Integrand size = 49, antiderivative size = 294 \[ \int \frac {\sqrt {c+\sqrt {a x^2+x \sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2}} \, dx=\frac {2 \sqrt {c+\sqrt {x \left (a x+\sqrt {-b+a^2 x^2}\right )}}}{a}-\frac {\sqrt {\sqrt {2} \sqrt {b}-2 \sqrt {a} c} \left (-\sqrt {b}+\sqrt {2} \sqrt {a} c\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c+\sqrt {x \left (a x+\sqrt {-b+a^2 x^2}\right )}}}{\sqrt {\sqrt {2} \sqrt {b}-2 \sqrt {a} c}}\right )}{a^{5/4} \left (-\sqrt {2} \sqrt {b}+2 \sqrt {a} c\right )}-\frac {\left (\sqrt {b}+\sqrt {2} \sqrt {a} c\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c+\sqrt {x \left (a x+\sqrt {-b+a^2 x^2}\right )}}}{\sqrt {\sqrt {2} \sqrt {b}+2 \sqrt {a} c}}\right )}{a^{5/4} \sqrt {\sqrt {2} \sqrt {b}+2 \sqrt {a} c}} \]
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)
Time = 4.70 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {c+\sqrt {a x^2+x \sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2}} \, dx=\frac {4 \sqrt {a} \sqrt {c+\sqrt {x \left (a x+\sqrt {-b+a^2 x^2}\right )}}-\sqrt {2 \sqrt {2} \sqrt {a} \sqrt {b}-4 a c} \arctan \left (\frac {\sqrt {2 \sqrt {2} \sqrt {a} \sqrt {b}-4 a c} \sqrt {c+\sqrt {x \left (a x+\sqrt {-b+a^2 x^2}\right )}}}{\sqrt {2} \sqrt {b}-2 \sqrt {a} c}\right )+\sqrt {2} \sqrt {-\sqrt {2} \sqrt {a} \sqrt {b}-2 a c} \arctan \left (\frac {\sqrt {2} \sqrt {-\sqrt {2} \sqrt {a} \sqrt {b}-2 a c} \sqrt {c+\sqrt {x \left (a x+\sqrt {-b+a^2 x^2}\right )}}}{\sqrt {2} \sqrt {b}+2 \sqrt {a} c}\right )}{2 a^{3/2}} \]
(4*Sqrt[a]*Sqrt[c + Sqrt[x*(a*x + Sqrt[-b + a^2*x^2])]] - Sqrt[2*Sqrt[2]*S qrt[a]*Sqrt[b] - 4*a*c]*ArcTan[(Sqrt[2*Sqrt[2]*Sqrt[a]*Sqrt[b] - 4*a*c]*Sq rt[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))
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.
\(\displaystyle \int \frac {\sqrt {\sqrt {x \sqrt {a^2 x^2-b}+a x^2}+c}}{\sqrt {a^2 x^2-b}} \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \frac {\sqrt {\sqrt {x \sqrt {a^2 x^2-b}+a x^2}+c}}{\sqrt {a^2 x^2-b}}dx\) |
3.29.52.3.1 Defintions of rubi rules used
\[\int \frac {\sqrt {c +\sqrt {a \,x^{2}+x \sqrt {a^{2} x^{2}-b}}}}{\sqrt {a^{2} x^{2}-b}}d x\]
Timed out. \[ \int \frac {\sqrt {c+\sqrt {a x^2+x \sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2}} \, dx=\text {Timed out} \]
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")
\[ \int \frac {\sqrt {c+\sqrt {a x^2+x \sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2}} \, dx=\int \frac {\sqrt {c + \sqrt {a x^{2} + x \sqrt {a^{2} x^{2} - b}}}}{\sqrt {a^{2} x^{2} - b}}\, dx \]
\[ \int \frac {\sqrt {c+\sqrt {a x^2+x \sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2}} \, dx=\int { \frac {\sqrt {c + \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b} x}}}{\sqrt {a^{2} x^{2} - b}} \,d x } \]
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")
Exception generated. \[ \int \frac {\sqrt {c+\sqrt {a x^2+x \sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2}} \, dx=\text {Exception raised: TypeError} \]
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 &
Timed out. \[ \int \frac {\sqrt {c+\sqrt {a x^2+x \sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2}} \, dx=\int \frac {\sqrt {c+\sqrt {x\,\sqrt {a^2\,x^2-b}+a\,x^2}}}{\sqrt {a^2\,x^2-b}} \,d x \]