Optimal. Leaf size=61 \[ \frac{1}{3} x^3 \left (a+b (c x)^n\right )^p \left (\frac{b (c x)^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{3}{n},-p;\frac{n+3}{n};-\frac{b (c x)^n}{a}\right ) \]
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Rubi [A] time = 0.0961511, antiderivative size = 61, 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 \[ \frac{1}{3} x^3 \left (a+b (c x)^n\right )^p \left (\frac{b (c x)^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{3}{n},-p;\frac{n+3}{n};-\frac{b (c x)^n}{a}\right ) \]
Antiderivative was successfully verified.
[In] Int[x^2*(a + b*(c*x)^n)^p,x]
[Out]
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Rubi in Sympy [A] time = 10.0853, size = 44, normalized size = 0.72 \[ \frac{x^{3} \left (1 + \frac{b \left (c x\right )^{n}}{a}\right )^{- p} \left (a + b \left (c x\right )^{n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{3}{n} \\ \frac{n + 3}{n} \end{matrix}\middle |{- \frac{b \left (c x\right )^{n}}{a}} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(a+b*(c*x)**n)**p,x)
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Mathematica [A] time = 0.045285, size = 61, normalized size = 1. \[ \frac{1}{3} x^3 \left (a+b (c x)^n\right )^p \left (\frac{b (c x)^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{3}{n},-p;1+\frac{3}{n};-\frac{b (c x)^n}{a}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^2*(a + b*(c*x)^n)^p,x]
[Out]
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Maple [F] time = 0.071, size = 0, normalized size = 0. \[ \int{x}^{2} \left ( a+b \left ( cx \right ) ^{n} \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(a+b*(c*x)^n)^p,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (\left (c x\right )^{n} b + a\right )}^{p} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x)^n*b + a)^p*x^2,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (\left (c x\right )^{n} b + a\right )}^{p} x^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x)^n*b + a)^p*x^2,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int x^{2} \left (a + b \left (c x\right )^{n}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(a+b*(c*x)**n)**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (\left (c x\right )^{n} b + a\right )}^{p} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x)^n*b + a)^p*x^2,x, algorithm="giac")
[Out]