Optimal. Leaf size=77 \[ \frac{a^2 x^3 \left (c x^n\right )^{-3/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^3}-\frac{a x^3 \left (c x^n\right )^{-2/n}}{b^2}+\frac{x^3 \left (c x^n\right )^{-1/n}}{2 b} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0705028, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{a^2 x^3 \left (c x^n\right )^{-3/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^3}-\frac{a x^3 \left (c x^n\right )^{-2/n}}{b^2}+\frac{x^3 \left (c x^n\right )^{-1/n}}{2 b} \]
Antiderivative was successfully verified.
[In] Int[x^2/(a + b*(c*x^n)^n^(-1)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{a^{2} x^{3} \left (c x^{n}\right )^{- \frac{3}{n}} \log{\left (a + b \left (c x^{n}\right )^{\frac{1}{n}} \right )}}{b^{3}} + \frac{x^{3} \left (c x^{n}\right )^{- \frac{3}{n}} \int ^{\left (c x^{n}\right )^{\frac{1}{n}}} x\, dx}{b} - \frac{x^{3} \left (c x^{n}\right )^{- \frac{3}{n}} \int ^{\left (c x^{n}\right )^{\frac{1}{n}}} a\, dx}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(a+b*(c*x**n)**(1/n)),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 4.74079, size = 0, normalized size = 0. \[ \int \frac{x^2}{a+b \left (c x^n\right )^{\frac{1}{n}}} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[x^2/(a + b*(c*x^n)^n^(-1)),x]
[Out]
_______________________________________________________________________________________
Maple [C] time = 0.099, size = 439, normalized size = 5.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(a+b*(c*x^n)^(1/n)),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 23.0289, size = 72, normalized size = 0.94 \[ \frac{a^{2} c^{-\frac{3}{n}} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{b^{3}} + \frac{{\left (b c^{\left (\frac{1}{n}\right )} x^{2} - 2 \, a x\right )} c^{-\frac{2}{n}}}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((c*x^n)^(1/n)*b + a),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.255823, size = 74, normalized size = 0.96 \[ \frac{b^{2} c^{\frac{2}{n}} x^{2} - 2 \, a b c^{\left (\frac{1}{n}\right )} x + 2 \, a^{2} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{2 \, b^{3} c^{\frac{3}{n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((c*x^n)^(1/n)*b + a),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{a + b \left (c x^{n}\right )^{\frac{1}{n}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(a+b*(c*x**n)**(1/n)),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((c*x^n)^(1/n)*b + a),x, algorithm="giac")
[Out]