3.1014 \(\int \frac{\sqrt{x (-a x+b \sqrt{\frac{a}{b^2}+\frac{a^2 x^2}{b^2}})}}{x \sqrt{\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}} \, dx\)

Optimal. Leaf size=46 \[ \frac{\sqrt{2} b \sin ^{-1}\left (\frac{a x-b \sqrt{\frac{a^2 x^2}{b^2}+\frac{a}{b^2}}}{\sqrt{a}}\right )}{\sqrt{a}} \]

[Out]

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

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Rubi [A]  time = 1.16756, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 57, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {2131, 2130, 216} \[ \frac{\sqrt{2} b \sin ^{-1}\left (\frac{a x-b \sqrt{\frac{a^2 x^2}{b^2}+\frac{a}{b^2}}}{\sqrt{a}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[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 2131

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]

Rule 2130

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 216

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]

Rubi steps

\begin{align*} \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{-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 ) \operatorname{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]  time = 0.18469, size = 161, normalized size = 3.5 \[ \frac{\sqrt{2} b^2 \sqrt{\frac{a \left (a x^2+1\right )}{b^2}} \sqrt{a x \left (a x-b \sqrt{\frac{a \left (a x^2+1\right )}{b^2}}\right )} \sqrt{x \left (b \sqrt{\frac{a \left (a x^2+1\right )}{b^2}}-a x\right )} \tanh ^{-1}\left (\frac{\sqrt{a x \left (a x-b \sqrt{\frac{a \left (a x^2+1\right )}{b^2}}\right )}}{\sqrt{2} a x}\right )}{a^2 \left (-b x^2 \sqrt{\frac{a \left (a x^2+1\right )}{b^2}}+a x^3+x\right )} \]

Antiderivative was successfully verified.

[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^2*Sqrt[(a*(1 + a*x^2))/b^2]*Sqrt[a*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])]*ArcTanh[Sqrt[a*x*(a*x - b*Sqrt[(a*(1 + a*x^2))/b^2])]/(Sqrt[2]*a*x)])/(a^2*(x + a*x^3 -
b*x^2*Sqrt[(a*(1 + a*x^2))/b^2]))

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Maple [F]  time = 0.023, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}\sqrt{x \left ( \sqrt{{\frac{a}{{b}^{2}}}+{\frac{{a}^{2}{x}^{2}}{{b}^{2}}}}b-ax \right ) }{\frac{1}{\sqrt{{\frac{a}{{b}^{2}}}+{\frac{{a}^{2}{x}^{2}}{{b}^{2}}}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [A]  time = 68.2958, size = 396, normalized size = 8.61 \begin{align*} \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 ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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