\(\int x^{-2+n} (a+b x)^{-n} \, dx\) [741]

Optimal result

Integrand size = 15, antiderivative size = 28 \[ \int x^{-2+n} (a+b x)^{-n} \, dx=-\frac {x^{-1+n} (a+b x)^{1-n}}{a (1-n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {37} \[ \int x^{-2+n} (a+b x)^{-n} \, dx=-\frac {x^{n-1} (a+b x)^{1-n}}{a (1-n)} \]

[In]

Int[x^(-2 + n)/(a + b*x)^n,x]

[Out]

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {x^{-1+n} (a+b x)^{1-n}}{a (1-n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int x^{-2+n} (a+b x)^{-n} \, dx=\frac {x^{-1+n} (a+b x)^{1-n}}{a (-1+n)} \]

[In]

Integrate[x^(-2 + n)/(a + b*x)^n,x]

[Out]

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

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04

method	result	size
gosper	\(\frac {x^{-1+n} \left (b x +a \right ) \left (b x +a \right )^{-n}}{a \left (-1+n \right )}\)	\(29\)
parallelrisch	\(\frac {\left (x^{2} x^{-2+n} b +x \,x^{-2+n} a \right ) \left (b x +a \right )^{-n}}{a \left (-1+n \right )}\)	\(38\)

[In]

int(x^(-2+n)/((b*x+a)^n),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18 \[ \int x^{-2+n} (a+b x)^{-n} \, dx=\frac {{\left (b x^{2} + a x\right )} x^{n - 2}}{{\left (a n - a\right )} {\left (b x + a\right )}^{n}} \]

[In]

integrate(x^(-2+n)/((b*x+a)^n),x, algorithm="fricas")

[Out]

(b*x^2 + a*x)*x^(n - 2)/((a*n - a)*(b*x + a)^n)

Sympy [A] (verification not implemented)

Time = 17.18 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int x^{-2+n} (a+b x)^{-n} \, dx=\frac {a^{- n} x^{n - 1} \left (1 + \frac {b x}{a}\right )^{1 - n} \Gamma \left (n - 1\right )}{\Gamma \left (n\right )} \]

[In]

integrate(x**(-2+n)/((b*x+a)**n),x)

[Out]

x**(n - 1)*(1 + b*x/a)**(1 - n)*gamma(n - 1)/(a**n*gamma(n))

Maxima [F]

\[ \int x^{-2+n} (a+b x)^{-n} \, dx=\int { \frac {x^{n - 2}}{{\left (b x + a\right )}^{n}} \,d x } \]

[In]

integrate(x^(-2+n)/((b*x+a)^n),x, algorithm="maxima")

[Out]

integrate(x^(n - 2)/(b*x + a)^n, x)

Giac [F]

\[ \int x^{-2+n} (a+b x)^{-n} \, dx=\int { \frac {x^{n - 2}}{{\left (b x + a\right )}^{n}} \,d x } \]

[In]

integrate(x^(-2+n)/((b*x+a)^n),x, algorithm="giac")

[Out]

integrate(x^(n - 2)/(b*x + a)^n, x)

Mupad [B] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int x^{-2+n} (a+b x)^{-n} \, dx=\frac {x^n\,\left (a+b\,x\right )}{a\,x\,\left (n-1\right )\,{\left (a+b\,x\right )}^n} \]

[In]

int(x^(n - 2)/(a + b*x)^n,x)

[Out]

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