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

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

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Rubi [A]  time = 0.02, antiderivative size = 44, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {x^{-m (p+1)} \left (a x^m+b x^{m p+m+1}\right )^{p+1}}{b (p+1) (m p+1)} \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 43, normalized size = 0.98 \begin {gather*} \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)} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.02, size = 0, normalized size = 0.00 \begin {gather*} \int \left (x^m \left (a+b x^{1+m p}\right )\right )^p \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.41, size = 61, normalized size = 1.39 \begin {gather*} \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 {gather*}

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left ({\left (b x^{m p + 1} + a\right )} x^{m}\right )^{p}\,{d x} \end {gather*}

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)

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maple [F]  time = 0.85, size = 0, normalized size = 0.00 \begin {gather*} \int \left (\left (b \,x^{m p +1}+a \right ) x^{m}\right )^{p}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left ({\left (b x^{m p + 1} + a\right )} x^{m}\right )^{p}\,{d x} \end {gather*}

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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