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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0248845, antiderivative size = 61, normalized size of antiderivative = 1., 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 = {1979, 2000} \[ \frac{x^{\frac{(1-n) (p+1)}{p}} \left (a x^{-\frac{1-n}{p}}+b x^{n-\frac{1-n}{p}}\right )^{p+1}}{b n (p+1)} \]

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 1979

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

Rule 2000

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*}

Mathematica [A]  time = 0.0250275, size = 45, normalized size = 0.74 \[ \frac{x^{1-n} \left (a+b x^n\right ) \left (x^{\frac{n-1}{p}} \left (a+b x^n\right )\right )^p}{b n (p+1)} \]

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))

________________________________________________________________________________________

Maple [F]  time = 0.395, size = 0, normalized size = 0. \begin{align*} \int \left ({x}^{{\frac{-1+n}{p}}} \left ( a+b{x}^{n} \right ) \right ) ^{p}\, dx \end{align*}

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)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left ({\left (b x^{n} + a\right )} x^{\frac{n - 1}{p}}\right )^{p}\,{d x} \end{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 0.907711, size = 107, normalized size = 1.75 \begin{align*} \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{align*}

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)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left ({\left (b x^{n} + a\right )} x^{\frac{n - 1}{p}}\right )^{p}\,{d x} \end{align*}

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)