Integrand size = 57, antiderivative size = 46 \[ \int \frac {\sqrt {x \left (-a x+b \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}\right )}}{x \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\frac {\sqrt {2} b \arcsin \left (\frac {a x-b \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a}}\right )}{\sqrt {a}} \]
Leaf count is larger than twice the leaf count of optimal. \(114\) vs. \(2(46)=92\).
Time = 0.01 (sec) , antiderivative size = 114, normalized size of antiderivative = 2.48 \[ \int \frac {\sqrt {x \left (-a x+b \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}\right )}}{x \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\frac {\sqrt {2} b \sqrt {x \left (-a x+b \sqrt {\frac {a \left (1+a x^2\right )}{b^2}}\right )} \sqrt {a x \left (a x+b \sqrt {\frac {a \left (1+a x^2\right )}{b^2}}\right )} \arctan \left (\frac {\sqrt {2} \sqrt {a x \left (a x+b \sqrt {\frac {a \left (1+a x^2\right )}{b^2}}\right )}}{\sqrt {a}}\right )}{a^{3/2} x} \]
(Sqrt[2]*b*Sqrt[x*(-(a*x) + b*Sqrt[(a*(1 + a*x^2))/b^2])]*Sqrt[a*x*(a*x + b*Sqrt[(a*(1 + a*x^2))/b^2])]*ArcTan[(Sqrt[2]*Sqrt[a*x*(a*x + b*Sqrt[(a*(1 + a*x^2))/b^2])])/Sqrt[a]])/(a^(3/2)*x)
Time = 1.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2556, 2555, 223}
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 |
\(\displaystyle -\frac {\sqrt {2} b \int \frac {1}{\sqrt {1-\frac {\left (b \sqrt {\frac {a^2 x^2}{b^2}+\frac {a}{b^2}}-a x\right )^2}{a}}}d\left (b \sqrt {\frac {a^2 x^2}{b^2}+\frac {a}{b^2}}-a x\right )}{a}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle -\frac {\sqrt {2} b \arcsin \left (\frac {b \sqrt {\frac {a^2 x^2}{b^2}+\frac {a}{b^2}}-a x}{\sqrt {a}}\right )}{\sqrt {a}}\) |
3.11.14.3.1 Defintions of rubi rules used
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
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]
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]
\[\int \frac {\sqrt {x \left (-a x +b \sqrt {\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}\right )}}{x \sqrt {\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}d x\]
Time = 5.11 (sec) , antiderivative size = 161, normalized size of antiderivative = 3.50 \[ \int \frac {\sqrt {x \left (-a x+b \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}\right )}}{x \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\left [\frac {1}{2} \, \sqrt {2} b \sqrt {-\frac {1}{a}} \log \left (4 \, a x^{2} - 4 \, b x \sqrt {\frac {a^{2} x^{2} + a}{b^{2}}} + 2 \, \sqrt {-a x^{2} + b x \sqrt {\frac {a^{2} x^{2} + a}{b^{2}}}} {\left (\sqrt {2} a x \sqrt {-\frac {1}{a}} - \sqrt {2} b \sqrt {-\frac {1}{a}} \sqrt {\frac {a^{2} x^{2} + a}{b^{2}}}\right )} + 1\right ), -\frac {\sqrt {2} b \arctan \left (\frac {\sqrt {2} \sqrt {-a x^{2} + b x \sqrt {\frac {a^{2} x^{2} + a}{b^{2}}}}}{2 \, \sqrt {a} x}\right )}{\sqrt {a}}\right ] \]
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")
[1/2*sqrt(2)*b*sqrt(-1/a)*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)*a*x*sqrt(-1/a) - sqrt( 2)*b*sqrt(-1/a)*sqrt((a^2*x^2 + a)/b^2)) + 1), -sqrt(2)*b*arctan(1/2*sqrt( 2)*sqrt(-a*x^2 + b*x*sqrt((a^2*x^2 + a)/b^2))/(sqrt(a)*x))/sqrt(a)]
\[ \int \frac {\sqrt {x \left (-a x+b \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}\right )}}{x \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\int \frac {\sqrt {- x \left (a x - b \sqrt {\frac {a^{2} x^{2}}{b^{2}} + \frac {a}{b^{2}}}\right )}}{x \sqrt {\frac {a \left (a x^{2} + 1\right )}{b^{2}}}}\, dx \]
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)
\[ \int \frac {\sqrt {x \left (-a x+b \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}\right )}}{x \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\int { \frac {\sqrt {-{\left (a x - \sqrt {\frac {a^{2} x^{2}}{b^{2}} + \frac {a}{b^{2}}} b\right )} x}}{\sqrt {\frac {a^{2} x^{2}}{b^{2}} + \frac {a}{b^{2}}} x} \,d x } \]
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")
\[ \int \frac {\sqrt {x \left (-a x+b \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}\right )}}{x \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\int { \frac {\sqrt {-{\left (a x - \sqrt {\frac {a^{2} x^{2}}{b^{2}} + \frac {a}{b^{2}}} b\right )} x}}{\sqrt {\frac {a^{2} x^{2}}{b^{2}} + \frac {a}{b^{2}}} x} \,d x } \]
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")
Timed out. \[ \int \frac {\sqrt {x \left (-a x+b \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}\right )}}{x \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\int \frac {\sqrt {-x\,\left (a\,x-b\,\sqrt {\frac {a}{b^2}+\frac {a^2\,x^2}{b^2}}\right )}}{x\,\sqrt {\frac {a}{b^2}+\frac {a^2\,x^2}{b^2}}} \,d x \]