3.2720 \(\int x^{-1+n} (a+b x^n)^p \, dx\)

Optimal. Leaf size=23 \[ \frac{\left (a+b x^n\right )^{p+1}}{b n (p+1)} \]

[Out]

(a + b*x^n)^(1 + p)/(b*n*(1 + p))

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Rubi [A]  time = 0.0083898, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {261} \[ \frac{\left (a+b x^n\right )^{p+1}}{b n (p+1)} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + n)*(a + b*x^n)^p,x]

[Out]

(a + b*x^n)^(1 + p)/(b*n*(1 + p))

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int x^{-1+n} \left (a+b x^n\right )^p \, dx &=\frac{\left (a+b x^n\right )^{1+p}}{b n (1+p)}\\ \end{align*}

Mathematica [A]  time = 0.0066652, size = 23, normalized size = 1. \[ \frac{\left (a+b x^n\right )^{p+1}}{b n (p+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + n)*(a + b*x^n)^p,x]

[Out]

(a + b*x^n)^(1 + p)/(b*n*(1 + p))

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Maple [A]  time = 0.039, size = 29, normalized size = 1.3 \begin{align*}{\frac{ \left ( a+b{x}^{n} \right ) \left ( a+b{x}^{n} \right ) ^{p}}{b \left ( 1+p \right ) n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1+n)*(a+b*x^n)^p,x)

[Out]

(a+b*x^n)/b/(1+p)/n*(a+b*x^n)^p

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*(a+b*x^n)^p,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.33997, size = 55, normalized size = 2.39 \begin{align*} \frac{{\left (b x^{n} + a\right )}{\left (b x^{n} + a\right )}^{p}}{b n p + b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*(a+b*x^n)^p,x, algorithm="fricas")

[Out]

(b*x^n + a)*(b*x^n + a)^p/(b*n*p + b*n)

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Sympy [A]  time = 132.362, size = 75, normalized size = 3.26 \begin{align*} \begin{cases} \frac{\log{\left (x \right )}}{a} & \text{for}\: b = 0 \wedge n = 0 \wedge p = -1 \\\frac{a^{p} x^{n}}{n} & \text{for}\: b = 0 \\\left (a + b\right )^{p} \log{\left (x \right )} & \text{for}\: n = 0 \\\frac{\log{\left (\frac{a}{b} + x^{n} \right )}}{b n} & \text{for}\: p = -1 \\\frac{a \left (a + b x^{n}\right )^{p}}{b n p + b n} + \frac{b x^{n} \left (a + b x^{n}\right )^{p}}{b n p + b n} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((log(x)/a, Eq(b, 0) & Eq(n, 0) & Eq(p, -1)), (a**p*x**n/n, Eq(b, 0)), ((a + b)**p*log(x), Eq(n, 0)),
 (log(a/b + x**n)/(b*n), Eq(p, -1)), (a*(a + b*x**n)**p/(b*n*p + b*n) + b*x**n*(a + b*x**n)**p/(b*n*p + b*n),
True))

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Giac [A]  time = 1.11277, size = 31, normalized size = 1.35 \begin{align*} \frac{{\left (b x^{n} + a\right )}^{p + 1}}{b n{\left (p + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*(a+b*x^n)^p,x, algorithm="giac")

[Out]

(b*x^n + a)^(p + 1)/(b*n*(p + 1))