Integrand size = 59, antiderivative size = 167 \[ \int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x^2 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\frac {2 \left (-1+2 a x^2\right ) \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{3 b x^2}-\frac {4 \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{3 x}-\frac {\sqrt {2} a \text {arctanh}\left (\sqrt {2} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}\right )}{b} \]
2/3*(2*a*x^2-1)*(a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2))^(1/2)/b/x^2-4/3*(-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)/x-2^( 1/2)*a*arctanh(2^(1/2)*(a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2))^(1/2))/b
Result contains complex when optimal does not.
Time = 8.98 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.89 \[ \int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x^2 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\frac {\left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )^2 \left (-4+8 a x^2-8 b x \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}+3 i \sqrt {2} \sqrt {a} x \sqrt {x \left (-a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )} \log \left (b \left (i a x-i b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}+\sqrt {2} \sqrt {a} \sqrt {x \left (-a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )}\right )\right )-3 i \sqrt {2} \sqrt {a} x \sqrt {x \left (-a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )} \log \left (-i a x+i b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}+\sqrt {2} \sqrt {a} \sqrt {x \left (-a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )}\right )\right )}{6 b \left (x \left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )\right )^{3/2}} \]
Integrate[Sqrt[-(a/b^2) + (a^2*x^2)/b^2]/(x^2*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^ 2) + (a^2*x^2)/b^2]]),x]
((a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])^2*(-4 + 8*a*x^2 - 8*b*x*Sqrt[(a*(-1 + a*x^2))/b^2] + (3*I)*Sqrt[2]*Sqrt[a]*x*Sqrt[x*(-(a*x) + b*Sqrt[(a*(-1 + a*x^2))/b^2])]*Log[b*(I*a*x - I*b*Sqrt[(a*(-1 + a*x^2))/b^2] + Sqrt[2]*Sqr t[a]*Sqrt[x*(-(a*x) + b*Sqrt[(a*(-1 + a*x^2))/b^2])])] - (3*I)*Sqrt[2]*Sqr t[a]*x*Sqrt[x*(-(a*x) + b*Sqrt[(a*(-1 + a*x^2))/b^2])]*Log[(-I)*a*x + I*b* Sqrt[(a*(-1 + a*x^2))/b^2] + Sqrt[2]*Sqrt[a]*Sqrt[x*(-(a*x) + b*Sqrt[(a*(- 1 + a*x^2))/b^2])]]))/(6*b*(x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2]))^(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.
| \(\displaystyle \int \frac {\sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}}{x^2 \sqrt {b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}} \, dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \frac {\sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}}{x^2 \sqrt {b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}}dx\) |
3.23.45.3.1 Defintions of rubi rules used
\[\int \frac {\sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}{x^{2} \sqrt {a \,x^{2}+b x \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}}d x\]
Time = 18.13 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x^2 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\frac {3 \, \sqrt {2} a x^{2} \log \left (4 \, a x^{2} - 4 \, b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} + 2 \, \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} {\left (2 \, \sqrt {2} b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} - \sqrt {2} {\left (2 \, a x^{2} - 1\right )}\right )} - 1\right ) + 4 \, {\left (2 \, a x^{2} - 2 \, b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} - 1\right )} \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}}}{6 \, b x^{2}} \]
integrate((-a/b^2+a^2*x^2/b^2)^(1/2)/x^2/(a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^( 1/2))^(1/2),x, algorithm="fricas")
1/6*(3*sqrt(2)*a*x^2*log(4*a*x^2 - 4*b*x*sqrt((a^2*x^2 - a)/b^2) + 2*sqrt( a*x^2 + b*x*sqrt((a^2*x^2 - a)/b^2))*(2*sqrt(2)*b*x*sqrt((a^2*x^2 - a)/b^2 ) - sqrt(2)*(2*a*x^2 - 1)) - 1) + 4*(2*a*x^2 - 2*b*x*sqrt((a^2*x^2 - a)/b^ 2) - 1)*sqrt(a*x^2 + b*x*sqrt((a^2*x^2 - a)/b^2)))/(b*x^2)
\[ \int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x^2 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\int \frac {\sqrt {\frac {a \left (a x^{2} - 1\right )}{b^{2}}}}{x^{2} \sqrt {x \left (a x + b \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}\right )}}\, dx \]
integrate((-a/b**2+a**2*x**2/b**2)**(1/2)/x**2/(a*x**2+b*x*(-a/b**2+a**2*x **2/b**2)**(1/2))**(1/2),x)
Integral(sqrt(a*(a*x**2 - 1)/b**2)/(x**2*sqrt(x*(a*x + b*sqrt(a**2*x**2/b* *2 - a/b**2)))), x)
\[ \int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x^2 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\int { \frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}}{\sqrt {a x^{2} + \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} b x} x^{2}} \,d x } \]
integrate((-a/b^2+a^2*x^2/b^2)^(1/2)/x^2/(a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^( 1/2))^(1/2),x, algorithm="maxima")
\[ \int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x^2 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\int { \frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}}{\sqrt {a x^{2} + \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} b x} x^{2}} \,d x } \]
integrate((-a/b^2+a^2*x^2/b^2)^(1/2)/x^2/(a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^( 1/2))^(1/2),x, algorithm="giac")
Timed out. \[ \int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x^2 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\int \frac {\sqrt {\frac {a^2\,x^2}{b^2}-\frac {a}{b^2}}}{x^2\,\sqrt {a\,x^2+b\,x\,\sqrt {\frac {a^2\,x^2}{b^2}-\frac {a}{b^2}}}} \,d x \]