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]
_______________________________________________________________________________________
Rubi [A] time = 0.970811, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 59, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.034 \[ \frac{\sqrt{2} b \sinh ^{-1}\left (\frac{b \sqrt{\frac{a^2 x^2}{b^2}-\frac{a}{b^2}}+a x}{\sqrt{a}}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]]/(x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 17.8563, 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((a*x**2+b*x*(-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]
_______________________________________________________________________________________
Mathematica [B] time = 1.1849, 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[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]]/(x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.05, size = 0, normalized size = 0. \[ \int{\frac{1}{x}\sqrt{a{x}^{2}+bx\sqrt{-{\frac{a}{{b}^{2}}}+{\frac{{a}^{2}{x}^{2}}{{b}^{2}}}}}{\frac{1}{\sqrt{-{\frac{a}{{b}^{2}}}+{\frac{{a}^{2}{x}^{2}}{{b}^{2}}}}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a*x^2+b*x*(-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]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a x^{2} + \sqrt{\frac{a^{2} x^{2}}{b^{2}} - \frac{a}{b^{2}}} b x}}{\sqrt{\frac{a^{2} x^{2}}{b^{2}} - \frac{a}{b^{2}}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a*x^2 + 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]
_______________________________________________________________________________________
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^2 + 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]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x \left (a x + b \sqrt{\frac{a^{2} x^{2}}{b^{2}} - \frac{a}{b^{2}}}\right )}}{x \sqrt{\frac{a \left (a x^{2} - 1\right )}{b^{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*x**2+b*x*(-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]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a x^{2} + \sqrt{\frac{a^{2} x^{2}}{b^{2}} - \frac{a}{b^{2}}} b x}}{\sqrt{\frac{a^{2} x^{2}}{b^{2}} - \frac{a}{b^{2}}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a*x^2 + 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]