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