3.5.54 \(\int x^{-1-n q-p (1+q)} (x^n (a+b x^p))^q \, dx\) [454]

Optimal. Leaf size=40 \[ -\frac {x^{-((n+p) (1+q))} \left (a x^n+b x^{n+p}\right )^{1+q}}{a p (1+q)} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2005, 2039} \begin {gather*} -\frac {x^{-((q+1) (n+p))} \left (a x^n+b x^{n+p}\right )^{q+1}}{a p (q+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]
time = 0.07, size = 38, normalized size = 0.95 \begin {gather*} -\frac {x^{-((n+p) (1+q))} \left (x^n \left (a+b x^p\right )\right )^{1+q}}{a p (1+q)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.18, size = 0, normalized size = 0.00 \[\int x^{-1-q n -p \left (q +1\right )} \left (x^{n} \left (a +b \,x^{p}\right )\right )^{q}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [A]
time = 3.47, size = 64, normalized size = 1.60 \begin {gather*} -\frac {{\left (b x x^{-{\left (n + p\right )} q - p - 1} x^{p} + a x x^{-{\left (n + p\right )} q - p - 1}\right )} {\left (b x^{n} x^{p} + a x^{n}\right )}^{q}}{a p q + a p} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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