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

Optimal. Leaf size=19 \[ \frac{x^n (a+b x)^{-n}}{a n} \]

[Out]

x^n/(a*n*(a + b*x)^n)

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Rubi [A]  time = 0.00262, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {37} \[ \frac{x^n (a+b x)^{-n}}{a n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + n)*(a + b*x)^(-1 - n),x]

[Out]

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

Mathematica [A]  time = 0.0047965, size = 19, normalized size = 1. \[ \frac{x^n (a+b x)^{-n}}{a n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^n/(a*n*(a + b*x)^n)

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Maple [A]  time = 0.001, size = 20, normalized size = 1.1 \begin{align*}{\frac{{x}^{n} \left ( bx+a \right ) ^{-n}}{an}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.13209, size = 30, normalized size = 1.58 \begin{align*} \frac{e^{\left (-n \log \left (b x + a\right ) + n \log \left (x\right )\right )}}{a n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(-n*log(b*x + a) + n*log(x))/(a*n)

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Fricas [A]  time = 1.6104, size = 68, normalized size = 3.58 \begin{align*} \frac{{\left (b x^{2} + a x\right )}{\left (b x + a\right )}^{-n - 1} x^{n - 1}}{a n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x^2 + a*x)*(b*x + a)^(-n - 1)*x^(n - 1)/(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**(-1+n)*(b*x+a)**(-1-n),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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