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

Optimal. Leaf size=24 \[ -\frac{(-a-b x)^{-n} (a+b x)^n}{x} \]

[Out]

-((a + b*x)^n/(x*(-a - b*x)^n))

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Rubi [A]  time = 0.0030462, antiderivative size = 24, 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} \[ -\frac{(-a-b x)^{-n} (a+b x)^n}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^n/(x^2*(-a - b*x)^n),x]

[Out]

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

Mathematica [A]  time = 0.0027004, size = 24, normalized size = 1. \[ -\frac{(-a-b x)^{-n} (a+b x)^n}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a + b*x)^n/(x*(-a - b*x)^n))

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Maple [A]  time = 0.002, size = 25, normalized size = 1. \begin{align*} -{\frac{ \left ( bx+a \right ) ^{n}}{x \left ( -bx-a \right ) ^{n}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.50331, size = 11, normalized size = 0.46 \begin{align*} -\frac{\left (-1\right )^{n}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-1)^n/x

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Fricas [A]  time = 1.83193, size = 19, normalized size = 0.79 \begin{align*} -\frac{\cos \left (\pi n\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(pi*n)/x

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Sympy [A]  time = 18.5831, size = 44, normalized size = 1.83 \begin{align*} \begin{cases} - \frac{\left (- a - b x\right )^{- n} \left (a + b x\right )^{n}}{x} + \frac{b \left (- a - b x\right )^{- n} \left (a + b x\right )^{n}}{a} & \text{for}\: a \neq 0 \\- \frac{\left (-1\right )^{- n}}{x} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-(-a - b*x)**(-n)*(a + b*x)**n/x + b*(-a - b*x)**(-n)*(a + b*x)**n/a, Ne(a, 0)), (-(-1)**(-n)/x, Tr
ue))

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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^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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