\(\int \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx\) [2828]

Optimal result

Integrand size = 49, antiderivative size = 286 \[ \int \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx=\frac {\left (-75 b-8 a c^2 x+60 a^2 x^2\right ) \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}+\left (16 c^3+6 a c x\right ) \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}+\sqrt {b+a^2 x^2} \left (\left (-8 c^2+60 a x\right ) \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}+6 c \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}\right )}{105 a \sqrt {a x+\sqrt {b+a^2 x^2}}}-\frac {b \text {arctanh}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\sqrt {c}}\right )}{a \sqrt {c}} \]

[Out]

1/105*((60*a^2*x^2-8*a*c^2*x-75*b)*(c+(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2)+(6*a*c*x+16*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)*((60*a*x-8*c^2)*(c+(a*x+(a^2*x^2+b)^(1/2
))^(1/2))^(1/2)+6*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/(a*x+(a^2*x^2+b)
^(1/2))^(1/2)-b*arctanh((c+(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2)/c^(1/2))/a/c^(1/2)

Rubi [F]

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

[In]

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

[Out]

Defer[Int][Sqrt[a*x + 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 {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.85 \[ \int \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx=\frac {\frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \left (-75 b+60 a^2 x^2+2 a x \left (-4 c^2+30 \sqrt {b+a^2 x^2}+3 c \sqrt {a x+\sqrt {b+a^2 x^2}}\right )+2 c \left (-4 c \sqrt {b+a^2 x^2}+8 c^2 \sqrt {a x+\sqrt {b+a^2 x^2}}+3 \sqrt {b+a^2 x^2} \sqrt {a x+\sqrt {b+a^2 x^2}}\right )\right )}{\sqrt {a x+\sqrt {b+a^2 x^2}}}-\frac {105 b \text {arctanh}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\sqrt {c}}\right )}{\sqrt {c}}}{105 a} \]

[In]

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

[Out]

((Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]*(-75*b + 60*a^2*x^2 + 2*a*x*(-4*c^2 + 30*Sqrt[b + a^2*x^2] + 3*c*Sqr
t[a*x + Sqrt[b + a^2*x^2]]) + 2*c*(-4*c*Sqrt[b + a^2*x^2] + 8*c^2*Sqrt[a*x + Sqrt[b + a^2*x^2]] + 3*Sqrt[b + a
^2*x^2]*Sqrt[a*x + Sqrt[b + a^2*x^2]])))/Sqrt[a*x + Sqrt[b + a^2*x^2]] - (105*b*ArcTanh[Sqrt[c + Sqrt[a*x + Sq
rt[b + a^2*x^2]]]/Sqrt[c]])/Sqrt[c])/(105*a)

Maple [F]

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

[In]

int((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)

[Out]

int((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)

none

Time = 0.33 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.26 \[ \int \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx=\left [\frac {105 \, b \sqrt {c} \log \left (2 \, {\left (a \sqrt {c} x - \sqrt {a^{2} x^{2} + b} \sqrt {c}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}} - 2 \, {\left (a c x - \sqrt {a^{2} x^{2} + b} c\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}} + b\right ) + 2 \, {\left (16 \, c^{4} + 6 \, a c^{2} x + 6 \, \sqrt {a^{2} x^{2} + b} c^{2} - {\left (8 \, c^{3} - 135 \, a c x + 75 \, \sqrt {a^{2} x^{2} + b} c\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}}\right )} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{210 \, a c}, \frac {105 \, b \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{c}\right ) + {\left (16 \, c^{4} + 6 \, a c^{2} x + 6 \, \sqrt {a^{2} x^{2} + b} c^{2} - {\left (8 \, c^{3} - 135 \, a c x + 75 \, \sqrt {a^{2} x^{2} + b} c\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b}}\right )} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{105 \, a c}\right ] \]

[In]

integrate((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")

[Out]

[1/210*(105*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
+ 6*a*c^2*x + 6*sqrt(a^2*x^2 + b)*c^2 - (8*c^3 - 135*a*c*x + 75*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), 1/105*(105*b*sqrt(-c)*arctan(sqrt(-c)*sqrt(c + sqrt(a*x
+ sqrt(a^2*x^2 + b)))/c) + (16*c^4 + 6*a*c^2*x + 6*sqrt(a^2*x^2 + b)*c^2 - (8*c^3 - 135*a*c*x + 75*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)]

Sympy [F]

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

[In]

integrate((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)

[Out]

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

Maxima [F]

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

[In]

integrate((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")

[Out]

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

Giac [F(-2)]

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

[In]

integrate((a*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")

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

[In]

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