3.1013 \(\int \frac{\sqrt{x (a x+b \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}})}}{x \sqrt{-\frac{a}{b^2}+\frac{a^2 x^2}{b^2}}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 1.17126, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 58, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.052, Rules used = {2131, 2130, 215} \[ \frac{\sqrt{2} b \sinh ^{-1}\left (\frac{b \sqrt{\frac{a^2 x^2}{b^2}-\frac{a}{b^2}}+a x}{\sqrt{a}}\right )}{\sqrt{a}} \]

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*ArcSinh[(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 215

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]

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 \sinh ^{-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 = 0.161254, size = 148, normalized size = 3.22 \[ \frac{\sqrt{2} x \sqrt{a x \left (b \sqrt{\frac{a \left (a x^2-1\right )}{b^2}}+a x\right )} \left (b x \sqrt{\frac{a \left (a x^2-1\right )}{b^2}}+a x^2-1\right ) \tanh ^{-1}\left (\frac{\sqrt{a x \left (b \sqrt{\frac{a \left (a x^2-1\right )}{b^2}}+a x\right )}}{\sqrt{2} a x}\right )}{\sqrt{\frac{a \left (a x^2-1\right )}{b^2}} \left (x \left (b \sqrt{\frac{a \left (a x^2-1\right )}{b^2}}+a x\right )\right )^{3/2}} \]

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

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Maple [F]  time = 0.02, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}\sqrt{x \left ( ax+\sqrt{-{\frac{a}{{b}^{2}}}+{\frac{{a}^{2}{x}^{2}}{{b}^{2}}}}b \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*x+(-a/b^2+a^2*x^2/b^2)^(1/2)*b))^(1/2)/x/(-a/b^2+a^2*x^2/b^2)^(1/2),x)

[Out]

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

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Fricas [A]  time = 71.9513, size = 385, normalized size = 8.37 \begin{align*} \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 (\frac{\sqrt{2} \sqrt{a x^{2} + b x \sqrt{\frac{a^{2} x^{2} - a}{b^{2}}}} \sqrt{-\frac{1}{a}}}{2 \, x}\right )\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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*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))*(sqr
t(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(1/2*sqr
t(2)*sqrt(a*x^2 + b*x*sqrt((a^2*x^2 - a)/b^2))*sqrt(-1/a)/x)]

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