\(\int \sqrt {b+a^2 x^2} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx\) [3083]

Optimal result

Integrand size = 41, antiderivative size = 515 \[ \int \sqrt {b+a^2 x^2} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx=\frac {\left (80080 b^2 c^3-2048 b c^7+1050 a b^2 c x-2304 a b c^5 x+197120 a^2 b c^3 x^2-4096 a^2 c^7 x^2-3072 a^3 c^5 x^3+35840 a^4 c^3 x^4\right ) \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}+\left (-840 b^2 c^2+1024 b c^6-1575 a b^2 x+1920 a b c^4 x+2048 a^2 c^6 x^2+2560 a^3 c^4 x^3\right ) \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}+\sqrt {b+a^2 x^2} \left (\left (1050 b^2 c-768 b c^5+179200 a b c^3 x-4096 a c^7 x-3072 a^2 c^5 x^2+35840 a^3 c^3 x^3\right ) \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}+\left (-1575 b^2+640 b c^4+2048 a c^6 x+2560 a^2 c^4 x^2\right ) \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}\right )}{80640 a^2 c^3 x \sqrt {b+a^2 x^2}+40320 a c^3 \left (b+2 a^2 x^2\right )}+\frac {5 b^2 \text {arctanh}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\sqrt {c}}\right )}{128 a c^{7/2}}-\frac {2 b \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\sqrt {c}}\right )}{a} \]

[Out]

((35840*a^4*c^3*x^4-3072*a^3*c^5*x^3-4096*a^2*c^7*x^2+197120*a^2*b*c^3*x^2-2304*a*b*c^5*x-2048*b*c^7+1050*a*b^
2*c*x+80080*b^2*c^3)*(c+(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2)+(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)*(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)*((35840*a^3*c^3*x^3-3072*a^2*c^5*x^2-4096*a*c^7*x+179200*a*b*c^3*x-768*b*c^5+1050*b^2*c)*(c+(a
*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2)+(2560*a^2*c^4*x^2+2048*a*c^6*x+640*b*c^4-1575*b^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)))/(80640*a^2*c^3*x*(a^2*x^2+b)^(1/2)+40320*a*c^3*(2*a^2*x^2+b))+
5/128*b^2*arctanh((c+(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2)/c^(1/2))/a/c^(7/2)-2*b*c^(1/2)*arctanh((c+(a*x+(a^2*
x^2+b)^(1/2))^(1/2))^(1/2)/c^(1/2))/a

Rubi [F]

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

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \sqrt {b+a^2 x^2} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.15 (sec) , antiderivative size = 455, normalized size of antiderivative = 0.88 \[ \int \sqrt {b+a^2 x^2} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx=\frac {\frac {\sqrt {c} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \left (512 a c^3 x \left (a x+\sqrt {b+a^2 x^2}\right ) \left (-8 c^4-6 a c^2 x+70 a^2 x^2+4 c^3 \sqrt {a x+\sqrt {b+a^2 x^2}}+5 a c x \sqrt {a x+\sqrt {b+a^2 x^2}}\right )-35 b^2 \left (-2288 c^3+24 c^2 \sqrt {a x+\sqrt {b+a^2 x^2}}-30 c \left (a x+\sqrt {b+a^2 x^2}\right )+45 \left (a x+\sqrt {b+a^2 x^2}\right )^{3/2}\right )+128 b c^3 \left (-16 c^4+8 c^3 \sqrt {a x+\sqrt {b+a^2 x^2}}-6 c^2 \left (3 a x+\sqrt {b+a^2 x^2}\right )+5 c \sqrt {a x+\sqrt {b+a^2 x^2}} \left (3 a x+\sqrt {b+a^2 x^2}\right )+140 a x \left (11 a x+10 \sqrt {b+a^2 x^2}\right )\right )\right )}{b+2 a x \left (a x+\sqrt {b+a^2 x^2}\right )}+1575 b^2 \text {arctanh}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\sqrt {c}}\right )-80640 b c^4 \text {arctanh}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\sqrt {c}}\right )}{40320 a c^{7/2}} \]

[In]

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

[Out]

