3.2919 \(\int \left (a+b (c x)^n\right )^p \, dx\)

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]

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

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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]

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

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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]

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

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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]

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

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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]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(((c*x)^n*b + a)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(((c*x)^n*b + a)^p, x)

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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]

Integral((a + b*(c*x)**n)**p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(((c*x)^n*b + a)^p, x)