Optimal. Leaf size=54 \[ \frac{2 x^{-n/2} (c x)^{n/2} \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{\sqrt{b} c n} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0781306, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ \frac{2 x^{-n/2} (c x)^{n/2} \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{\sqrt{b} c n} \]
Antiderivative was successfully verified.
[In] Int[(c*x)^(-1 + n/2)/Sqrt[a + b*x^n],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 8.64597, size = 42, normalized size = 0.78 \[ \frac{2 x^{- \frac{n}{2}} \left (c x\right )^{\frac{n}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} x^{\frac{n}{2}}}{\sqrt{a + b x^{n}}} \right )}}{\sqrt{b} c n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x)**(-1+1/2*n)/(a+b*x**n)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0759691, size = 54, normalized size = 1. \[ \frac{2 x^{-n/2} (c x)^{n/2} \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{\sqrt{b} c n} \]
Antiderivative was successfully verified.
[In] Integrate[(c*x)^(-1 + n/2)/Sqrt[a + b*x^n],x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.069, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{-1+{\frac{n}{2}}}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x)^(-1+1/2*n)/(a+b*x^n)^(1/2),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((c*x)^(1/2*n - 1)/sqrt(b*x^n + a),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [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((c*x)^(1/2*n - 1)/sqrt(b*x^n + a),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((c*x)**(-1+1/2*n)/(a+b*x**n)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{\frac{1}{2} \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x)^(1/2*n - 1)/sqrt(b*x^n + a),x, algorithm="giac")
[Out]