3.5.52 \(\int (x^{\frac {-1+n}{p}} (a+b x^n))^p \, dx\) [452]

Optimal. Leaf size=61 \[ \frac {x^{\frac {(1-n) (1+p)}{p}} \left (b x^{n-\frac {1-n}{p}}+a x^{-\frac {1-n}{p}}\right )^{1+p}}{b n (1+p)} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2004, 2025} \begin {gather*} \frac {x^{\frac {(1-n) (p+1)}{p}} \left (a x^{-\frac {1-n}{p}}+b x^{n-\frac {1-n}{p}}\right )^{p+1}}{b n (p+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^((-1 + n)/p)*(a + b*x^n))^p,x]

[Out]

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

Rule 2004

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && GeneralizedBinomialQ[u, x] &&  !Gene
ralizedBinomialMatchQ[u, x]

Rule 2025

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + b*x^n)^(p + 1)/(b*(n - j)*(p + 1)*x
^(n - 1)), x] /; FreeQ[{a, b, j, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && EqQ[j*p - n + j + 1, 0]

Rubi steps

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

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Mathematica [A]
time = 0.04, size = 45, normalized size = 0.74 \begin {gather*} \frac {x^{1-n} \left (a+b x^n\right ) \left (x^{\frac {-1+n}{p}} \left (a+b x^n\right )\right )^p}{b n (1+p)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \left (x^{\frac {-1+n}{p}} \left (a +b \,x^{n}\right )\right )^{p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 2.40, size = 54, normalized size = 0.89 \begin {gather*} \frac {{\left (b x x^{n} + a x\right )} {\left (b x^{n} x^{\frac {n - 1}{p}} + a x^{\frac {n - 1}{p}}\right )}^{p}}{{\left (b n p + b n\right )} x^{n}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (x^{\frac {n - 1}{p}} \left (a + b x^{n}\right )\right )^{p}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (x^{\frac {n-1}{p}}\,\left (a+b\,x^n\right )\right )}^p \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^((n - 1)/p)*(a + b*x^n))^p,x)

[Out]

int((x^((n - 1)/p)*(a + b*x^n))^p, x)

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