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

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 = 1.84009, 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 \[ \frac{\sqrt{2} b \sin ^{-1}\left (\frac{a x-b \sqrt{\frac{a^2 x^2}{b^2}+\frac{a}{b^2}}}{\sqrt{a}}\right )}{\sqrt{a}} \]

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]

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Rubi in Sympy [A]  time = 26.3129, size = 42, normalized size = 0.91 \[ - \frac{\sqrt{2} b \operatorname{asin}{\left (\frac{- a x + b \sqrt{\frac{a^{2} x^{2}}{b^{2}} + \frac{a}{b^{2}}}}{\sqrt{a}} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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]

-sqrt(2)*b*asin((-a*x + b*sqrt(a**2*x**2/b**2 + a/b**2))/sqrt(a))/sqrt(a)

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Mathematica [B]  time = 0.320454, size = 213, normalized size = 4.63 \[ \frac{b^2 \sqrt{\frac{a \left (a x^2+1\right )}{b^2}} \sqrt{a x \left (a x-b \sqrt{\frac{a \left (a x^2+1\right )}{b^2}}\right )} \sqrt{x \left (b \sqrt{\frac{a \left (a x^2+1\right )}{b^2}}-a x\right )} \left (\log \left (1-\frac{\sqrt{a x \left (a x-b \sqrt{\frac{a \left (a x^2+1\right )}{b^2}}\right )}}{\sqrt{2} a x}\right )-\log \left (\frac{\sqrt{a x \left (a x-b \sqrt{\frac{a \left (a x^2+1\right )}{b^2}}\right )}}{\sqrt{2} a x}+1\right )\right )}{\sqrt{2} a^2 x \left (b x \sqrt{\frac{a \left (a x^2+1\right )}{b^2}}-a x^2-1\right )} \]

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]

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

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Maple [F]  time = 0.035, size = 0, normalized size = 0. \[ \int{\frac{1}{x}\sqrt{x \left ( -ax+b\sqrt{{\frac{a}{{b}^{2}}}+{\frac{{a}^{2}{x}^{2}}{{b}^{2}}}} \right ) }{\frac{1}{\sqrt{{\frac{a}{{b}^{2}}}+{\frac{{a}^{2}{x}^{2}}{{b}^{2}}}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-{\left (a x - \sqrt{\frac{a^{2} x^{2}}{b^{2}} + \frac{a}{b^{2}}} b\right )} x}}{\sqrt{\frac{a^{2} x^{2}}{b^{2}} + \frac{a}{b^{2}}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  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, 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 [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  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, algorithm="fricas")

[Out]

Timed out

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

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/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-{\left (a x - \sqrt{\frac{a^{2} x^{2}}{b^{2}} + \frac{a}{b^{2}}} b\right )} x}}{\sqrt{\frac{a^{2} x^{2}}{b^{2}} + \frac{a}{b^{2}}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  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, 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)