\(\int \frac {\sqrt {-b+a^2 x^2}}{\sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx\) [3079]

Optimal result

Integrand size = 68, antiderivative size = 507 \[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {\left (945 b^3-4224 b^2 c^4-504 a b^2 c^2 x+3072 a b c^6 x-1890 a^2 b^2 x^2+7680 a^2 b c^4 x^2-4096 a^3 c^6 x^3\right ) \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\left (432 b^2 c^3-2048 b c^7+630 a b^2 c x-2304 a b c^5 x+4096 a^2 c^7 x^2+3072 a^3 c^5 x^3\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\sqrt {-b+a^2 x^2} \left (\left (-504 b^2 c^2+1024 b c^6-1890 a b^2 x+7680 a b c^4 x-4096 a^2 c^6 x^2\right ) \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\left (630 b^2 c-768 b c^5+4096 a c^7 x+3072 a^2 c^5 x^2\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{3840 a c^5 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/2}}+\frac {63 b^2 \text {arctanh}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {c}}\right )}{256 a c^{11/2}}-\frac {b \text {arctanh}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {c}}\right )}{a c^{3/2}} \] Output:

1/3840*((-4096*a^3*c^6*x^3+7680*a^2*b*c^4*x^2+3072*a*b*c^6*x-1890*a^2*b^2* 
x^2-504*a*b^2*c^2*x-4224*b^2*c^4+945*b^3)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/2) 
)^(1/2)+(3072*a^3*c^5*x^3+4096*a^2*c^7*x^2-2304*a*b*c^5*x-2048*b*c^7+630*a 
*b^2*c*x+432*b^2*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^2*x^2-b)^(1/2)*((-4096*a^2*c^6*x^2+7680*a*b*c^4*x+102 
4*b*c^6-1890*a*b^2*x-504*b^2*c^2)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/2))^(1/2)+ 
(3072*a^2*c^5*x^2+4096*a*c^7*x-768*b*c^5+630*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/c^5/(a*x+(a^2*x^2-b)^( 
1/2))^(5/2)+63/256*b^2*arctanh((c+(a*x+(a^2*x^2-b)^(1/2))^(1/2))^(1/2)/c^( 
1/2))/a/c^(11/2)-b*arctanh((c+(a*x+(a^2*x^2-b)^(1/2))^(1/2))^(1/2)/c^(1/2) 
)/a/c^(3/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 1.62 (sec) , antiderivative size = 507, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {\left (945 b^3-4224 b^2 c^4-504 a b^2 c^2 x+3072 a b c^6 x-1890 a^2 b^2 x^2+7680 a^2 b c^4 x^2-4096 a^3 c^6 x^3\right ) \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\left (432 b^2 c^3-2048 b c^7+630 a b^2 c x-2304 a b c^5 x+4096 a^2 c^7 x^2+3072 a^3 c^5 x^3\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\sqrt {-b+a^2 x^2} \left (\left (-504 b^2 c^2+1024 b c^6-1890 a b^2 x+7680 a b c^4 x-4096 a^2 c^6 x^2\right ) \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\left (630 b^2 c-768 b c^5+4096 a c^7 x+3072 a^2 c^5 x^2\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{3840 a c^5 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/2}}+\frac {63 b^2 \text {arctanh}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {c}}\right )}{256 a c^{11/2}}-\frac {b \text {arctanh}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {c}}\right )}{a c^{3/2}} \] Input:

Integrate[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:

((945*b^3 - 4224*b^2*c^4 - 504*a*b^2*c^2*x + 3072*a*b*c^6*x - 1890*a^2*b^2 
*x^2 + 7680*a^2*b*c^4*x^2 - 4096*a^3*c^6*x^3)*Sqrt[c + Sqrt[a*x + Sqrt[-b 
+ a^2*x^2]]] + (432*b^2*c^3 - 2048*b*c^7 + 630*a*b^2*c*x - 2304*a*b*c^5*x 
+ 4096*a^2*c^7*x^2 + 3072*a^3*c^5*x^3)*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]*((-504*b^2*c^2 + 
 1024*b*c^6 - 1890*a*b^2*x + 7680*a*b*c^4*x - 4096*a^2*c^6*x^2)*Sqrt[c + S 
qrt[a*x + Sqrt[-b + a^2*x^2]]] + (630*b^2*c - 768*b*c^5 + 4096*a*c^7*x + 3 
072*a^2*c^5*x^2)*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[- 
b + a^2*x^2]]]))/(3840*a*c^5*(a*x + Sqrt[-b + a^2*x^2])^(5/2)) + (63*b^2*A 
rcTanh[Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]/Sqrt[c]])/(256*a*c^(11/2)) 
 - (b*ArcTanh[Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]/Sqrt[c]])/(a*c^(3/2 
))
 

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
 

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)
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 776, normalized size of antiderivative = 1.53 \[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx =\text {Too large to display} \] 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:

[1/7680*(15*(256*b^2*c^4 - 63*b^3)*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*(2048*b*c^8 + 864*a^2*b*c^4*x^2 - 432*b^2*c^4 + 6*(128*a*b*c^ 
6 + 105*a*b^2*c^2)*x + 6*(128*b*c^6 - 144*a*b*c^4*x - 105*b^2*c^2)*sqrt(a^ 
2*x^2 - b) - (1536*a^3*c^5*x^3 + 1024*b*c^7 + 1008*a^2*b*c^3*x^2 - 504*b^2 
*c^3 - 3*(1664*a*b*c^5 - 315*a*b^2*c)*x - 3*(512*a^2*c^5*x^2 - 1408*b*c^5 
+ 336*a*b*c^3*x + 315*b^2*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*b*c^6), 1/3840*(15*(256*b 
^2*c^4 - 63*b^3)*sqrt(-c)*arctan((sqrt(a^2*x^2 - b)*sqrt(-c)*c - sqrt(a*x 
+ sqrt(a^2*x^2 - b))*((c^2 - a*x)*sqrt(-c) + sqrt(a^2*x^2 - b)*sqrt(-c)) + 
 (c^3 - a*c*x)*sqrt(-c))*sqrt(c + sqrt(a*x + sqrt(a^2*x^2 - b)))/(c^4 - 2* 
a*c^2*x + b)) + (2048*b*c^8 + 864*a^2*b*c^4*x^2 - 432*b^2*c^4 + 6*(128*a*b 
*c^6 + 105*a*b^2*c^2)*x + 6*(128*b*c^6 - 144*a*b*c^4*x - 105*b^2*c^2)*sqrt 
(a^2*x^2 - b) - (1536*a^3*c^5*x^3 + 1024*b*c^7 + 1008*a^2*b*c^3*x^2 - 504* 
b^2*c^3 - 3*(1664*a*b*c^5 - 315*a*b^2*c)*x - 3*(512*a^2*c^5*x^2 - 1408*b*c 
^5 + 336*a*b*c^3*x + 315*b^2*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*b*c^6)]
 

Sympy [F]

\[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int \frac {\sqrt {a^{2} x^{2} - b}}{\sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} - b}}} \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}\, dx \] 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**2*x**2 - b)/(sqrt(c + sqrt(a*x + sqrt(a**2*x**2 - b)))*sq 
rt(a*x + sqrt(a**2*x**2 - b))), x)
 

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)
 

Giac [F(-1)]

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

Timed out
 

Mupad [F(-1)]

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

Reduce [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:

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)