3.2918 \(\int x \left (a+b (c x)^n\right )^p \, dx\)

Optimal. Leaf size=61 \[ \frac{1}{2} x^2 \left (a+b (c x)^n\right )^p \left (\frac{b (c x)^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{2}{n},-p;\frac{n+2}{n};-\frac{b (c x)^n}{a}\right ) \]

[Out]

(x^2*(a + b*(c*x)^n)^p*Hypergeometric2F1[2/n, -p, (2 + n)/n, -((b*(c*x)^n)/a)])/
(2*(1 + (b*(c*x)^n)/a)^p)

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Rubi [A]  time = 0.0862629, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{1}{2} x^2 \left (a+b (c x)^n\right )^p \left (\frac{b (c x)^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{2}{n},-p;\frac{n+2}{n};-\frac{b (c x)^n}{a}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x*(a + b*(c*x)^n)^p,x]

[Out]

(x^2*(a + b*(c*x)^n)^p*Hypergeometric2F1[2/n, -p, (2 + n)/n, -((b*(c*x)^n)/a)])/
(2*(1 + (b*(c*x)^n)/a)^p)

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Rubi in Sympy [A]  time = 9.20675, size = 44, normalized size = 0.72 \[ \frac{x^{2} \left (1 + \frac{b \left (c x\right )^{n}}{a}\right )^{- p} \left (a + b \left (c x\right )^{n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{2}{n} \\ \frac{n + 2}{n} \end{matrix}\middle |{- \frac{b \left (c x\right )^{n}}{a}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(a+b*(c*x)**n)**p,x)

[Out]

x**2*(1 + b*(c*x)**n/a)**(-p)*(a + b*(c*x)**n)**p*hyper((-p, 2/n), ((n + 2)/n,),
-b*(c*x)**n/a)/2

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Mathematica [A]  time = 0.0391579, size = 61, normalized size = 1. \[ \frac{1}{2} x^2 \left (a+b (c x)^n\right )^p \left (\frac{b (c x)^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{2}{n},-p;1+\frac{2}{n};-\frac{b (c x)^n}{a}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x*(a + b*(c*x)^n)^p,x]

[Out]

(x^2*(a + b*(c*x)^n)^p*Hypergeometric2F1[2/n, -p, 1 + 2/n, -((b*(c*x)^n)/a)])/(2
*(1 + (b*(c*x)^n)/a)^p)

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Maple [F]  time = 0.069, size = 0, normalized size = 0. \[ \int x \left ( a+b \left ( cx \right ) ^{n} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(a+b*(c*x)^n)^p,x)

[Out]

int(x*(a+b*(c*x)^n)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (\left (c x\right )^{n} b + a\right )}^{p} x\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((c*x)^n*b + a)^p*x,x, algorithm="maxima")

[Out]

integrate(((c*x)^n*b + a)^p*x, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (\left (c x\right )^{n} b + a\right )}^{p} x, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((c*x)^n*b + a)^p*x,x, algorithm="fricas")

[Out]

integral(((c*x)^n*b + a)^p*x, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x \left (a + b \left (c x\right )^{n}\right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(a+b*(c*x)**n)**p,x)

[Out]

Integral(x*(a + b*(c*x)**n)**p, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (\left (c x\right )^{n} b + a\right )}^{p} x\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((c*x)^n*b + a)^p*x,x, algorithm="giac")

[Out]

integrate(((c*x)^n*b + a)^p*x, x)