3.32.2 \(\int \frac {(b^2+a x^2)^2}{(-b^2+a x^2)^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx\) [3102]

3.32.2.1 Optimal result

Integrand size = 44, antiderivative size = 569 \[ \int \frac {\left (b^2+a x^2\right )^2}{\left (-b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\frac {2 x \left (-2 b^2+a x^2\right )}{\left (-b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {2 \sqrt {2} \sqrt {b} \arctan \left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2} \sqrt {b}}\right )}{\sqrt {a}}-\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \sqrt {b} \arctan \left (\frac {\sqrt {a} x}{\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {b}}\right )}{\sqrt {a}}-\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \sqrt {b} \arctan \left (\frac {\sqrt {a} x}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {b}}\right )}{\sqrt {a}}-\frac {\sqrt {-1+\sqrt {2}} \left (2+\sqrt {2}\right ) \sqrt {b} \text {arctanh}\left (\frac {-\frac {a x}{\sqrt {b+\sqrt {b^2+a x^2}}}+\sqrt {a} \sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {a} \sqrt {b}}\right )}{2 \sqrt {a}}+\frac {\sqrt {-1+\sqrt {2}} \left (2+\sqrt {2}\right ) \sqrt {b} \text {arctanh}\left (\frac {-\frac {a x}{\sqrt {b+\sqrt {b^2+a x^2}}}+\sqrt {a} \sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {a} \sqrt {b}}\right )}{2 \sqrt {a}} \]

output

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

3.32.2.2 Mathematica [A] (verified)

Time = 1.04 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.42 \[ \int \frac {\left (b^2+a x^2\right )^2}{\left (-b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx=-\frac {2 x \left (-2 b^2+a x^2\right )}{\left (b^2-a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {2} \sqrt {b} \arctan \left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}}-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {-1+\sqrt {2}} \sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {a}}-\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \sqrt {b} \text {arctanh}\left (\frac {\sqrt {1+\sqrt {2}} \sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}} \]

input

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

output

(-2*x*(-2*b^2 + a*x^2))/((b^2 - a*x^2)*Sqrt[b + Sqrt[b^2 + a*x^2]]) - (Sqr 
t[2]*Sqrt[b]*ArcTan[(Sqrt[a]*x)/(Sqrt[2]*Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2 
]])])/Sqrt[a] - (Sqrt[b]*ArcTan[(Sqrt[-1 + Sqrt[2]]*Sqrt[a]*x)/(Sqrt[b]*Sq 
rt[b + Sqrt[b^2 + a*x^2]])])/(Sqrt[2*(1 + Sqrt[2])]*Sqrt[a]) - (Sqrt[(1 + 
Sqrt[2])/2]*Sqrt[b]*ArcTanh[(Sqrt[1 + Sqrt[2]]*Sqrt[a]*x)/(Sqrt[b]*Sqrt[b 
+ Sqrt[b^2 + a*x^2]])])/Sqrt[a]
 

3.32.2.3 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.

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {4 b^2}{\left (b^2-a x^2\right ) \sqrt {\sqrt {a x^2+b^2}+b}}+\frac {1}{\sqrt {\sqrt {a x^2+b^2}+b}}+\frac {4 b^4}{\left (b^2-a x^2\right )^2 \sqrt {\sqrt {a x^2+b^2}+b}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle a b^2 \int \frac {1}{\left (\sqrt {a} b-a x\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}}dx+a b^2 \int \frac {1}{\left (\sqrt {a} b+a x\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}}dx-b \int \frac {1}{\left (b-\sqrt {a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}}dx-b \int \frac {1}{\left (b+\sqrt {a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}}dx+\int \frac {1}{\sqrt {b+\sqrt {b^2+a x^2}}}dx\)

input

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

output

$Aborted
 

3.32.2.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.32.2.4 Maple [F]

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

input

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

output

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

3.32.2.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.32.2.6 Sympy [F]

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

input

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

output

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

3.32.2.7 Maxima [F]

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

input

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

output

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

3.32.2.8 Giac [F]

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

input

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

output

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

3.32.2.9 Mupad [F(-1)]

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

input

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

output

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