\(\int x^{-1+n-p (1+q)} (a x^n+b x^p)^q \, dx\) [453]

Optimal result

Integrand size = 25, antiderivative size = 39 \[ \int x^{-1+n-p (1+q)} \left (a x^n+b x^p\right )^q \, dx=\frac {x^{-p (1+q)} \left (a x^n+b x^p\right )^{1+q}}{a (n-p) (1+q)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2039} \[ \int x^{-1+n-p (1+q)} \left (a x^n+b x^p\right )^q \, dx=\frac {x^{-p (q+1)} \left (a x^n+b x^p\right )^{q+1}}{a (q+1) (n-p)} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.03 \[ \int x^{-1+n-p (1+q)} \left (a x^n+b x^p\right )^q \, dx=-\frac {x^{-p (1+q)} \left (a x^n+b x^p\right )^{1+q}}{a (-n+p) (1+q)} \]

[In]

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

[Out]

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

Maple [F]

\[\int x^{-1+n -p \left (q +1\right )} \left (a \,x^{n}+b \,x^{p}\right )^{q}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.95 \[ \int x^{-1+n-p (1+q)} \left (a x^n+b x^p\right )^q \, dx=\frac {{\left (a x x^{-p q + n - p - 1} x^{n} + b x x^{-p q + n - p - 1} x^{p}\right )} {\left (a x^{n} + b x^{p}\right )}^{q}}{{\left (a n - a p + {\left (a n - a p\right )} q\right )} x^{n}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

\[ \int x^{-1+n-p (1+q)} \left (a x^n+b x^p\right )^q \, dx=\int { {\left (a x^{n} + b x^{p}\right )}^{q} x^{-p {\left (q + 1\right )} + n - 1} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int x^{-1+n-p (1+q)} \left (a x^n+b x^p\right )^q \, dx=\int { {\left (a x^{n} + b x^{p}\right )}^{q} x^{-p {\left (q + 1\right )} + n - 1} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int x^{-1+n-p (1+q)} \left (a x^n+b x^p\right )^q \, dx=\int x^{n-p\,\left (q+1\right )-1}\,{\left (a\,x^n+b\,x^p\right )}^q \,d x \]

[In]

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

[Out]

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