Optimal. Leaf size=53 \[ \frac{2 \sqrt{b (c x)^n-a}}{n}-\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )}{n} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.106826, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ \frac{2 \sqrt{b (c x)^n-a}}{n}-\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )}{n} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[-a + b*(c*x)^n]/x,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 5.22331, size = 41, normalized size = 0.77 \[ - \frac{2 \sqrt{a} \operatorname{atan}{\left (\frac{\sqrt{- a + b \left (c x\right )^{n}}}{\sqrt{a}} \right )}}{n} + \frac{2 \sqrt{- a + b \left (c x\right )^{n}}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-a+b*(c*x)**n)**(1/2)/x,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0321183, size = 50, normalized size = 0.94 \[ \frac{2 \left (\sqrt{b (c x)^n-a}-\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )\right )}{n} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[-a + b*(c*x)^n]/x,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.005, size = 46, normalized size = 0.9 \[ -2\,{\frac{\sqrt{a}}{n}\arctan \left ({\frac{\sqrt{-a+b \left ( cx \right ) ^{n}}}{\sqrt{a}}} \right ) }+2\,{\frac{\sqrt{-a+b \left ( cx \right ) ^{n}}}{n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-a+b*(c*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((c*x)^n*b - a)/x,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.284697, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{-a} \log \left (\frac{\left (c x\right )^{n} b - 2 \, \sqrt{\left (c x\right )^{n} b - a} \sqrt{-a} - 2 \, a}{\left (c x\right )^{n}}\right ) + 2 \, \sqrt{\left (c x\right )^{n} b - a}}{n}, -\frac{2 \,{\left (\sqrt{a} \arctan \left (\frac{\sqrt{\left (c x\right )^{n} b - a}}{\sqrt{a}}\right ) - \sqrt{\left (c x\right )^{n} b - a}\right )}}{n}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((c*x)^n*b - a)/x,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- a + b \left (c x\right )^{n}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-a+b*(c*x)**n)**(1/2)/x,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\left (c x\right )^{n} b - a}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((c*x)^n*b - a)/x,x, algorithm="giac")
[Out]