\(\int \frac {\sqrt {x (a x+b \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}})}}{x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx\) [373]

Optimal result

Integrand size = 58, antiderivative size = 46 \[ \int \frac {\sqrt {x \left (a x+b \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}\right )}}{x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\frac {\sqrt {2} b \text {arcsinh}\left (\frac {a x+b \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a}}\right )}{\sqrt {a}} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(107\) vs. \(2(46)=92\).

Time = 0.01 (sec) , antiderivative size = 107, normalized size of antiderivative = 2.33 \[ \int \frac {\sqrt {x \left (a x+b \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}\right )}}{x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=-\frac {\sqrt {2} b \sqrt {x \left (-a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )} \sqrt {x \left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )} \arctan \left (\sqrt {2} \sqrt {x \left (-a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )}\right )}{a x} \] Input:

Integrate[Sqrt[x*(a*x + b*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])]/(x*Sqrt[-(a/b^2 
) + (a^2*x^2)/b^2]),x]
 

Output:

-((Sqrt[2]*b*Sqrt[x*(-(a*x) + b*Sqrt[(a*(-1 + a*x^2))/b^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])]])/(a*x))
 

Rubi [A] (verified)

Time = 1.71 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.052, Rules used = {2556, 2555, 222}

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.

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

\(\Big \downarrow \) 2556

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

\(\Big \downarrow \) 2555

\(\displaystyle \frac {\sqrt {2} b \int \frac {1}{\sqrt {\frac {\left (\sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}} b+a x\right )^2}{a}+1}}d\left (\sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}} b+a x\right )}{a}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {\sqrt {2} b \text {arcsinh}\left (\frac {b \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x}{\sqrt {a}}\right )}{\sqrt {a}}\)

Input:

Int[Sqrt[x*(a*x + b*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])]/(x*Sqrt[-(a/b^2) + (a 
^2*x^2)/b^2]),x]
 

Output:

(Sqrt[2]*b*ArcSinh[(a*x + b*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])/Sqrt[a]])/Sqrt 
[a]
 

Defintions of rubi rules used

rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt 
[a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
 

rule 2555

Int[Sqrt[(a_.)*(x_)^2 + (b_.)*(x_)*Sqrt[(c_) + (d_.)*(x_)^2]]/((x_)*Sqrt[(c 
_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[Sqrt[2]*(b/a)   Subst[Int[1/Sqrt[1 + 
 x^2/a], x], x, a*x + b*Sqrt[c + d*x^2]], x] /; FreeQ[{a, b, c, d}, x] && E 
qQ[a^2 - b^2*d, 0] && EqQ[b^2*c + a, 0]
 

rule 2556

Int[Sqrt[(e_.)*(x_)*((a_.)*(x_) + (b_.)*Sqrt[(c_) + (d_.)*(x_)^2])]/((x_)*S 
qrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Int[Sqrt[a*e*x^2 + b*e*x*Sqrt[c + d 
*x^2]]/(x*Sqrt[c + d*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a^2 - b^ 
2*d, 0] && EqQ[b^2*c*e + a, 0]
 

Maple [F]

\[\int \frac {\sqrt {x \left (a x +b \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}\right )}}{x \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}d x\]

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (38) = 76\).

Time = 5.29 (sec) , antiderivative size = 191, normalized size of antiderivative = 4.15 \[ \int \frac {\sqrt {x \left (a x+b \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}\right )}}{x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\left [\frac {\sqrt {2} b \log \left (-4 \, a x^{2} - 4 \, b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} - 2 \, \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} {\left (\sqrt {2} \sqrt {a} x + \frac {\sqrt {2} b \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}}{\sqrt {a}}\right )} + 1\right )}{2 \, \sqrt {a}}, \sqrt {2} b \sqrt {-\frac {1}{a}} \arctan \left (-\sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} {\left (\sqrt {2} a x \sqrt {-\frac {1}{a}} - \sqrt {2} b \sqrt {-\frac {1}{a}} \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}\right )}\right )\right ] \] Input:

integrate((x*(a*x+b*(-a/b^2+a^2*x^2/b^2)^(1/2)))^(1/2)/x/(-a/b^2+a^2*x^2/b 
^2)^(1/2),x, algorithm="fricas")
 

Output:

[1/2*sqrt(2)*b*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))*(sqrt(2)*sqrt(a)*x + sqrt(2)*b*sqrt((a^2*x 
^2 - a)/b^2)/sqrt(a)) + 1)/sqrt(a), sqrt(2)*b*sqrt(-1/a)*arctan(-sqrt(a*x^ 
2 + b*x*sqrt((a^2*x^2 - a)/b^2))*(sqrt(2)*a*x*sqrt(-1/a) - sqrt(2)*b*sqrt( 
-1/a)*sqrt((a^2*x^2 - a)/b^2)))]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((x*(a*x+b*(-a/b^2+a^2*x^2/b^2)^(1/2)))^(1/2)/x/(-a/b^2+a^2*x^2/b 
^2)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((x*(a*x+b*(-a/b^2+a^2*x^2/b^2)^(1/2)))^(1/2)/x/(-a/b^2+a^2*x^2/b 
^2)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.39 \[ \int \frac {\sqrt {x \left (a x+b \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}\right )}}{x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\frac {\sqrt {a}\, \sqrt {2}\, b \left (-\mathrm {log}\left (\sqrt {\sqrt {a}\, \sqrt {a \,x^{2}-1}+a x}-\sqrt {x}\, \sqrt {a}\, \sqrt {2}\right )+\mathrm {log}\left (\sqrt {\sqrt {a}\, \sqrt {a \,x^{2}-1}+a x}+\sqrt {x}\, \sqrt {a}\, \sqrt {2}\right )\right )}{2 a} \] Input:

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

Output:

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