Integrand size = 68, antiderivative size = 383 \[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {-6 c \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{5/6}+7 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{5/6}}{3 a c^2 \sqrt {a x+\sqrt {-b+a^2 x^2}}}-\frac {7 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {3} \sqrt [6]{c}}\right )}{6 \sqrt {3} a c^{13/6}}+\frac {7 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {3} \sqrt [6]{c}}\right )}{6 \sqrt {3} a c^{13/6}}-\frac {7 \text {arctanh}\left (\frac {\sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [6]{c}}\right )}{9 a c^{13/6}}-\frac {7 \text {arctanh}\left (\frac {\sqrt [6]{c}+\frac {\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [6]{c}}}{\sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}\right )}{18 a c^{13/6}} \]
1/3*(-6*c*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(5/6)+7*(a*x+(a^2*x^2-b)^(1/2) )^(1/4)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(5/6))/a/c^2/(a*x+(a^2*x^2-b)^(1 /2))^(1/2)+7/18*arctan(-1/3*3^(1/2)+2/3*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^ (1/6)*3^(1/2)/c^(1/6))*3^(1/2)/a/c^(13/6)+7/18*arctan(1/3*3^(1/2)+2/3*(c+( a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/6)*3^(1/2)/c^(1/6))*3^(1/2)/a/c^(13/6)-7/ 9*arctanh((c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/6)/c^(1/6))/a/c^(13/6)-7/18 *arctanh((c^(1/6)+(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3)/c^(1/6))/(c+(a*x +(a^2*x^2-b)^(1/2))^(1/4))^(1/6))/a/c^(13/6)
Time = 10.65 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.40 \[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {-36 c^{7/6} \sqrt {a x+\sqrt {-b+a^2 x^2}} \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{5/6}+42 \sqrt [6]{c} \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4} \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{5/6}-7 \sqrt {3} a x \arctan \left (\frac {1-\frac {2 \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [6]{c}}}{\sqrt {3}}\right )-7 \sqrt {-3 b+3 a^2 x^2} \arctan \left (\frac {1-\frac {2 \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [6]{c}}}{\sqrt {3}}\right )+7 \sqrt {3} a x \arctan \left (\frac {1+\frac {2 \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [6]{c}}}{\sqrt {3}}\right )+7 \sqrt {-3 b+3 a^2 x^2} \arctan \left (\frac {1+\frac {2 \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [6]{c}}}{\sqrt {3}}\right )-14 \left (a x+\sqrt {-b+a^2 x^2}\right ) \text {arctanh}\left (\frac {\sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [6]{c}}\right )-7 \left (a x+\sqrt {-b+a^2 x^2}\right ) \text {arctanh}\left (\frac {\sqrt [6]{c} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}\right )}{18 a c^{13/6} \left (a x+\sqrt {-b+a^2 x^2}\right )} \]
Integrate[1/(Sqrt[-b + a^2*x^2]*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6)),x]
(-36*c^(7/6)*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*(c + (a*x + Sqrt[-b + a^2*x^2] )^(1/4))^(5/6) + 42*c^(1/6)*(a*x + Sqrt[-b + a^2*x^2])^(3/4)*(c + (a*x + S qrt[-b + a^2*x^2])^(1/4))^(5/6) - 7*Sqrt[3]*a*x*ArcTan[(1 - (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6))/c^(1/6))/Sqrt[3]] - 7*Sqrt[-3*b + 3*a^2 *x^2]*ArcTan[(1 - (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6))/c^(1/6) )/Sqrt[3]] + 7*Sqrt[3]*a*x*ArcTan[(1 + (2*(c + (a*x + Sqrt[-b + a^2*x^2])^ (1/4))^(1/6))/c^(1/6))/Sqrt[3]] + 7*Sqrt[-3*b + 3*a^2*x^2]*ArcTan[(1 + (2* (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6))/c^(1/6))/Sqrt[3]] - 14*(a*x + Sqrt[-b + a^2*x^2])*ArcTanh[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6) /c^(1/6)] - 7*(a*x + Sqrt[-b + a^2*x^2])*ArcTanh[(c^(1/6)*(c + (a*x + Sqrt [-b + a^2*x^2])^(1/4))^(1/6))/(c^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^( 1/4))^(1/3))])/(18*a*c^(13/6)*(a*x + Sqrt[-b + a^2*x^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.
