Integrand size = 59, antiderivative size = 196 \[ \int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\frac {\left (-2-a x^2\right ) \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{b x}+\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}-\frac {\sqrt {a} \log \left (-a x-b \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}+\sqrt {2} \sqrt {a} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}\right )}{\sqrt {2} b} \]
(-a*x^2-2)*(a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2))^(1/2)/b/x+(-a/b^2+a^2*x^ 2/b^2)^(1/2)*(a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2))^(1/2)-1/2*a^(1/2)*ln(- a*x-b*(-a/b^2+a^2*x^2/b^2)^(1/2)+2^(1/2)*a^(1/2)*(a*x^2+b*x*(-a/b^2+a^2*x^ 2/b^2)^(1/2))^(1/2))*2^(1/2)/b
Time = 6.90 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=-\frac {\sqrt {x \left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )} \left (4+2 a x^2-2 b x \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}+\sqrt {2} \sqrt {x \left (-a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )} \arctan \left (\sqrt {2} \sqrt {x \left (-a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )}\right )\right )}{2 b x} \]
Integrate[Sqrt[-(a/b^2) + (a^2*x^2)/b^2]/(x*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]]),x]
-1/2*(Sqrt[x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])]*(4 + 2*a*x^2 - 2*b*x*Sq rt[(a*(-1 + a*x^2))/b^2] + Sqrt[2]*Sqrt[x*(-(a*x) + b*Sqrt[(a*(-1 + a*x^2) )/b^2])]*ArcTan[Sqrt[2]*Sqrt[x*(-(a*x) + b*Sqrt[(a*(-1 + a*x^2))/b^2])]])) /(b*x)
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.
| \(\displaystyle \int \frac {\sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}}{x \sqrt {b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}} \, dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \frac {\sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}}{x \sqrt {b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}}dx\) |
3.25.28.3.1 Defintions of rubi rules used
\[\int \frac {\sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}{x \sqrt {a \,x^{2}+b x \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}}d x\]
Time = 14.72 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.51 \[ \int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\left [\frac {\sqrt {2} \sqrt {a} x \log \left (-4 \, a^{2} x^{2} - 4 \, a b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} - 2 \, {\left (\sqrt {2} a^{\frac {3}{2}} x + \sqrt {2} \sqrt {a} b \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}\right )} \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} + a\right ) - 4 \, \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} {\left (a x^{2} - b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} + 2\right )}}{4 \, b x}, -\frac {\sqrt {2} \sqrt {-a} x \arctan \left (\frac {\sqrt {2} \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} \sqrt {-a}}{2 \, a x}\right ) + 2 \, \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} {\left (a x^{2} - b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} + 2\right )}}{2 \, b x}\right ] \]
integrate((-a/b^2+a^2*x^2/b^2)^(1/2)/x/(a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/ 2))^(1/2),x, algorithm="fricas")
[1/4*(sqrt(2)*sqrt(a)*x*log(-4*a^2*x^2 - 4*a*b*x*sqrt((a^2*x^2 - a)/b^2) - 2*(sqrt(2)*a^(3/2)*x + sqrt(2)*sqrt(a)*b*sqrt((a^2*x^2 - a)/b^2))*sqrt(a* x^2 + b*x*sqrt((a^2*x^2 - a)/b^2)) + a) - 4*sqrt(a*x^2 + b*x*sqrt((a^2*x^2 - a)/b^2))*(a*x^2 - b*x*sqrt((a^2*x^2 - a)/b^2) + 2))/(b*x), -1/2*(sqrt(2 )*sqrt(-a)*x*arctan(1/2*sqrt(2)*sqrt(a*x^2 + b*x*sqrt((a^2*x^2 - a)/b^2))* sqrt(-a)/(a*x)) + 2*sqrt(a*x^2 + b*x*sqrt((a^2*x^2 - a)/b^2))*(a*x^2 - b*x *sqrt((a^2*x^2 - a)/b^2) + 2))/(b*x)]
\[ \int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\int \frac {\sqrt {\frac {a \left (a x^{2} - 1\right )}{b^{2}}}}{x \sqrt {x \left (a x + b \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}\right )}}\, dx \]
integrate((-a/b**2+a**2*x**2/b**2)**(1/2)/x/(a*x**2+b*x*(-a/b**2+a**2*x**2 /b**2)**(1/2))**(1/2),x)
\[ \int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\int { \frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}}{\sqrt {a x^{2} + \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} b x} x} \,d x } \]
integrate((-a/b^2+a^2*x^2/b^2)^(1/2)/x/(a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/ 2))^(1/2),x, algorithm="maxima")
\[ \int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\int { \frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}}{\sqrt {a x^{2} + \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} b x} x} \,d x } \]
integrate((-a/b^2+a^2*x^2/b^2)^(1/2)/x/(a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/ 2))^(1/2),x, algorithm="giac")
Timed out. \[ \int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\int \frac {\sqrt {\frac {a^2\,x^2}{b^2}-\frac {a}{b^2}}}{x\,\sqrt {a\,x^2+b\,x\,\sqrt {\frac {a^2\,x^2}{b^2}-\frac {a}{b^2}}}} \,d x \]