1/2))^(1/2), x)
 

Reduce [F]

\[ \int \frac {\left (b^2+a x^2\right )^{3/2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\frac {-2 \sqrt {a \,x^{2}+b^{2}}\, \sqrt {\sqrt {a \,x^{2}+b^{2}}+b}\, a b \,x^{2}-14 \sqrt {a \,x^{2}+b^{2}}\, \sqrt {\sqrt {a \,x^{2}+b^{2}}+b}\, b^{3}+10 \sqrt {b}\, \sqrt {a}\, \sqrt {2}\, b^{3} i x -2 \left (\int \frac {\sqrt {\sqrt {a \,x^{2}+b^{2}}+b}}{a \,x^{4}+b^{2} x^{2}}d x \right ) b^{6} x +7 \left (\int \frac {\sqrt {\sqrt {a \,x^{2}+b^{2}}+b}}{a \,x^{2}+b^{2}}d x \right ) a \,b^{4} x +5 \left (\int \frac {\sqrt {\sqrt {a \,x^{2}+b^{2}}+b}\, x^{4}}{a \,x^{2}+b^{2}}d x \right ) a^{3} x +14 \left (\int \frac {\sqrt {\sqrt {a \,x^{2}+b^{2}}+b}\, x^{2}}{a \,x^{2}+b^{2}}d x \right ) a^{2} b^{2} x -12 \left (\int \frac {\sqrt {a \,x^{2}+b^{2}}\, \sqrt {\sqrt {a \,x^{2}+b^{2}}+b}}{a \,x^{4}+b^{2} x^{2}}d x \right ) b^{5} x}{5 a x} \] Input:

int((a*x^2+b^2)^(3/2)/(b+(a*x^2+b^2)^(1/2))^(1/2),x)
 

Output:

( - 2*sqrt(a*x**2 + b**2)*sqrt(sqrt(a*x**2 + b**2) + b)*a*b*x**2 - 14*sqrt 
(a*x**2 + b**2)*sqrt(sqrt(a*x**2 + b**2) + b)*b**3 + 10*sqrt(b)*sqrt(a)*sq 
rt(2)*b**3*i*x - 2*int(sqrt(sqrt(a*x**2 + b**2) + b)/(a*x**4 + b**2*x**2), 
x)*b**6*x + 7*int(sqrt(sqrt(a*x**2 + b**2) + b)/(a*x**2 + b**2),x)*a*b**4* 
x + 5*int((sqrt(sqrt(a*x**2 + b**2) + b)*x**4)/(a*x**2 + b**2),x)*a**3*x + 
 14*int((sqrt(sqrt(a*x**2 + b**2) + b)*x**2)/(a*x**2 + b**2),x)*a**2*b**2* 
x - 12*int((sqrt(a*x**2 + b**2)*sqrt(sqrt(a*x**2 + b**2) + b))/(a*x**4 + b 
**2*x**2),x)*b**5*x)/(5*a*x)