Integrand size = 68, antiderivative size = 507 \[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {\left (945 b^3-4224 b^2 c^4-504 a b^2 c^2 x+3072 a b c^6 x-1890 a^2 b^2 x^2+7680 a^2 b c^4 x^2-4096 a^3 c^6 x^3\right ) \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\left (432 b^2 c^3-2048 b c^7+630 a b^2 c x-2304 a b c^5 x+4096 a^2 c^7 x^2+3072 a^3 c^5 x^3\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\sqrt {-b+a^2 x^2} \left (\left (-504 b^2 c^2+1024 b c^6-1890 a b^2 x+7680 a b c^4 x-4096 a^2 c^6 x^2\right ) \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\left (630 b^2 c-768 b c^5+4096 a c^7 x+3072 a^2 c^5 x^2\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{3840 a c^5 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/2}}+\frac {63 b^2 \text {arctanh}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {c}}\right )}{256 a c^{11/2}}-\frac {b \text {arctanh}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {c}}\right )}{a c^{3/2}} \]
1/3840*((-4096*a^3*c^6*x^3+7680*a^2*b*c^4*x^2+3072*a*b*c^6*x-1890*a^2*b^2* x^2-504*a*b^2*c^2*x-4224*b^2*c^4+945*b^3)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/2) )^(1/2)+(3072*a^3*c^5*x^3+4096*a^2*c^7*x^2-2304*a*b*c^5*x-2048*b*c^7+630*a *b^2*c*x+432*b^2*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)*((-4096*a^2*c^6*x^2+7680*a*b*c^4*x+102 4*b*c^6-1890*a*b^2*x-504*b^2*c^2)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/2))^(1/2)+ (3072*a^2*c^5*x^2+4096*a*c^7*x-768*b*c^5+630*b^2*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/c^5/(a*x+(a^2*x^2-b)^( 1/2))^(5/2)+63/256*b^2*arctanh((c+(a*x+(a^2*x^2-b)^(1/2))^(1/2))^(1/2)/c^( 1/2))/a/c^(11/2)-b*arctanh((c+(a*x+(a^2*x^2-b)^(1/2))^(1/2))^(1/2)/c^(1/2) )/a/c^(3/2)
Time = 1.57 (sec) , antiderivative size = 507, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {\left (945 b^3-4224 b^2 c^4-504 a b^2 c^2 x+3072 a b c^6 x-1890 a^2 b^2 x^2+7680 a^2 b c^4 x^2-4096 a^3 c^6 x^3\right ) \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\left (432 b^2 c^3-2048 b c^7+630 a b^2 c x-2304 a b c^5 x+4096 a^2 c^7 x^2+3072 a^3 c^5 x^3\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\sqrt {-b+a^2 x^2} \left (\left (-504 b^2 c^2+1024 b c^6-1890 a b^2 x+7680 a b c^4 x-4096 a^2 c^6 x^2\right ) \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\left (630 b^2 c-768 b c^5+4096 a c^7 x+3072 a^2 c^5 x^2\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{3840 a c^5 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/2}}+\frac {63 b^2 \text {arctanh}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {c}}\right )}{256 a c^{11/2}}-\frac {b \text {arctanh}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {c}}\right )}{a c^{3/2}} \]
Integrate[Sqrt[-b + a^2*x^2]/(Sqrt[a*x + Sqrt[-b + a^2*x^2]]*Sqrt[c + Sqrt [a*x + Sqrt[-b + a^2*x^2]]]),x]
((945*b^3 - 4224*b^2*c^4 - 504*a*b^2*c^2*x + 3072*a*b*c^6*x - 1890*a^2*b^2 *x^2 + 7680*a^2*b*c^4*x^2 - 4096*a^3*c^6*x^3)*Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]] + (432*b^2*c^3 - 2048*b*c^7 + 630*a*b^2*c*x - 2304*a*b*c^5*x + 4096*a^2*c^7*x^2 + 3072*a^3*c^5*x^3)*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]*((-504*b^2*c^2 + 1024*b*c^6 - 1890*a*b^2*x + 7680*a*b*c^4*x - 4096*a^2*c^6*x^2)*Sqrt[c + S qrt[a*x + Sqrt[-b + a^2*x^2]]] + (630*b^2*c - 768*b*c^5 + 4096*a*c^7*x + 3 072*a^2*c^5*x^2)*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[- b + a^2*x^2]]]))/(3840*a*c^5*(a*x + Sqrt[-b + a^2*x^2])^(5/2)) + (63*b^2*A rcTanh[Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]/Sqrt[c]])/(256*a*c^(11/2)) - (b*ArcTanh[Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]/Sqrt[c]])/(a*c^(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.
