\(\int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx\) [2428]

Optimal result

Integrand size = 59, antiderivative size = 196 \[ \int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\frac {\left (-2-a x^2\right ) \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{b x}+\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}-\frac {\sqrt {a} \log \left (-a x-b \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}+\sqrt {2} \sqrt {a} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}\right )}{\sqrt {2} b} \] Output:

(-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
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 6.92 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=-\frac {\sqrt {x \left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )} \left (4+2 a x^2-2 b x \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}+\sqrt {2} \sqrt {x \left (-a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )} \arctan \left (\sqrt {2} \sqrt {x \left (-a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )}\right )\right )}{2 b x} \] Input:

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]
 

Output:

-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)
 

Rubi [F]

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

\(\displaystyle \int \frac {\sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}}{x \sqrt {b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}}dx\)

Input:

Int[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]
 

Output:

$Aborted
 

Maple [F]

\[\int \frac {\sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}{x \sqrt {a \,x^{2}+b x \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}}d x\]

Input:

int((-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)
 

Output:

int((-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)
 

Fricas [A] (verification not implemented)

Time = 13.22 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.65 \[ \int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\left [\frac {\sqrt {2} \sqrt {a} x \log \left (-4 \, a^{2} x^{2} - 4 \, a b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} - 2 \, {\left (\sqrt {2} a^{\frac {3}{2}} x + \sqrt {2} \sqrt {a} b \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}\right )} \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} + a\right ) - 4 \, \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} {\left (a x^{2} - b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} + 2\right )}}{4 \, b x}, \frac {\sqrt {2} \sqrt {-a} x \arctan \left (-\frac {{\left (\sqrt {2} \sqrt {-a} a x - \sqrt {2} \sqrt {-a} b \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}\right )} \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}}}{a}\right ) - 2 \, \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} {\left (a x^{2} - b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} + 2\right )}}{2 \, b x}\right ] \] Input:

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")
 

Output:

[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(-(sqrt(2)*sqrt(-a)*a*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) - 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)]
 

Sympy [F]

\[ \int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\int \frac {\sqrt {\frac {a \left (a x^{2} - 1\right )}{b^{2}}}}{x \sqrt {x \left (a x + b \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}\right )}}\, dx \] Input:

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)
 

Output:

Integral(sqrt(a*(a*x**2 - 1)/b**2)/(x*sqrt(x*(a*x + b*sqrt(a**2*x**2/b**2 
- a/b**2)))), x)
 

Maxima [F]

\[ \int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\int { \frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}}{\sqrt {a x^{2} + \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} b x} x} \,d x } \] Input:

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")
 

Output:

integrate(sqrt(a^2*x^2/b^2 - a/b^2)/(sqrt(a*x^2 + sqrt(a^2*x^2/b^2 - a/b^2 
)*b*x)*x), x)
 

Giac [F]

\[ \int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\int { \frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}}{\sqrt {a x^{2} + \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} b x} x} \,d x } \] Input:

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")
 

Output:

integrate(sqrt(a^2*x^2/b^2 - a/b^2)/(sqrt(a*x^2 + sqrt(a^2*x^2/b^2 - a/b^2 
)*b*x)*x), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\int \frac {\sqrt {\frac {a^2\,x^2}{b^2}-\frac {a}{b^2}}}{x\,\sqrt {a\,x^2+b\,x\,\sqrt {\frac {a^2\,x^2}{b^2}-\frac {a}{b^2}}}} \,d x \] Input:

int(((a^2*x^2)/b^2 - a/b^2)^(1/2)/(x*(a*x^2 + b*x*((a^2*x^2)/b^2 - a/b^2)^ 
(1/2))^(1/2)),x)
 

Output:

int(((a^2*x^2)/b^2 - a/b^2)^(1/2)/(x*(a*x^2 + b*x*((a^2*x^2)/b^2 - a/b^2)^ 
(1/2))^(1/2)), x)
 

Reduce [F]

\[ \int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\frac {2 \sqrt {x}\, \sqrt {a}\, \sqrt {a \,x^{2}-1}\, \sqrt {\sqrt {a}\, \sqrt {a \,x^{2}-1}+a x}+\sqrt {a}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {a}\, \sqrt {a \,x^{2}-1}+a x}-\sqrt {x}\, \sqrt {a}\, \sqrt {2}\right )-\sqrt {a}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {a}\, \sqrt {a \,x^{2}-1}+a x}+\sqrt {x}\, \sqrt {a}\, \sqrt {2}\right )-4 \left (\int \frac {\sqrt {x}\, \sqrt {\sqrt {a}\, \sqrt {a \,x^{2}-1}+a x}\, x^{2}}{a \,x^{2}-1}d x \right ) a^{2}-3 \left (\int \frac {\sqrt {x}\, \sqrt {\sqrt {a}\, \sqrt {a \,x^{2}-1}+a x}}{a \,x^{4}-x^{2}}d x \right )+7 \left (\int \frac {\sqrt {x}\, \sqrt {\sqrt {a}\, \sqrt {a \,x^{2}-1}+a x}}{a \,x^{2}-1}d x \right ) a}{3 b} \] Input:

int((-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)
 

Output:

(2*sqrt(x)*sqrt(a)*sqrt(a*x**2 - 1)*sqrt(sqrt(a)*sqrt(a*x**2 - 1) + a*x) + 
 sqrt(a)*sqrt(2)*log(sqrt(sqrt(a)*sqrt(a*x**2 - 1) + a*x) - sqrt(x)*sqrt(a 
)*sqrt(2)) - sqrt(a)*sqrt(2)*log(sqrt(sqrt(a)*sqrt(a*x**2 - 1) + a*x) + sq 
rt(x)*sqrt(a)*sqrt(2)) - 4*int((sqrt(x)*sqrt(sqrt(a)*sqrt(a*x**2 - 1) + a* 
x)*x**2)/(a*x**2 - 1),x)*a**2 - 3*int((sqrt(x)*sqrt(sqrt(a)*sqrt(a*x**2 - 
1) + a*x))/(a*x**4 - x**2),x) + 7*int((sqrt(x)*sqrt(sqrt(a)*sqrt(a*x**2 - 
1) + a*x))/(a*x**2 - 1),x)*a)/(3*b)