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

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

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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} \[ \frac {\left (a+b x^k\right )^{n+1}}{b k (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + k)*(a + b*x^k)^n,x]

[Out]

(a + b*x^k)^(1 + n)/(b*k*(1 + n))

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

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

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + k)*(a + b*x^k)^n,x]

[Out]

(a + b*x^k)^(1 + n)/(b*k*(1 + n))

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{-1+k} \left (a+b x^k\right )^n \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[x^(-1 + k)*(a + b*x^k)^n,x]

[Out]

Could not integrate

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fricas [A]  time = 0.80, size = 27, normalized size = 1.17 \[ \frac {{\left (b x^{k} + a\right )} {\left (b x^{k} + a\right )}^{n}}{b k n + b k} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.98, size = 23, normalized size = 1.00 \[ \frac {{\left (b x^{k} + a\right )}^{n + 1}}{b k {\left (n + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x^k + a)^(n + 1)/(b*k*(n + 1))

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maple [A]  time = 0.32, size = 29, normalized size = 1.26

method	result	size
risch	\(\frac {\left (a +b \,x^{k}\right ) \left (a +b \,x^{k}\right )^{n}}{b \left (1+n \right ) k}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1+k)*(a+b*x^k)^n,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.43, size = 23, normalized size = 1.00 \[ \frac {{\left (b x^{k} + a\right )}^{n + 1}}{b k {\left (n + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x^k + a)^(n + 1)/(b*k*(n + 1))

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mupad [B]  time = 0.73, size = 23, normalized size = 1.00 \[ \frac {{\left (a+b\,x^k\right )}^{n+1}}{b\,k\,\left (n+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(k - 1)*(a + b*x^k)^n,x)

[Out]

(a + b*x^k)^(n + 1)/(b*k*(n + 1))

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sympy [A]  time = 60.37, size = 75, normalized size = 3.26 \[ \begin {cases} \frac {\log {\relax (x )}}{a} & \text {for}\: b = 0 \wedge k = 0 \wedge n = -1 \\\frac {a^{n} x^{k}}{k} & \text {for}\: b = 0 \\\left (a + b\right )^{n} \log {\relax (x )} & \text {for}\: k = 0 \\\frac {\log {\left (\frac {a}{b} + x^{k} \right )}}{b k} & \text {for}\: n = -1 \\\frac {a \left (a + b x^{k}\right )^{n}}{b k n + b k} + \frac {b x^{k} \left (a + b x^{k}\right )^{n}}{b k n + b k} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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