3.31.27 \(\int \frac {\sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx\) [3027]

3.31.27.1 Optimal result

Integrand size = 68, antiderivative size = 431 \[ \int \frac {\sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {\left (-840 b^2 c^2+1024 b c^6-1575 a b^2 x+1920 a b c^4 x-2048 a^2 c^6 x^2-2560 a^3 c^4 x^3\right ) \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\left (1050 b^2 c-768 b c^5-11760 a b c^3 x+2048 a c^7 x+1536 a^2 c^5 x^2+2240 a^3 c^3 x^3\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\sqrt {-b+a^2 x^2} \left (\left (-1575 b^2+640 b c^4-2048 a c^6 x-2560 a^2 c^4 x^2\right ) \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\left (-10640 b c^3+2048 c^7+1536 a c^5 x+2240 a^2 c^3 x^2\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{5040 a c^3 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/2}}+\frac {5 b^2 \text {arctanh}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {c}}\right )}{16 a c^{7/2}} \]

output

1/5040*((-2560*a^3*c^4*x^3-2048*a^2*c^6*x^2+1920*a*b*c^4*x+1024*b*c^6-1575 
*a*b^2*x-840*b^2*c^2)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/2))^(1/2)+(2240*a^3*c^ 
3*x^3+1536*a^2*c^5*x^2+2048*a*c^7*x-11760*a*b*c^3*x-768*b*c^5+1050*b^2*c)* 
(a*x+(a^2*x^2-b)^(1/2))^(1/2)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/2))^(1/2)+(a^2 
*x^2-b)^(1/2)*((-2560*a^2*c^4*x^2-2048*a*c^6*x+640*b*c^4-1575*b^2)*(c+(a*x 
+(a^2*x^2-b)^(1/2))^(1/2))^(1/2)+(2240*a^2*c^3*x^2+1536*a*c^5*x+2048*c^7-1 
0640*b*c^3)*(a*x+(a^2*x^2-b)^(1/2))^(1/2)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/2) 
)^(1/2)))/a/c^3/(a*x+(a^2*x^2-b)^(1/2))^(3/2)+5/16*b^2*arctanh((c+(a*x+(a^ 
2*x^2-b)^(1/2))^(1/2))^(1/2)/c^(1/2))/a/c^(7/2)
 

3.31.27.2 Mathematica [A] (verified)

Time = 1.21 (sec) , antiderivative size = 407, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {\frac {\sqrt {c} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}} \left (-105 b^2 \left (8 c^2-10 c \sqrt {a x+\sqrt {-b+a^2 x^2}}+15 \left (a x+\sqrt {-b+a^2 x^2}\right )\right )-16 b c^3 \left (-64 c^3+48 c^2 \sqrt {a x+\sqrt {-b+a^2 x^2}}-40 c \left (3 a x+\sqrt {-b+a^2 x^2}\right )+35 \sqrt {a x+\sqrt {-b+a^2 x^2}} \left (21 a x+19 \sqrt {-b+a^2 x^2}\right )\right )+64 c^3 \left (a x+\sqrt {-b+a^2 x^2}\right ) \left (32 c^4 \sqrt {a x+\sqrt {-b+a^2 x^2}}+8 a c^2 x \left (-4 c+3 \sqrt {a x+\sqrt {-b+a^2 x^2}}\right )+5 a^2 x^2 \left (-8 c+7 \sqrt {a x+\sqrt {-b+a^2 x^2}}\right )\right )\right )}{\left (a x+\sqrt {-b+a^2 x^2}\right )^{3/2}}+1575 b^2 \text {arctanh}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {c}}\right )}{5040 a c^{7/2}} \]

input

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

output

((Sqrt[c]*Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]*(-105*b^2*(8*c^2 - 10*c 
*Sqrt[a*x + Sqrt[-b + a^2*x^2]] + 15*(a*x + Sqrt[-b + a^2*x^2])) - 16*b*c^ 
3*(-64*c^3 + 48*c^2*Sqrt[a*x + Sqrt[-b + a^2*x^2]] - 40*c*(3*a*x + Sqrt[-b 
 + a^2*x^2]) + 35*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*(21*a*x + 19*Sqrt[-b + a^ 
2*x^2])) + 64*c^3*(a*x + Sqrt[-b + a^2*x^2])*(32*c^4*Sqrt[a*x + Sqrt[-b + 
a^2*x^2]] + 8*a*c^2*x*(-4*c + 3*Sqrt[a*x + Sqrt[-b + a^2*x^2]]) + 5*a^2*x^ 
2*(-8*c + 7*Sqrt[a*x + Sqrt[-b + a^2*x^2]]))))/(a*x + Sqrt[-b + a^2*x^2])^ 
(3/2) + 1575*b^2*ArcTanh[Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]/Sqrt[c]] 
)/(5040*a*c^(7/2))
 

3.31.27.3 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.

