Optimal. Leaf size=45 \[ \frac{2 \sqrt{a+b x^n}}{n}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{n} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0799839, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{2 \sqrt{a+b x^n}}{n}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{n} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*x^n]/x,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 7.61599, size = 37, normalized size = 0.82 \[ - \frac{2 \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{n}}}{\sqrt{a}} \right )}}{n} + \frac{2 \sqrt{a + b x^{n}}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*x**n)**(1/2)/x,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0331685, size = 42, normalized size = 0.93 \[ \frac{2 \left (\sqrt{a+b x^n}-\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )\right )}{n} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b*x^n]/x,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.002, size = 36, normalized size = 0.8 \[{\frac{1}{n} \left ( 2\,\sqrt{a+b{x}^{n}}-2\,\sqrt{a}{\it Artanh} \left ({\frac{\sqrt{a+b{x}^{n}}}{\sqrt{a}}} \right ) \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*x^n)^(1/2)/x,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^n + a)/x,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.229079, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{a} \log \left (\frac{b x^{n} - 2 \, \sqrt{b x^{n} + a} \sqrt{a} + 2 \, a}{x^{n}}\right ) + 2 \, \sqrt{b x^{n} + a}}{n}, -\frac{2 \,{\left (\sqrt{-a} \arctan \left (\frac{\sqrt{b x^{n} + a}}{\sqrt{-a}}\right ) - \sqrt{b x^{n} + a}\right )}}{n}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^n + a)/x,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b*x**n)**(1/2)/x,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{n} + a}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^n + a)/x,x, algorithm="giac")
[Out]