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

Optimal. Leaf size=46 \[ x \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b x^n}{a}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0316681, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ x \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b x^n}{a}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^n)^p,x]

[Out]

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

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Rubi in Sympy [A]  time = 3.70442, size = 36, normalized size = 0.78 \[ x \left (1 + \frac{b x^{n}}{a}\right )^{- p} \left (a + b x^{n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**n)**p,x)

[Out]

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

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Mathematica [A]  time = 0.0267131, size = 46, normalized size = 1. \[ x \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b x^n}{a}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^n)^p,x]

[Out]

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

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Maple [F]  time = 0.002, size = 0, normalized size = 0. \[ \int \left ( a+b{x}^{n} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^n)^p,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^n + a)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^n + a)^p, x)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**n)**p,x)

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^n + a)^p, x)