3.5.46 \(\int x^n (x^m (a+b x^{1+n+m p}))^p \, dx\) [446]

Optimal. Leaf size=46 \[ \frac {x^{-m (1+p)} \left (a x^m+b x^{1+m+n+m p}\right )^{1+p}}{b (1+p) (1+n+m p)} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2005, 2039} \begin {gather*} \frac {x^{-m (p+1)} \left (a x^m+b x^{m p+m+n+1}\right )^{p+1}}{b (p+1) (m p+n+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2005

Int[(u_)^(p_.)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[(c*x)^m*ExpandToSum[u, x]^p, x] /; FreeQ[{c, m, p}, x] &&
GeneralizedBinomialQ[u, x] &&  !GeneralizedBinomialMatchQ[u, x]

Rule 2039

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

Rubi steps

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

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Mathematica [A]
time = 0.06, size = 45, normalized size = 0.98 \begin {gather*} \frac {x^{-m (1+p)} \left (x^m \left (a+b x^{1+n+m p}\right )\right )^{1+p}}{b (1+p) (1+n+m p)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.20, size = 0, normalized size = 0.00 \[\int x^{n} \left (x^{m} \left (a +b \,x^{m p +n +1}\right )\right )^{p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [A]
time = 2.08, size = 76, normalized size = 1.65 \begin {gather*} \frac {{\left (b x x^{m p + n + 1} x^{n} + a x x^{n}\right )} {\left (b x^{m p + n + 1} x^{m} + a x^{m}\right )}^{p}}{{\left (b m p^{2} + b n + {\left (b m + b n + b\right )} p + b\right )} x^{m p + n + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3434 deep

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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