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

Optimal. Leaf size=26 \[ -\frac{(-a-b x)^{-n} (a+b x)^n}{2 x^2} \]

[Out]

-(a + b*x)^n/(2*x^2*(-a - b*x)^n)

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Rubi [A]  time = 0.0030823, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {23, 30} \[ -\frac{(-a-b x)^{-n} (a+b x)^n}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^n/(x^3*(-a - b*x)^n),x]

[Out]

-(a + b*x)^n/(2*x^2*(-a - b*x)^n)

Rule 23

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((c_) + (d_.)*(v_))^(n_), x_Symbol] :> Dist[(a + b*v)^m/(c + d*v)^m, Int[u*
(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] &&  !(IntegerQ[m] || IntegerQ[n
] || GtQ[b/d, 0])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0025799, size = 26, normalized size = 1. \[ -\frac{(-a-b x)^{-n} (a+b x)^n}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^n/(x^3*(-a - b*x)^n),x]

[Out]

-(a + b*x)^n/(2*x^2*(-a - b*x)^n)

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Maple [A]  time = 0.001, size = 25, normalized size = 1. \begin{align*} -{\frac{ \left ( bx+a \right ) ^{n}}{2\,{x}^{2} \left ( -bx-a \right ) ^{n}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.06961, size = 11, normalized size = 0.42 \begin{align*} -\frac{\left (-1\right )^{n}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(-1)^n/x^2

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Fricas [A]  time = 1.84959, size = 27, normalized size = 1.04 \begin{align*} -\frac{\cos \left (\pi n\right )}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*cos(pi*n)/x^2

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Sympy [A]  time = 22.4169, size = 20, normalized size = 0.77 \begin{align*} - \frac{\left (- a - b x\right )^{- n} \left (a + b x\right )^{n}}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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