Optimal. Leaf size=52 \[ x \left (a+b (c x)^n\right )^p \left (\frac{b (c x)^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b (c x)^n}{a}\right ) \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0579275, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ x \left (a+b (c x)^n\right )^p \left (\frac{b (c x)^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b (c x)^n}{a}\right ) \]
Antiderivative was successfully verified.
[In] Int[(a + b*(c*x)^n)^p,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 5.31379, size = 41, normalized size = 0.79 \[ x \left (1 + \frac{b \left (c x\right )^{n}}{a}\right )^{- p} \left (a + b \left (c x\right )^{n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b \left (c x\right )^{n}}{a}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*(c*x)**n)**p,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0295204, size = 52, normalized size = 1. \[ x \left (a+b (c x)^n\right )^p \left (\frac{b (c x)^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b (c x)^n}{a}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*(c*x)^n)^p,x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.087, size = 0, normalized size = 0. \[ \int \left ( a+b \left ( cx \right ) ^{n} \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*(c*x)^n)^p,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (\left (c x\right )^{n} b + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x)^n*b + a)^p,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (\left (c x\right )^{n} b + a\right )}^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x)^n*b + a)^p,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (a + b \left (c x\right )^{n}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b*(c*x)**n)**p,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (\left (c x\right )^{n} b + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x)^n*b + a)^p,x, algorithm="giac")
[Out]