Integrand size = 45, antiderivative size = 495 \[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {\left (-1680 b^2 c^3+6144 b c^7-2450 a b^2 c x+6912 a b c^5 x-12288 a^2 c^7 x^2-9216 a^3 c^5 x^3\right ) \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\left (1960 b^2 c^2-3072 b c^6+3675 a b^2 x-5760 a b c^4 x+6144 a^2 c^6 x^2+7680 a^3 c^4 x^3\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\sqrt {-b+a^2 x^2} \left (\left (-2450 b^2 c+2304 b c^5-12288 a c^7 x-9216 a^2 c^5 x^2\right ) \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\left (3675 b^2-1920 b c^4+6144 a c^6 x+7680 a^2 c^4 x^2\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{26880 a^2 c^4 x \sqrt {-b+a^2 x^2}+13440 a c^4 \left (-b+2 a^2 x^2\right )}-\frac {35 b^2 \text {arctanh}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {c}}\right )}{128 a c^{9/2}}+\frac {2 b \text {arctanh}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {c}}\right )}{a \sqrt {c}} \]
((-9216*a^3*c^5*x^3-12288*a^2*c^7*x^2+6912*a*b*c^5*x+6144*b*c^7-2450*a*b^2 *c*x-1680*b^2*c^3)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/2))^(1/2)+(7680*a^3*c^4*x ^3+6144*a^2*c^6*x^2-5760*a*b*c^4*x-3072*b*c^6+3675*a*b^2*x+1960*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)*((-9216*a^2*c^5*x^2-12288*a*c^7*x+2304*b*c^5-2450*b^2*c)*(c+( a*x+(a^2*x^2-b)^(1/2))^(1/2))^(1/2)+(7680*a^2*c^4*x^2+6144*a*c^6*x-1920*b* c^4+3675*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)))/(26880*a^2*c^4*x*(a^2*x^2-b)^(1/2)+13440*a*c^4*(2*a^2*x^2-b))- 35/128*b^2*arctanh((c+(a*x+(a^2*x^2-b)^(1/2))^(1/2))^(1/2)/c^(1/2))/a/c^(9 /2)+2*b*arctanh((c+(a*x+(a^2*x^2-b)^(1/2))^(1/2))^(1/2)/c^(1/2))/a/c^(1/2)
Time = 1.01 (sec) , antiderivative size = 447, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {\frac {\sqrt {c} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}} \left (1536 a c^4 x \left (a x+\sqrt {-b+a^2 x^2}\right ) \left (-8 c^3-6 a c x+4 c^2 \sqrt {a x+\sqrt {-b+a^2 x^2}}+5 a x \sqrt {a x+\sqrt {-b+a^2 x^2}}\right )+35 b^2 \left (-48 c^3+56 c^2 \sqrt {a x+\sqrt {-b+a^2 x^2}}-70 c \left (a x+\sqrt {-b+a^2 x^2}\right )+105 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/2}\right )-384 b c^4 \left (-16 c^3+8 c^2 \sqrt {a x+\sqrt {-b+a^2 x^2}}-6 c \left (3 a x+\sqrt {-b+a^2 x^2}\right )+5 \sqrt {a x+\sqrt {-b+a^2 x^2}} \left (3 a x+\sqrt {-b+a^2 x^2}\right )\right )\right )}{-b+2 a x \left (a x+\sqrt {-b+a^2 x^2}\right )}-3675 b^2 \text {arctanh}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {c}}\right )+26880 b c^4 \text {arctanh}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {c}}\right )}{13440 a c^{9/2}} \]
((Sqrt[c]*Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]*(1536*a*c^4*x*(a*x + Sq rt[-b + a^2*x^2])*(-8*c^3 - 6*a*c*x + 4*c^2*Sqrt[a*x + Sqrt[-b + a^2*x^2]] + 5*a*x*Sqrt[a*x + Sqrt[-b + a^2*x^2]]) + 35*b^2*(-48*c^3 + 56*c^2*Sqrt[a *x + Sqrt[-b + a^2*x^2]] - 70*c*(a*x + Sqrt[-b + a^2*x^2]) + 105*(a*x + Sq rt[-b + a^2*x^2])^(3/2)) - 384*b*c^4*(-16*c^3 + 8*c^2*Sqrt[a*x + Sqrt[-b + a^2*x^2]] - 6*c*(3*a*x + Sqrt[-b + a^2*x^2]) + 5*Sqrt[a*x + Sqrt[-b + a^2 *x^2]]*(3*a*x + Sqrt[-b + a^2*x^2]))))/(-b + 2*a*x*(a*x + Sqrt[-b + a^2*x^ 2])) - 3675*b^2*ArcTanh[Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]/Sqrt[c]] + 26880*b*c^4*ArcTanh[Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]/Sqrt[c]])/( 13440*a*c^(9/2))
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.
