\(\int \frac {\sqrt {c+\sqrt {a x^2+x \sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2}} \, dx\) [2852]

Optimal result

Integrand size = 49, antiderivative size = 294 \[ \int \frac {\sqrt {c+\sqrt {a x^2+x \sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2}} \, dx=\frac {2 \sqrt {c+\sqrt {x \left (a x+\sqrt {-b+a^2 x^2}\right )}}}{a}-\frac {\sqrt {\sqrt {2} \sqrt {b}-2 \sqrt {a} c} \left (-\sqrt {b}+\sqrt {2} \sqrt {a} c\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c+\sqrt {x \left (a x+\sqrt {-b+a^2 x^2}\right )}}}{\sqrt {\sqrt {2} \sqrt {b}-2 \sqrt {a} c}}\right )}{a^{5/4} \left (-\sqrt {2} \sqrt {b}+2 \sqrt {a} c\right )}-\frac {\left (\sqrt {b}+\sqrt {2} \sqrt {a} c\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c+\sqrt {x \left (a x+\sqrt {-b+a^2 x^2}\right )}}}{\sqrt {\sqrt {2} \sqrt {b}+2 \sqrt {a} c}}\right )}{a^{5/4} \sqrt {\sqrt {2} \sqrt {b}+2 \sqrt {a} c}} \]

[Out]

2*(c+(x*(a*x+(a^2*x^2-b)^(1/2)))^(1/2))^(1/2)/a-(2^(1/2)*b^(1/2)-2*a^(1/2)*c)^(1/2)*(-b^(1/2)+2^(1/2)*a^(1/2)*
c)*arctan(2^(1/2)*a^(1/4)*(c+(x*(a*x+(a^2*x^2-b)^(1/2)))^(1/2))^(1/2)/(2^(1/2)*b^(1/2)-2*a^(1/2)*c)^(1/2))/a^(
5/4)/(-2^(1/2)*b^(1/2)+2*a^(1/2)*c)-(b^(1/2)+2^(1/2)*a^(1/2)*c)*arctanh(2^(1/2)*a^(1/4)*(c+(x*(a*x+(a^2*x^2-b)
^(1/2)))^(1/2))^(1/2)/(2^(1/2)*b^(1/2)+2*a^(1/2)*c)^(1/2))/a^(5/4)/(2^(1/2)*b^(1/2)+2*a^(1/2)*c)^(1/2)

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 4.70 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {c+\sqrt {a x^2+x \sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2}} \, dx=\frac {4 \sqrt {a} \sqrt {c+\sqrt {x \left (a x+\sqrt {-b+a^2 x^2}\right )}}-\sqrt {2 \sqrt {2} \sqrt {a} \sqrt {b}-4 a c} \arctan \left (\frac {\sqrt {2 \sqrt {2} \sqrt {a} \sqrt {b}-4 a c} \sqrt {c+\sqrt {x \left (a x+\sqrt {-b+a^2 x^2}\right )}}}{\sqrt {2} \sqrt {b}-2 \sqrt {a} c}\right )+\sqrt {2} \sqrt {-\sqrt {2} \sqrt {a} \sqrt {b}-2 a c} \arctan \left (\frac {\sqrt {2} \sqrt {-\sqrt {2} \sqrt {a} \sqrt {b}-2 a c} \sqrt {c+\sqrt {x \left (a x+\sqrt {-b+a^2 x^2}\right )}}}{\sqrt {2} \sqrt {b}+2 \sqrt {a} c}\right )}{2 a^{3/2}} \]

[In]

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

[Out]

(4*Sqrt[a]*Sqrt[c + Sqrt[x*(a*x + Sqrt[-b + a^2*x^2])]] - Sqrt[2*Sqrt[2]*Sqrt[a]*Sqrt[b] - 4*a*c]*ArcTan[(Sqrt
[2*Sqrt[2]*Sqrt[a]*Sqrt[b] - 4*a*c]*Sqrt[c + Sqrt[x*(a*x + Sqrt[-b + a^2*x^2])]])/(Sqrt[2]*Sqrt[b] - 2*Sqrt[a]
*c)] + Sqrt[2]*Sqrt[-(Sqrt[2]*Sqrt[a]*Sqrt[b]) - 2*a*c]*ArcTan[(Sqrt[2]*Sqrt[-(Sqrt[2]*Sqrt[a]*Sqrt[b]) - 2*a*
c]*Sqrt[c + Sqrt[x*(a*x + Sqrt[-b + a^2*x^2])]])/(Sqrt[2]*Sqrt[b] + 2*Sqrt[a]*c)])/(2*a^(3/2))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

integrate((c+(a*x^2+x*(a^2*x^2-b)^(1/2))^(1/2))^(1/2)/(a^2*x^2-b)^(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. \[ \int \frac {\sqrt {c+\sqrt {a x^2+x \sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2}} \, dx=\int \frac {\sqrt {c+\sqrt {x\,\sqrt {a^2\,x^2-b}+a\,x^2}}}{\sqrt {a^2\,x^2-b}} \,d x \]

[In]

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

[Out]

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