\(\int \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx\) [2850]

Optimal result

Integrand size = 27, antiderivative size = 293 \[ \int \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx=\frac {\left (-6 b c-16 a c^3 x+48 a^2 c x^2\right ) \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}+\left (-15 b+8 a c^2 x\right ) \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}+\sqrt {b+a^2 x^2} \left (\left (-16 c^3+48 a c x\right ) \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}+8 c^2 \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}\right )}{60 a^2 c x+60 a c \sqrt {b+a^2 x^2}}+\frac {b \text {arctanh}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\sqrt {c}}\right )}{4 a c^{3/2}} \]

[Out]

((48*a^2*c*x^2-16*a*c^3*x-6*b*c)*(c+(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2)+(8*a*c^2*x-15*b)*(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)*((48*a*c*x-16*c^3)*(c+(a*x+(a^2*x^2+b)^(1/
2))^(1/2))^(1/2)+8*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)))/(60*a^2*c*x+60*
a*c*(a^2*x^2+b)^(1/2))+1/4*b*arctanh((c+(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2)/c^(1/2))/a/c^(3/2)

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.63 \[ \int \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx=\frac {\frac {\sqrt {c} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \left (-3 b \left (2 c+5 \sqrt {a x+\sqrt {b+a^2 x^2}}\right )+8 c \left (a x+\sqrt {b+a^2 x^2}\right ) \left (-2 c^2+6 a x+c \sqrt {a x+\sqrt {b+a^2 x^2}}\right )\right )}{a x+\sqrt {b+a^2 x^2}}+15 b \text {arctanh}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\sqrt {c}}\right )}{60 a c^{3/2}} \]

[In]

Integrate[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]]]*(-3*b*(2*c + 5*Sqrt[a*x + Sqrt[b + a^2*x^2]]) + 8*c*(a*x + S
qrt[b + a^2*x^2])*(-2*c^2 + 6*a*x + c*Sqrt[a*x + Sqrt[b + a^2*x^2]])))/(a*x + Sqrt[b + a^2*x^2]) + 15*b*ArcTan
h[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/Sqrt[c]])/(60*a*c^(3/2))

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.23 \[ \int \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx=\left [\frac {15 \, b \sqrt {c} \log \left (-2 \, {\left (a \sqrt {c} x - \sqrt {a^{2} x^{2} + b} \sqrt {c}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}} - 2 \, {\left (a c x - \sqrt {a^{2} x^{2} + b} c\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}} + b\right ) - 2 \, {\left (16 \, c^{4} - 54 \, a c^{2} x + 6 \, \sqrt {a^{2} x^{2} + b} c^{2} - {\left (8 \, c^{3} + 15 \, a c x - 15 \, \sqrt {a^{2} x^{2} + b} c\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}}\right )} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{120 \, a c^{2}}, -\frac {15 \, b \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{c}\right ) + {\left (16 \, c^{4} - 54 \, a c^{2} x + 6 \, \sqrt {a^{2} x^{2} + b} c^{2} - {\left (8 \, c^{3} + 15 \, a c x - 15 \, \sqrt {a^{2} x^{2} + b} c\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}}\right )} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{60 \, a c^{2}}\right ] \]

[In]

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

[Out]

[1/120*(15*b*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 + s
qrt(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*(16*c^4
- 54*a*c^2*x + 6*sqrt(a^2*x^2 + b)*c^2 - (8*c^3 + 15*a*c*x - 15*sqrt(a^2*x^2 + b)*c)*sqrt(a*x + sqrt(a^2*x^2
+ b)))*sqrt(c + sqrt(a*x + sqrt(a^2*x^2 + b))))/(a*c^2), -1/60*(15*b*sqrt(-c)*arctan(sqrt(-c)*sqrt(c + sqrt(a*
x + sqrt(a^2*x^2 + b)))/c) + (16*c^4 - 54*a*c^2*x + 6*sqrt(a^2*x^2 + b)*c^2 - (8*c^3 + 15*a*c*x - 15*sqrt(a^2*
x^2 + b)*c)*sqrt(a*x + sqrt(a^2*x^2 + b)))*sqrt(c + sqrt(a*x + sqrt(a^2*x^2 + b))))/(a*c^2)]

Sympy [F]

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

[In]

integrate((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))), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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