| \(\displaystyle \int \frac {\sqrt {a^2 x^2-b}}{\sqrt {\sqrt {a^2 x^2-b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+c}} \, dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \frac {\sqrt {a^2 x^2-b}}{\sqrt {\sqrt {a^2 x^2-b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+c}}dx\) |
Int[Sqrt[-b + a^2*x^2]/(Sqrt[a*x + Sqrt[-b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[-b + a^2*x^2]]]),x]
3.31.79.3.1 Defintions of rubi rules used
\[\int \frac {\sqrt {a^{2} x^{2}-b}}{\sqrt {a x +\sqrt {a^{2} x^{2}-b}}\, \sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}-b}}}}d x\]
int((a^2*x^2-b)^(1/2)/(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),x)
int((a^2*x^2-b)^(1/2)/(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),x)
Time = 0.31 (sec) , antiderivative size = 679, normalized size of antiderivative = 1.34 \[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\left [\frac {15 \, {\left (256 \, b^{2} c^{4} - 63 \, b^{3}\right )} \sqrt {c} \log \left (-2 \, {\left (a \sqrt {c} x - \sqrt {a^{2} x^{2} - b} \sqrt {c}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} - b}}} + 2 \, {\left (a c x - \sqrt {a^{2} x^{2} - b} c\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}} + b\right ) + 2 \, {\left (2048 \, b c^{8} + 864 \, a^{2} b c^{4} x^{2} - 432 \, b^{2} c^{4} + 6 \, {\left (128 \, a b c^{6} + 105 \, a b^{2} c^{2}\right )} x + 6 \, {\left (128 \, b c^{6} - 144 \, a b c^{4} x - 105 \, b^{2} c^{2}\right )} \sqrt {a^{2} x^{2} - b} - {\left (1536 \, a^{3} c^{5} x^{3} + 1024 \, b c^{7} + 1008 \, a^{2} b c^{3} x^{2} - 504 \, b^{2} c^{3} - 3 \, {\left (1664 \, a b c^{5} - 315 \, a b^{2} c\right )} x - 3 \, {\left (512 \, a^{2} c^{5} x^{2} - 1408 \, b c^{5} + 336 \, a b c^{3} x + 315 \, b^{2} c\right )} \sqrt {a^{2} x^{2} - b}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}}\right )} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}}{7680 \, a b c^{6}}, \frac {15 \, {\left (256 \, b^{2} c^{4} - 63 \, b^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}}{c}\right ) + {\left (2048 \, b c^{8} + 864 \, a^{2} b c^{4} x^{2} - 432 \, b^{2} c^{4} + 6 \, {\left (128 \, a b c^{6} + 105 \, a b^{2} c^{2}\right )} x + 6 \, {\left (128 \, b c^{6} - 144 \, a b c^{4} x - 105 \, b^{2} c^{2}\right )} \sqrt {a^{2} x^{2} - b} - {\left (1536 \, a^{3} c^{5} x^{3} + 1024 \, b c^{7} + 1008 \, a^{2} b c^{3} x^{2} - 504 \, b^{2} c^{3} - 3 \, {\left (1664 \, a b c^{5} - 315 \, a b^{2} c\right )} x - 3 \, {\left (512 \, a^{2} c^{5} x^{2} - 1408 \, b c^{5} + 336 \, a b c^{3} x + 315 \, b^{2} c\right )} \sqrt {a^{2} x^{2} - b}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}}\right )} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}}{3840 \, a b c^{6}}\right ] \]
integrate((a^2*x^2-b)^(1/2)/(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),x, algorithm="fricas")
[1/7680*(15*(256*b^2*c^4 - 63*b^3)*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*(2048*b*c^8 + 864*a^2*b*c^4*x^2 - 432*b^2*c^4 + 6*(128*a*b*c^ 6 + 105*a*b^2*c^2)*x + 6*(128*b*c^6 - 144*a*b*c^4*x - 105*b^2*c^2)*sqrt(a^ 2*x^2 - b) - (1536*a^3*c^5*x^3 + 1024*b*c^7 + 1008*a^2*b*c^3*x^2 - 504*b^2 *c^3 - 3*(1664*a*b*c^5 - 315*a*b^2*c)*x - 3*(512*a^2*c^5*x^2 - 1408*b*c^5 + 336*a*b*c^3*x + 315*b^2*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*b*c^6), 1/3840*(15*(256*b ^2*c^4 - 63*b^3)*sqrt(-c)*arctan(sqrt(-c)*sqrt(c + sqrt(a*x + sqrt(a^2*x^2 - b)))/c) + (2048*b*c^8 + 864*a^2*b*c^4*x^2 - 432*b^2*c^4 + 6*(128*a*b*c^ 6 + 105*a*b^2*c^2)*x + 6*(128*b*c^6 - 144*a*b*c^4*x - 105*b^2*c^2)*sqrt(a^ 2*x^2 - b) - (1536*a^3*c^5*x^3 + 1024*b*c^7 + 1008*a^2*b*c^3*x^2 - 504*b^2 *c^3 - 3*(1664*a*b*c^5 - 315*a*b^2*c)*x - 3*(512*a^2*c^5*x^2 - 1408*b*c^5 + 336*a*b*c^3*x + 315*b^2*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*b*c^6)]
\[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int \frac {\sqrt {a^{2} x^{2} - b}}{\sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} - b}}} \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}\, dx \]
integrate((a**2*x**2-b)**(1/2)/(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),x)
Integral(sqrt(a**2*x**2 - b)/(sqrt(c + sqrt(a*x + sqrt(a**2*x**2 - b)))*sq rt(a*x + sqrt(a**2*x**2 - b))), x)
\[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int { \frac {\sqrt {a^{2} x^{2} - b}}{\sqrt {a x + \sqrt {a^{2} x^{2} - b}} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}} \,d x } \]
integrate((a^2*x^2-b)^(1/2)/(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),x, algorithm="maxima")
integrate(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)))), x)
Exception generated. \[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\text {Exception raised: TypeError} \]
integrate((a^2*x^2-b)^(1/2)/(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),x, algorithm="giac")
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. \[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int \frac {\sqrt {a^2\,x^2-b}}{\sqrt {a\,x+\sqrt {a^2\,x^2-b}}\,\sqrt {c+\sqrt {a\,x+\sqrt {a^2\,x^2-b}}}} \,d x \]
int((a^2*x^2 - b)^(1/2)/((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)),x)