| \(\displaystyle \int \frac {\sqrt {a^2 x^2-b}}{\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+c}} \, dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \frac {\sqrt {a^2 x^2-b}}{\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+c}}dx\) |
3.31.69.3.1 Defintions of rubi rules used
\[\int \frac {\sqrt {a^{2} x^{2}-b}}{\sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}-b}}}}d x\]
Time = 0.32 (sec) , antiderivative size = 581, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\left [\frac {105 \, {\left (256 \, b c^{4} - 35 \, b^{2}\right )} \sqrt {c} \log \left (2 \, {\left (a \sqrt {c} x - \sqrt {a^{2} x^{2} - b} \sqrt {c}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} - b}}} + 2 \, {\left (a c x - \sqrt {a^{2} x^{2} - b} c\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}} + b\right ) - 2 \, {\left (6144 \, c^{8} + 3360 \, a^{2} c^{4} x^{2} - 1680 \, b c^{4} + 2 \, {\left (1152 \, a c^{6} + 1225 \, a b c^{2}\right )} x + 2 \, {\left (1152 \, c^{6} - 1680 \, a c^{4} x - 1225 \, b c^{2}\right )} \sqrt {a^{2} x^{2} - b} - {\left (3072 \, c^{7} + 3920 \, a^{2} c^{3} x^{2} - 1960 \, b c^{3} + 15 \, {\left (128 \, a c^{5} + 245 \, a b c\right )} x + 5 \, {\left (384 \, c^{5} - 784 \, a c^{3} x - 735 \, b c\right )} \sqrt {a^{2} x^{2} - b}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}}\right )} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}}{26880 \, a c^{5}}, -\frac {105 \, {\left (256 \, b c^{4} - 35 \, b^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}}{c}\right ) + {\left (6144 \, c^{8} + 3360 \, a^{2} c^{4} x^{2} - 1680 \, b c^{4} + 2 \, {\left (1152 \, a c^{6} + 1225 \, a b c^{2}\right )} x + 2 \, {\left (1152 \, c^{6} - 1680 \, a c^{4} x - 1225 \, b c^{2}\right )} \sqrt {a^{2} x^{2} - b} - {\left (3072 \, c^{7} + 3920 \, a^{2} c^{3} x^{2} - 1960 \, b c^{3} + 15 \, {\left (128 \, a c^{5} + 245 \, a b c\right )} x + 5 \, {\left (384 \, c^{5} - 784 \, a c^{3} x - 735 \, b c\right )} \sqrt {a^{2} x^{2} - b}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}}\right )} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}}{13440 \, a c^{5}}\right ] \]
[1/26880*(105*(256*b*c^4 - 35*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*(6144*c^8 + 3360*a^2*c^4*x^2 - 1680*b*c^4 + 2*(1152*a*c^6 + 12 25*a*b*c^2)*x + 2*(1152*c^6 - 1680*a*c^4*x - 1225*b*c^2)*sqrt(a^2*x^2 - b) - (3072*c^7 + 3920*a^2*c^3*x^2 - 1960*b*c^3 + 15*(128*a*c^5 + 245*a*b*c)* x + 5*(384*c^5 - 784*a*c^3*x - 735*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^5), -1/13440 *(105*(256*b*c^4 - 35*b^2)*sqrt(-c)*arctan(sqrt(-c)*sqrt(c + sqrt(a*x + sq rt(a^2*x^2 - b)))/c) + (6144*c^8 + 3360*a^2*c^4*x^2 - 1680*b*c^4 + 2*(1152 *a*c^6 + 1225*a*b*c^2)*x + 2*(1152*c^6 - 1680*a*c^4*x - 1225*b*c^2)*sqrt(a ^2*x^2 - b) - (3072*c^7 + 3920*a^2*c^3*x^2 - 1960*b*c^3 + 15*(128*a*c^5 + 245*a*b*c)*x + 5*(384*c^5 - 784*a*c^3*x - 735*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^5 )]
\[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int \frac {\sqrt {a^{2} x^{2} - b}}{\sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}}\, dx \]
\[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int { \frac {\sqrt {a^{2} x^{2} - b}}{\sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}} \,d x } \]
Exception generated. \[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int \frac {\sqrt {a^2\,x^2-b}}{\sqrt {c+\sqrt {a\,x+\sqrt {a^2\,x^2-b}}}} \,d x \]