\(\int \frac {\sqrt {x (-a x+b \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}})}}{x \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx\) [1014]

Optimal result

Integrand size = 57, antiderivative size = 46 \[ \int \frac {\sqrt {x \left (-a x+b \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}\right )}}{x \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\frac {\sqrt {2} b \arcsin \left (\frac {a x-b \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a}}\right )}{\sqrt {a}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2156, 2155, 222} \[ \int \frac {\sqrt {x \left (-a x+b \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}\right )}}{x \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\frac {\sqrt {2} b \arcsin \left (\frac {a x-b \sqrt {\frac {a^2 x^2}{b^2}+\frac {a}{b^2}}}{\sqrt {a}}\right )}{\sqrt {a}} \]

[In]

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]

[Out]

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

Rule 222

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]

Rule 2155

Int[Sqrt[(a_.)*(x_)^2 + (b_.)*(x_)*Sqrt[(c_) + (d_.)*(x_)^2]]/((x_)*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> D
ist[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]
&& EqQ[a^2 - b^2*d, 0] && EqQ[b^2*c + a, 0]

Rule 2156

Int[Sqrt[(e_.)*(x_)*((a_.)*(x_) + (b_.)*Sqrt[(c_) + (d_.)*(x_)^2])]/((x_)*Sqrt[(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]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {-a x^2+b x \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 \\ & = -\frac {\left (\sqrt {2} b\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a}}} \, dx,x,-a x+b \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}\right )}{a} \\ & = \frac {\sqrt {2} b \sin ^{-1}\left (\frac {a x-b \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a}}\right )}{\sqrt {a}} \\ \end{align*}

Mathematica [B] (verified)

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 \[ \int \frac {\sqrt {x \left (-a x+b \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}\right )}}{x \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\frac {\sqrt {2} b \sqrt {x \left (-a x+b \sqrt {\frac {a \left (1+a x^2\right )}{b^2}}\right )} \sqrt {a x \left (a x+b \sqrt {\frac {a \left (1+a x^2\right )}{b^2}}\right )} \arctan \left (\frac {\sqrt {2} \sqrt {a x \left (a x+b \sqrt {\frac {a \left (1+a x^2\right )}{b^2}}\right )}}{\sqrt {a}}\right )}{a^{3/2} x} \]

[In]

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]

[Out]

(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)

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 5.11 (sec) , antiderivative size = 161, normalized size of antiderivative = 3.50 \[ \int \frac {\sqrt {x \left (-a x+b \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}\right )}}{x \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\left [\frac {1}{2} \, \sqrt {2} b \sqrt {-\frac {1}{a}} \log \left (4 \, a x^{2} - 4 \, b x \sqrt {\frac {a^{2} x^{2} + a}{b^{2}}} + 2 \, \sqrt {-a x^{2} + b x \sqrt {\frac {a^{2} x^{2} + a}{b^{2}}}} {\left (\sqrt {2} a x \sqrt {-\frac {1}{a}} - \sqrt {2} b \sqrt {-\frac {1}{a}} \sqrt {\frac {a^{2} x^{2} + a}{b^{2}}}\right )} + 1\right ), -\frac {\sqrt {2} b \arctan \left (\frac {\sqrt {2} \sqrt {-a x^{2} + b x \sqrt {\frac {a^{2} x^{2} + a}{b^{2}}}}}{2 \, \sqrt {a} x}\right )}{\sqrt {a}}\right ] \]

[In]

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")

[Out]

[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*sqr
t(2)*sqrt(-a*x^2 + b*x*sqrt((a^2*x^2 + a)/b^2))/(sqrt(a)*x))/sqrt(a)]

Sympy [F]

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

[In]

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)

[Out]

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]

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

[In]

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")

[Out]

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]

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

[In]

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")

[Out]

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

[In]

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)

[Out]

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)