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

Optimal. Leaf size=28 \[ -\frac{x^{n-1} (a+b x)^{1-n}}{a (1-n)} \]

[Out]

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

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Rubi [A]  time = 0.0027961, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {37} \[ -\frac{x^{n-1} (a+b x)^{1-n}}{a (1-n)} \]

Antiderivative was successfully verified.

[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*} \int x^{-2+n} (a+b x)^{-n} \, dx &=-\frac{x^{-1+n} (a+b x)^{1-n}}{a (1-n)}\\ \end{align*}

Mathematica [A]  time = 0.0063209, size = 25, normalized size = 0.89 \[ \frac{x^{n-1} (a+b x)^{1-n}}{a (n-1)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-2 + n)/(a + b*x)^n,x]

[Out]

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

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Maple [A]  time = 0.003, size = 29, normalized size = 1. \begin{align*}{\frac{{x}^{-1+n} \left ( bx+a \right ) }{a \left ( -1+n \right ) \left ( bx+a \right ) ^{n}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{n - 2}}{{\left (b x + a\right )}^{n}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^(n - 2)/(b*x + a)^n, x)

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Fricas [A]  time = 1.6588, size = 66, normalized size = 2.36 \begin{align*} \frac{{\left (b x^{2} + a x\right )} x^{n - 2}}{{\left (a n - a\right )}{\left (b x + a\right )}^{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{n - 2}}{{\left (b x + a\right )}^{n}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^(n - 2)/(b*x + a)^n, x)