3.32.45 \(\int \frac {(b^2+a x^2)^2 \sqrt {b+\sqrt {b^2+a x^2}}}{(-b^2+a x^2)^2} \, dx\) [3145]

3.32.45.1 Optimal result

Integrand size = 44, antiderivative size = 1310 \[ \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 {-2 i a^{3/2} x \left (-9 b^7-46 a b^5 x^2-20 a^2 b^3 x^4+24 a^3 b x^6\right )+2 i \left (a b^8-13 a^2 b^6 x^2-75 a^3 b^4 x^4-52 a^4 b^2 x^6+16 a^5 x^8\right )+\sqrt {b^2+a x^2} \left (-2 i a^{3/2} x \left (-43 a b^4 x^2-60 a^2 b^2 x^4+16 a^3 x^6\right )+2 i \left (-a b^7-27 a^2 b^5 x^2-32 a^3 b^3 x^4+24 a^4 b x^6\right )\right )}{\frac {3 i a^2 x \left (-b^2+a x^2\right ) \left (5 b^4+20 a b^2 x^2+16 a^2 x^4\right )}{\sqrt {b+\sqrt {b^2+a x^2}}}+3 i a^{3/2} \left (-b^2+a x^2\right ) \left (-5 b^4-20 a b^2 x^2-16 a^2 x^4\right ) \sqrt {b+\sqrt {b^2+a x^2}}+\sqrt {b^2+a x^2} \left (\frac {3 i a^2 x \left (-b^2+a x^2\right ) \left (4 b^3+8 a b x^2\right )}{\sqrt {b+\sqrt {b^2+a x^2}}}+3 i a^{3/2} \left (-b^2+a x^2\right ) \left (4 b^3+8 a b x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}\right )}-\frac {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 {-1+\sqrt {2}} \sqrt {a}}+\frac {5 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 {2 \left (-1+\sqrt {2}\right )} \sqrt {a}}+\frac {3 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 {1+\sqrt {2}} \sqrt {a}}+\frac {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 {2 \left (1+\sqrt {2}\right )} \sqrt {a}}+\frac {3 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 {-1+\sqrt {2}} \sqrt {a}}-\frac {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 {2 \left (-1+\sqrt {2}\right )} \sqrt {a}}-\frac {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 {1+\sqrt {2}} \sqrt {a}}-\frac {5 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 {2 \left (1+\sqrt {2}\right )} \sqrt {a}} \]

output

(-2*I*a^(3/2)*x*(24*a^3*b*x^6-20*a^2*b^3*x^4-46*a*b^5*x^2-9*b^7)+2*I*(16*a 
^5*x^8-52*a^4*b^2*x^6-75*a^3*b^4*x^4-13*a^2*b^6*x^2+a*b^8)+(a*x^2+b^2)^(1/ 
2)*(-2*I*a^(3/2)*x*(16*a^3*x^6-60*a^2*b^2*x^4-43*a*b^4*x^2)+2*I*(24*a^4*b* 
x^6-32*a^3*b^3*x^4-27*a^2*b^5*x^2-a*b^7)))/(3*I*a^2*x*(a*x^2-b^2)*(16*a^2* 
x^4+20*a*b^2*x^2+5*b^4)/(b+(a*x^2+b^2)^(1/2))^(1/2)+3*I*a^(3/2)*(a*x^2-b^2 
)*(-16*a^2*x^4-20*a*b^2*x^2-5*b^4)*(b+(a*x^2+b^2)^(1/2))^(1/2)+(a*x^2+b^2) 
^(1/2)*(3*I*a^2*x*(a*x^2-b^2)*(8*a*b*x^2+4*b^3)/(b+(a*x^2+b^2)^(1/2))^(1/2 
)+3*I*a^(3/2)*(a*x^2-b^2)*(8*a*b*x^2+4*b^3)*(b+(a*x^2+b^2)^(1/2))^(1/2)))- 
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))/(2^(1/ 
2)-1)^(1/2)/a^(1/2)+5*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))/(-2+2*2^(1/2))^(1/2)/a^(1/2)+3*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))/(1+2^(1/2))^(1/2)/a^(1/2)+b^(3/2)*arcta 
n(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))/(2+2*2^(1/2))^(1/2)/a^( 
1/2)+3*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))/ 
(2^(1/2)-1)^(1/2)/a^(1/2)-b^(3/2)*arctanh(a^(1/2)*x/(-2+2*2^(1/2))^(1/2...
 

3.32.45.2 Mathematica [A] (verified)

Time = 1.11 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.18 \[ \int \frac {\left (b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}}{\left (-b^2+a x^2\right )^2} \, dx=\frac {1}{6} \left (\frac {4 x \left (5 b^3-2 a b x^2+4 b^2 \sqrt {b^2+a x^2}-a x^2 \sqrt {b^2+a x^2}\right )}{\left (b^2-a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {3 \left (6+\sqrt {2}\right ) b^{3/2} \arctan \left (\frac {\sqrt {-1+\sqrt {2}} \sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {1+\sqrt {2}} \sqrt {a}}+\frac {3 \left (-6+\sqrt {2}\right ) \sqrt {1+\sqrt {2}} b^{3/2} \text {arctanh}\left (\frac {\sqrt {1+\sqrt {2}} \sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}}\right ) \]

input

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

output

((4*x*(5*b^3 - 2*a*b*x^2 + 4*b^2*Sqrt[b^2 + a*x^2] - a*x^2*Sqrt[b^2 + a*x^ 
2]))/((b^2 - a*x^2)*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (3*(6 + Sqrt[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]) + (3*(-6 + Sqrt[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])/6
 

3.32.45.3 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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 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
 

3.32.45.3.1 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] 
]
 

3.32.45.4 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)
 

3.32.45.5 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
 

3.32.45.6 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)
 

3.32.45.7 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)
 

3.32.45.8 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)
 

3.32.45.9 Mupad [F(-1)]

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

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)