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

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

[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 = 0.0738037, antiderivative size = 40, normalized size of antiderivative = 1., 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 = {1980, 2014} \[ -\frac{x^{(q+1) (-(n+p))} \left (a x^n+b x^{n+p}\right )^{q+1}}{a p (q+1)} \]

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 1980

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 2014

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] && N
eQ[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*}

Mathematica [A]  time = 0.0397407, size = 38, normalized size = 0.95 \[ -\frac{x^{(q+1) (-(n+p))} \left (x^n \left (a+b x^p\right )\right )^{q+1}}{a p (q+1)} \]

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 = 0.47, size = 0, normalized size = 0. \begin{align*} \int{x}^{-1-nq-p \left ( 1+q \right ) } \left ({x}^{n} \left ( a+b{x}^{p} \right ) \right ) ^{q}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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 = 0.858222, size = 135, normalized size = 3.38 \begin{align*} -\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{align*}

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

[Out]

Timed out

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

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)