3.11.25 \(\int \frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\sqrt {b+a^2 x^2}} \, dx\) [1025]

3.11.25.1 Optimal result

Integrand size = 41, antiderivative size = 77 \[ \int \frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\sqrt {b+a^2 x^2}} \, dx=\frac {4 \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{a}-\frac {4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\sqrt {c}}\right )}{a} \]

output

4*(c+(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2)/a-4*c^(1/2)*arctanh((c+(a*x+(a^2 
*x^2+b)^(1/2))^(1/2))^(1/2)/c^(1/2))/a
 

3.11.25.2 Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\sqrt {b+a^2 x^2}} \, dx=\frac {4 \left (\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}-\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\sqrt {c}}\right )\right )}{a} \]

input

Integrate[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/Sqrt[b + a^2*x^2],x]
 

output

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

3.11.25.3 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.

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

\(\Big \downarrow \) 7299

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

input

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

output

$Aborted
 

3.11.25.3.1 Defintions of rubi rules used

rule 7299

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

3.11.25.4 Maple [A] (verified)

Time = 1.19 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.79

method	result	size
derivativedivides	\(\frac {4 \sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}+b}}}-4 \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}+b}}}}{\sqrt {c}}\right )}{a}\)	\(61\)
default	\(\frac {4 \sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}+b}}}-4 \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}+b}}}}{\sqrt {c}}\right )}{a}\)	\(61\)

input

int((c+(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2)/(a^2*x^2+b)^(1/2),x,method=_RE 
TURNVERBOSE)
 

output

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

3.11.25.5 Fricas [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 202, normalized size of antiderivative = 2.62 \[ \int \frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\sqrt {b+a^2 x^2}} \, dx=\left [\frac {2 \, {\left (\sqrt {c} \log \left (2 \, {\left (a \sqrt {c} x - \sqrt {a^{2} x^{2} + b} \sqrt {c}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}} - 2 \, {\left (a c x - \sqrt {a^{2} x^{2} + b} c\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}} + b\right ) + 2 \, \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}\right )}}{a}, \frac {4 \, {\left (\sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{c}\right ) + \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}\right )}}{a}\right ] \]

input

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

output

[2*(sqrt(c)*log(2*(a*sqrt(c)*x - sqrt(a^2*x^2 + b)*sqrt(c))*sqrt(a*x + sqr 
t(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*sqrt(c + sqrt(a*x + 
 sqrt(a^2*x^2 + b))))/a, 4*(sqrt(-c)*arctan(sqrt(-c)*sqrt(c + sqrt(a*x + s 
qrt(a^2*x^2 + b)))/c) + sqrt(c + sqrt(a*x + sqrt(a^2*x^2 + b))))/a]
 

3.11.25.6 Sympy [F]

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

input

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

output

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

3.11.25.7 Maxima [F]

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

input

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

output

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

3.11.25.8 Giac [F(-2)]

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

input

integrate((c+(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2)/(a^2*x^2+b)^(1/2),x, alg 
orithm="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 &
 

3.11.25.9 Mupad [F(-1)]

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

input

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

output

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