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 = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 1.00 \[ \frac {\left (a+b x^n\right )^{p+1}}{b n (p+1)} \]

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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fricas [A]  time = 0.64, size = 27, normalized size = 1.17 \[ \frac {{\left (b x^{n} + a\right )} {\left (b x^{n} + a\right )}^{p}}{b n p + b n} \]

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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giac [A]  time = 0.16, size = 23, normalized size = 1.00 \[ \frac {{\left (b x^{n} + a\right )}^{p + 1}}{b n {\left (p + 1\right )}} \]

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))

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maple [A]  time = 0.05, size = 29, normalized size = 1.26 \[ \frac {\left (b \,x^{n}+a \right ) \left (b \,x^{n}+a \right )^{p}}{\left (p +1\right ) b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.49, size = 23, normalized size = 1.00 \[ \frac {{\left (b x^{n} + a\right )}^{p + 1}}{b n {\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x^n + a)^(p + 1)/(b*n*(p + 1))

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mupad [B]  time = 1.59, size = 23, normalized size = 1.00 \[ \frac {{\left (a+b\,x^n\right )}^{p+1}}{b\,n\,\left (p+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(n - 1)*(a + b*x^n)^p,x)

[Out]

(a + b*x^n)^(p + 1)/(b*n*(p + 1))

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sympy [A]  time = 46.61, size = 75, normalized size = 3.26 \[ \begin {cases} \frac {\log {\relax (x )}}{a} & \text {for}\: b = 0 \wedge n = 0 \wedge p = -1 \\\frac {a^{p} x^{n}}{n} & \text {for}\: b = 0 \\\left (a + b\right )^{p} \log {\relax (x )} & \text {for}\: n = 0 \\\frac {\log {\left (\frac {a}{b} + x^{n} \right )}}{b n} & \text {for}\: p = -1 \\\frac {a \left (a + b x^{n}\right )^{p}}{b n p + b n} + \frac {b x^{n} \left (a + b x^{n}\right )^{p}}{b n p + b n} & \text {otherwise} \end {cases} \]

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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