3.322 \(\int \frac{1}{(a x+b x)^2} \, dx\)

Optimal. Leaf size=10 \[ -\frac{1}{x (a+b)^2} \]

[Out]

-(1/((a + b)^2*x))

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Rubi [A]  time = 0.0032584, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6, 12, 30} \[ -\frac{1}{x (a+b)^2} \]

Antiderivative was successfully verified.

[In]

Int[(a*x + b*x)^(-2),x]

[Out]

-(1/((a + b)^2*x))

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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

Mathematica [A]  time = 0.0030123, size = 10, normalized size = 1. \[ -\frac{1}{x (a+b)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*x + b*x)^(-2),x]

[Out]

-(1/((a + b)^2*x))

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Maple [A]  time = 0., size = 11, normalized size = 1.1 \begin{align*} -{\frac{1}{ \left ( a+b \right ) ^{2}x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.11015, size = 22, normalized size = 2.2 \begin{align*} -\frac{1}{{\left (a x + b x\right )}{\left (a + b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+b*x)^2,x, algorithm="maxima")

[Out]

-1/((a*x + b*x)*(a + b))

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Fricas [A]  time = 0.800912, size = 38, normalized size = 3.8 \begin{align*} -\frac{1}{{\left (a^{2} + 2 \, a b + b^{2}\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+b*x)^2,x, algorithm="fricas")

[Out]

-1/((a^2 + 2*a*b + b^2)*x)

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Sympy [A]  time = 0.111881, size = 15, normalized size = 1.5 \begin{align*} - \frac{1}{x \left (a^{2} + 2 a b + b^{2}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+b*x)**2,x)

[Out]

-1/(x*(a**2 + 2*a*b + b**2))

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Giac [A]  time = 1.12956, size = 22, normalized size = 2.2 \begin{align*} -\frac{1}{{\left (a x + b x\right )}{\left (a + b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+b*x)^2,x, algorithm="giac")

[Out]

-1/((a*x + b*x)*(a + b))