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

\(\Big \downarrow \) 7299

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

input

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

output

$Aborted
 

3.31.27.3.1 Defintions of rubi rules used

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

3.31.27.4 Maple [F]

\[\int \frac {\sqrt {a^{2} x^{2}-b}\, \sqrt {a x +\sqrt {a^{2} x^{2}-b}}}{\sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}-b}}}}d x\]

input

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

output

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

3.31.27.5 Fricas [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 564, normalized size of antiderivative = 1.31 \[ \int \frac {\sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\left [\frac {1575 \, b^{2} \sqrt {c} \log \left (2 \, {\left (a \sqrt {c} x - \sqrt {a^{2} x^{2} - b} \sqrt {c}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} - b}}} + 2 \, {\left (a c x - \sqrt {a^{2} x^{2} - b} c\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}} + b\right ) + 2 \, {\left (2048 \, c^{8} + 1120 \, a^{2} c^{4} x^{2} - 10640 \, b c^{4} + 6 \, {\left (128 \, a c^{6} + 175 \, a b c^{2}\right )} x + 2 \, {\left (384 \, c^{6} + 560 \, a c^{4} x - 525 \, b c^{2}\right )} \sqrt {a^{2} x^{2} - b} - {\left (1024 \, c^{7} + 1680 \, a^{2} c^{3} x^{2} - 840 \, b c^{3} + 5 \, {\left (128 \, a c^{5} + 315 \, a b c\right )} x + 5 \, {\left (128 \, c^{5} - 336 \, a c^{3} x - 315 \, b c\right )} \sqrt {a^{2} x^{2} - b}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}}\right )} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}}{10080 \, a c^{4}}, -\frac {1575 \, b^{2} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}}{c}\right ) - {\left (2048 \, c^{8} + 1120 \, a^{2} c^{4} x^{2} - 10640 \, b c^{4} + 6 \, {\left (128 \, a c^{6} + 175 \, a b c^{2}\right )} x + 2 \, {\left (384 \, c^{6} + 560 \, a c^{4} x - 525 \, b c^{2}\right )} \sqrt {a^{2} x^{2} - b} - {\left (1024 \, c^{7} + 1680 \, a^{2} c^{3} x^{2} - 840 \, b c^{3} + 5 \, {\left (128 \, a c^{5} + 315 \, a b c\right )} x + 5 \, {\left (128 \, c^{5} - 336 \, a c^{3} x - 315 \, b c\right )} \sqrt {a^{2} x^{2} - b}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}}\right )} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}}{5040 \, a c^{4}}\right ] \]

input

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

output

[1/10080*(1575*b^2*sqrt(c)*log(2*(a*sqrt(c)*x - sqrt(a^2*x^2 - b)*sqrt(c)) 
*sqrt(a*x + sqrt(a^2*x^2 - b))*sqrt(c + sqrt(a*x + sqrt(a^2*x^2 - b))) + 2 
*(a*c*x - sqrt(a^2*x^2 - b)*c)*sqrt(a*x + sqrt(a^2*x^2 - b)) + b) + 2*(204 
8*c^8 + 1120*a^2*c^4*x^2 - 10640*b*c^4 + 6*(128*a*c^6 + 175*a*b*c^2)*x + 2 
*(384*c^6 + 560*a*c^4*x - 525*b*c^2)*sqrt(a^2*x^2 - b) - (1024*c^7 + 1680* 
a^2*c^3*x^2 - 840*b*c^3 + 5*(128*a*c^5 + 315*a*b*c)*x + 5*(128*c^5 - 336*a 
*c^3*x - 315*b*c)*sqrt(a^2*x^2 - b))*sqrt(a*x + sqrt(a^2*x^2 - b)))*sqrt(c 
 + sqrt(a*x + sqrt(a^2*x^2 - b))))/(a*c^4), -1/5040*(1575*b^2*sqrt(-c)*arc 
tan(sqrt(-c)*sqrt(c + sqrt(a*x + sqrt(a^2*x^2 - b)))/c) - (2048*c^8 + 1120 
*a^2*c^4*x^2 - 10640*b*c^4 + 6*(128*a*c^6 + 175*a*b*c^2)*x + 2*(384*c^6 + 
560*a*c^4*x - 525*b*c^2)*sqrt(a^2*x^2 - b) - (1024*c^7 + 1680*a^2*c^3*x^2 
- 840*b*c^3 + 5*(128*a*c^5 + 315*a*b*c)*x + 5*(128*c^5 - 336*a*c^3*x - 315 
*b*c)*sqrt(a^2*x^2 - b))*sqrt(a*x + sqrt(a^2*x^2 - b)))*sqrt(c + sqrt(a*x 
+ sqrt(a^2*x^2 - b))))/(a*c^4)]
 

3.31.27.6 Sympy [F]

\[ \int \frac {\sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int \frac {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} \sqrt {a^{2} x^{2} - b}}{\sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}}\, dx \]

input

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

output

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

3.31.27.7 Maxima [F]

\[ \int \frac {\sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int { \frac {\sqrt {a^{2} x^{2} - b} \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}{\sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}} \,d x } \]

input

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

output

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

3.31.27.8 Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\text {Exception raised: TypeError} \]

input

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

output

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(con 
st gen &
 

3.31.27.9 Mupad [F(-1)]

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

input

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

output

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