((Sqrt[c]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]*(512*a*c^3*x*(a*x + Sqrt[b + a^2*x^2])*(-8*c^4 - 6*a*c^2*x +
70*a^2*x^2 + 4*c^3*Sqrt[a*x + Sqrt[b + a^2*x^2]] + 5*a*c*x*Sqrt[a*x + Sqrt[b + a^2*x^2]]) - 35*b^2*(-2288*c^3
+ 24*c^2*Sqrt[a*x + Sqrt[b + a^2*x^2]] - 30*c*(a*x + Sqrt[b + a^2*x^2]) + 45*(a*x + Sqrt[b + a^2*x^2])^(3/2))
+ 128*b*c^3*(-16*c^4 + 8*c^3*Sqrt[a*x + Sqrt[b + a^2*x^2]] - 6*c^2*(3*a*x + Sqrt[b + a^2*x^2]) + 5*c*Sqrt[a*x
+ Sqrt[b + a^2*x^2]]*(3*a*x + Sqrt[b + a^2*x^2]) + 140*a*x*(11*a*x + 10*Sqrt[b + a^2*x^2]))))/(b + 2*a*x*(a*x
+ Sqrt[b + a^2*x^2])) + 1575*b^2*ArcTanh[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/Sqrt[c]] - 80640*b*c^4*ArcTa
nh[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/Sqrt[c]])/(40320*a*c^(7/2))

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 554, normalized size of antiderivative = 1.08 \[ \int \sqrt {b+a^2 x^2} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx=\left [\frac {315 \, {\left (256 \, b c^{4} - 5 \, b^{2}\right )} \sqrt {c} \log \left (2 \, {\left (a \sqrt {c} x - \sqrt {a^{2} x^{2} + b} \sqrt {c}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}} - 2 \, {\left (a c x - \sqrt {a^{2} x^{2} + b} c\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}} + b\right ) - 2 \, {\left (2048 \, c^{8} + 1120 \, a^{2} c^{4} x^{2} - 80080 \, b c^{4} + 6 \, {\left (128 \, a c^{6} + 175 \, a b c^{2}\right )} x + 2 \, {\left (384 \, c^{6} - 9520 \, a c^{4} x - 525 \, b c^{2}\right )} \sqrt {a^{2} x^{2} + b} - {\left (1024 \, c^{7} - 1680 \, a^{2} c^{3} x^{2} - 840 \, b c^{3} + 5 \, {\left (128 \, a c^{5} + 315 \, a b c\right )} x + 5 \, {\left (128 \, c^{5} + 336 \, a c^{3} x - 315 \, b c\right )} \sqrt {a^{2} x^{2} + b}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}}\right )} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{80640 \, a c^{4}}, \frac {315 \, {\left (256 \, b c^{4} - 5 \, b^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{c}\right ) - {\left (2048 \, c^{8} + 1120 \, a^{2} c^{4} x^{2} - 80080 \, b c^{4} + 6 \, {\left (128 \, a c^{6} + 175 \, a b c^{2}\right )} x + 2 \, {\left (384 \, c^{6} - 9520 \, a c^{4} x - 525 \, b c^{2}\right )} \sqrt {a^{2} x^{2} + b} - {\left (1024 \, c^{7} - 1680 \, a^{2} c^{3} x^{2} - 840 \, b c^{3} + 5 \, {\left (128 \, a c^{5} + 315 \, a b c\right )} x + 5 \, {\left (128 \, c^{5} + 336 \, a c^{3} x - 315 \, b c\right )} \sqrt {a^{2} x^{2} + b}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}}\right )} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{40320 \, a c^{4}}\right ] \]

[In]

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

[Out]

[1/80640*(315*(256*b*c^4 - 5*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*(2048*c^8 + 1120*a^2*c^4*x^2 - 80080*b*c^4 + 6*(128*a*c^6 + 175*a*b*c^2)*x + 2*(384*c^6 - 9520*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 + sq
rt(a^2*x^2 + b))))/(a*c^4), 1/40320*(315*(256*b*c^4 - 5*b^2)*sqrt(-c)*arctan(sqrt(-c)*sqrt(c + sqrt(a*x + sqrt
(a^2*x^2 + b)))/c) - (2048*c^8 + 1120*a^2*c^4*x^2 - 80080*b*c^4 + 6*(128*a*c^6 + 175*a*b*c^2)*x + 2*(384*c^6 -
9520*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)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(c
onst gen &

Mupad [F(-1)]

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

[In]

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

[Out]

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