3.740 \(\int x^{-3+n} (a+b x)^{-n} \, dx\)

Optimal. Leaf size=64 \[ \frac{b x^{n-1} (a+b x)^{1-n}}{a^2 (1-n) (2-n)}-\frac{x^{n-2} (a+b x)^{1-n}}{a (2-n)} \]

[Out]

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

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Rubi [A]  time = 0.0091874, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {45, 37} \[ \frac{b x^{n-1} (a+b x)^{1-n}}{a^2 (1-n) (2-n)}-\frac{x^{n-2} (a+b x)^{1-n}}{a (2-n)} \]

Antiderivative was successfully verified.

[In]

Int[x^(-3 + n)/(a + b*x)^n,x]

[Out]

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

Rule 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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^{-3+n} (a+b x)^{-n} \, dx &=-\frac{x^{-2+n} (a+b x)^{1-n}}{a (2-n)}-\frac{b \int x^{-2+n} (a+b x)^{-n} \, dx}{a (2-n)}\\ &=-\frac{x^{-2+n} (a+b x)^{1-n}}{a (2-n)}+\frac{b x^{-1+n} (a+b x)^{1-n}}{a^2 (1-n) (2-n)}\\ \end{align*}

Mathematica [A]  time = 0.0155736, size = 39, normalized size = 0.61 \[ \frac{x^{n-2} (a+b x)^{1-n} (a (n-1)+b x)}{a^2 (n-2) (n-1)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-3 + n)/(a + b*x)^n,x]

[Out]

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

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Maple [A]  time = 0.004, size = 44, normalized size = 0.7 \begin{align*}{\frac{{x}^{-2+n} \left ( an+bx-a \right ) \left ( bx+a \right ) }{ \left ( bx+a \right ) ^{n} \left ( -2+n \right ) \left ( -1+n \right ){a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-3+n)/((b*x+a)^n),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{n - 3}}{{\left (b x + a\right )}^{n}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-3+n)/((b*x+a)^n),x, algorithm="maxima")

[Out]

integrate(x^(n - 3)/(b*x + a)^n, x)

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Fricas [A]  time = 1.64152, size = 126, normalized size = 1.97 \begin{align*} \frac{{\left (a b n x^{2} + b^{2} x^{3} +{\left (a^{2} n - a^{2}\right )} x\right )} x^{n - 3}}{{\left (a^{2} n^{2} - 3 \, a^{2} n + 2 \, a^{2}\right )}{\left (b x + a\right )}^{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-3+n)/((b*x+a)^n),x, algorithm="fricas")

[Out]

(a*b*n*x^2 + b^2*x^3 + (a^2*n - a^2)*x)*x^(n - 3)/((a^2*n^2 - 3*a^2*n + 2*a^2)*(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**(-3+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 - 3}}{{\left (b x + a\right )}^{n}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-3+n)/((b*x+a)^n),x, algorithm="giac")

[Out]

integrate(x^(n - 3)/(b*x + a)^n, x)