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

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

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Rubi [A]  time = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {x^{n-1} (a+b x)^{1-n}}{a (1-n)} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 0.89 \begin {gather*} \frac {x^{n-1} (a+b x)^{1-n}}{a (n-1)} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.03, size = 0, normalized size = 0.00 \begin {gather*} \int x^{-2+n} (a+b x)^{-n} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^(-2 + n)/(a + b*x)^n,x]

[Out]

Defer[IntegrateAlgebraic][x^(-2 + n)/(a + b*x)^n, x]

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fricas [A]  time = 0.97, size = 33, normalized size = 1.18 \begin {gather*} \frac {{\left (b x^{2} + a x\right )} x^{n - 2}}{{\left (a n - a\right )} {\left (b x + a\right )}^{n}} \end {gather*}

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{n - 2}}{{\left (b x + a\right )}^{n}}\,{d x} \end {gather*}

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)

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maple [A]  time = 0.00, size = 29, normalized size = 1.04 \begin {gather*} \frac {\left (b x +a \right ) x^{n -1} \left (b x +a \right )^{-n}}{\left (n -1\right ) a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{n - 2}}{{\left (b x + a\right )}^{n}}\,{d x} \end {gather*}

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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mupad [B]  time = 0.35, size = 29, normalized size = 1.04 \begin {gather*} \frac {x^n\,\left (a+b\,x\right )}{a\,x\,\left (n-1\right )\,{\left (a+b\,x\right )}^n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(n - 2)/(a + b*x)^n,x)

[Out]

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

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sympy [F(-1)]  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**(-2+n)/((b*x+a)**n),x)

[Out]

Timed out

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