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

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

[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 = 1.83898, 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 \[ \frac{\sqrt{2} b \sinh ^{-1}\left (\frac{b \sqrt{\frac{a^2 x^2}{b^2}-\frac{a}{b^2}}+a x}{\sqrt{a}}\right )}{\sqrt{a}} \]

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]

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Rubi in Sympy [A]  time = 25.9554, size = 41, normalized size = 0.89 \[ \frac{\sqrt{2} b \operatorname{asinh}{\left (\frac{a x + b \sqrt{\frac{a^{2} x^{2}}{b^{2}} - \frac{a}{b^{2}}}}{\sqrt{a}} \right )}}{\sqrt{a}} \]

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

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Mathematica [B]  time = 0.698278, size = 199, normalized size = 4.33 \[ -\frac{x \sqrt{a x \left (b \sqrt{\frac{a \left (a x^2-1\right )}{b^2}}+a x\right )} \left (b x \sqrt{\frac{a \left (a x^2-1\right )}{b^2}}+a x^2-1\right ) \left (\log \left (1-\frac{\sqrt{a x \left (b \sqrt{\frac{a \left (a x^2-1\right )}{b^2}}+a x\right )}}{\sqrt{2} a x}\right )-\log \left (\frac{\sqrt{a x \left (b \sqrt{\frac{a \left (a x^2-1\right )}{b^2}}+a x\right )}}{\sqrt{2} a x}+1\right )\right )}{\sqrt{2} \sqrt{\frac{a \left (a x^2-1\right )}{b^2}} \left (x \left (b \sqrt{\frac{a \left (a x^2-1\right )}{b^2}}+a x\right )\right )^{3/2}} \]

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]

-((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])*(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]*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 = 0.033, 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)