| \(\displaystyle \int \frac {1}{\sqrt {a^2 x^2-b} \sqrt {\sqrt {a^2 x^2-b}+a x} \sqrt [6]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}} \, dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \frac {1}{\sqrt {a^2 x^2-b} \sqrt {\sqrt {a^2 x^2-b}+a x} \sqrt [6]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}dx\) |
Int[1/(Sqrt[-b + a^2*x^2]*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*(c + (a*x + Sqrt[ -b + a^2*x^2])^(1/4))^(1/6)),x]
3.30.80.3.1 Defintions of rubi rules used
\[\int \frac {1}{\sqrt {a^{2} x^{2}-b}\, \sqrt {a x +\sqrt {a^{2} x^{2}-b}}\, {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}}}d x\]
int(1/(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))^(1/4))^(1/6),x)
int(1/(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))^(1/4))^(1/6),x)
Time = 0.34 (sec) , antiderivative size = 563, normalized size of antiderivative = 1.47 \[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=-\frac {14 \, a b c^{2} \left (\frac {1}{a^{6} c^{13}}\right )^{\frac {1}{6}} \log \left (a^{5} c^{11} \left (\frac {1}{a^{6} c^{13}}\right )^{\frac {5}{6}} + {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}}\right ) - 14 \, a b c^{2} \left (\frac {1}{a^{6} c^{13}}\right )^{\frac {1}{6}} \log \left (-a^{5} c^{11} \left (\frac {1}{a^{6} c^{13}}\right )^{\frac {5}{6}} + {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}}\right ) - 7 \, {\left (\sqrt {-3} a b c^{2} - a b c^{2}\right )} \left (\frac {1}{a^{6} c^{13}}\right )^{\frac {1}{6}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} a^{5} c^{11} + a^{5} c^{11}\right )} \left (\frac {1}{a^{6} c^{13}}\right )^{\frac {5}{6}} + {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}}\right ) + 7 \, {\left (\sqrt {-3} a b c^{2} - a b c^{2}\right )} \left (\frac {1}{a^{6} c^{13}}\right )^{\frac {1}{6}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} a^{5} c^{11} + a^{5} c^{11}\right )} \left (\frac {1}{a^{6} c^{13}}\right )^{\frac {5}{6}} + {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}}\right ) - 7 \, {\left (\sqrt {-3} a b c^{2} + a b c^{2}\right )} \left (\frac {1}{a^{6} c^{13}}\right )^{\frac {1}{6}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} a^{5} c^{11} - a^{5} c^{11}\right )} \left (\frac {1}{a^{6} c^{13}}\right )^{\frac {5}{6}} + {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}}\right ) + 7 \, {\left (\sqrt {-3} a b c^{2} + a b c^{2}\right )} \left (\frac {1}{a^{6} c^{13}}\right )^{\frac {1}{6}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} a^{5} c^{11} - a^{5} c^{11}\right )} \left (\frac {1}{a^{6} c^{13}}\right )^{\frac {5}{6}} + {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}}\right ) - 12 \, {\left (7 \, {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}} {\left (a x - \sqrt {a^{2} x^{2} - b}\right )} - 6 \, {\left (a c x - \sqrt {a^{2} x^{2} - b} c\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}}\right )} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {5}{6}}}{36 \, a b c^{2}} \]
integrate(1/(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))^(1/4))^(1/6),x, algorithm="fricas")
-1/36*(14*a*b*c^2*(1/(a^6*c^13))^(1/6)*log(a^5*c^11*(1/(a^6*c^13))^(5/6) + (c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/6)) - 14*a*b*c^2*(1/(a^6*c^13))^ (1/6)*log(-a^5*c^11*(1/(a^6*c^13))^(5/6) + (c + (a*x + sqrt(a^2*x^2 - b))^ (1/4))^(1/6)) - 7*(sqrt(-3)*a*b*c^2 - a*b*c^2)*(1/(a^6*c^13))^(1/6)*log(1/ 2*(sqrt(-3)*a^5*c^11 + a^5*c^11)*(1/(a^6*c^13))^(5/6) + (c + (a*x + sqrt(a ^2*x^2 - b))^(1/4))^(1/6)) + 7*(sqrt(-3)*a*b*c^2 - a*b*c^2)*(1/(a^6*c^13)) ^(1/6)*log(-1/2*(sqrt(-3)*a^5*c^11 + a^5*c^11)*(1/(a^6*c^13))^(5/6) + (c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/6)) - 7*(sqrt(-3)*a*b*c^2 + a*b*c^2)* (1/(a^6*c^13))^(1/6)*log(1/2*(sqrt(-3)*a^5*c^11 - a^5*c^11)*(1/(a^6*c^13)) ^(5/6) + (c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/6)) + 7*(sqrt(-3)*a*b*c^ 2 + a*b*c^2)*(1/(a^6*c^13))^(1/6)*log(-1/2*(sqrt(-3)*a^5*c^11 - a^5*c^11)* (1/(a^6*c^13))^(5/6) + (c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/6)) - 12*( 7*(a*x + sqrt(a^2*x^2 - b))^(3/4)*(a*x - sqrt(a^2*x^2 - b)) - 6*(a*c*x - s qrt(a^2*x^2 - b)*c)*sqrt(a*x + sqrt(a^2*x^2 - b)))*(c + (a*x + sqrt(a^2*x^ 2 - b))^(1/4))^(5/6))/(a*b*c^2)
\[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int \frac {1}{\sqrt [6]{c + \sqrt [4]{a x + \sqrt {a^{2} x^{2} - b}}} \sqrt {a x + \sqrt {a^{2} x^{2} - b}} \sqrt {a^{2} x^{2} - b}}\, dx \]
integrate(1/(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))**(1/4))**(1/6),x)
Integral(1/((c + (a*x + sqrt(a**2*x**2 - b))**(1/4))**(1/6)*sqrt(a*x + sqr t(a**2*x**2 - b))*sqrt(a**2*x**2 - b)), x)
\[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int { \frac {1}{\sqrt {a^{2} x^{2} - b} \sqrt {a x + \sqrt {a^{2} x^{2} - b}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}}} \,d x } \]
integrate(1/(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))^(1/4))^(1/6),x, algorithm="maxima")
integrate(1/(sqrt(a^2*x^2 - b)*sqrt(a*x + sqrt(a^2*x^2 - b))*(c + (a*x + s qrt(a^2*x^2 - b))^(1/4))^(1/6)), x)
Timed out. \[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\text {Timed out} \]
integrate(1/(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))^(1/4))^(1/6),x, algorithm="giac")
Timed out. \[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int \frac {1}{\sqrt {a\,x+\sqrt {a^2\,x^2-b}}\,{\left (c+{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}\right )}^{1/6}\,\sqrt {a^2\,x^2-b}} \,d x \]
int(1/((a*x + (a^2*x^2 - b)^(1/2))^(1/2)*(c + (a*x + (a^2*x^2 - b)^(1/2))^ (1/4))^(1/6)*(a^2*x^2 - b)^(1/2)),x)