3.445 \(\int (x^m (a+b x^{1+m p}))^p \, dx\)

Optimal. Leaf size=44 \[ \frac{x^{-m (p+1)} \left (a x^m+b x^{m p+m+1}\right )^{p+1}}{b (p+1) (m p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0150384, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {1979, 2000} \[ \frac{x^{-m (p+1)} \left (a x^m+b x^{m p+m+1}\right )^{p+1}}{b (p+1) (m p+1)} \]

Antiderivative was successfully verified.

[In]

Int[(x^m*(a + b*x^(1 + m*p)))^p,x]

[Out]

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

Mathematica [A]  time = 0.0025564, size = 43, normalized size = 0.98 \[ \frac{x^{-m (p+1)} \left (x^m \left (a+b x^{m p+1}\right )\right )^{p+1}}{b (p+1) (m p+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^m*(a + b*x^(1 + m*p)))^p,x]

[Out]

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

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Maple [F]  time = 0.388, size = 0, normalized size = 0. \begin{align*} \int \left ({x}^{m} \left ( a+b{x}^{mp+1} \right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((b*x^(m*p + 1) + a)*x^m)^p, x)

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Fricas [A]  time = 0.758676, size = 131, normalized size = 2.98 \begin{align*} \frac{{\left (b x x^{m p + 1} + a x\right )}{\left (b x^{m p + 1} x^{m} + a x^{m}\right )}^{p}}{{\left (b m p^{2} +{\left (b m + b\right )} p + b\right )} x^{m p + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x*x^(m*p + 1) + a*x)*(b*x^(m*p + 1)*x^m + a*x^m)^p/((b*m*p^2 + (b*m + b)*p + b)*x^(m*p + 1))

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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**m*(a+b*x**(m*p+1)))**p,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((b*x^(m*p + 1) + a)*x^m)^p, x)