3.323 \(\int \frac{1}{(a x+b x)^3} \, dx\)

Optimal. Leaf size=12 \[ -\frac{1}{2 x^2 (a+b)^3} \]

[Out]

-1/(2*(a + b)^3*x^2)

________________________________________________________________________________________

Rubi [A]  time = 0.0030896, antiderivative size = 12, 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} \[ -\frac{1}{2 x^2 (a+b)^3} \]

Antiderivative was successfully verified.

[In]

Int[(a*x + b*x)^(-3),x]

[Out]

-1/(2*(a + b)^3*x^2)

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

Mathematica [A]  time = 0.0034175, size = 12, normalized size = 1. \[ -\frac{1}{2 x^2 (a+b)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*x + b*x)^(-3),x]

[Out]

-1/(2*(a + b)^3*x^2)

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 11, normalized size = 0.9 \begin{align*} -{\frac{1}{2\, \left ( a+b \right ) ^{3}{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x+b*x)^3,x)

[Out]

-1/2/(a+b)^3/x^2

________________________________________________________________________________________

Maxima [A]  time = 0.995037, size = 22, normalized size = 1.83 \begin{align*} -\frac{1}{2 \,{\left (a x + b x\right )}^{2}{\left (a + b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+b*x)^3,x, algorithm="maxima")

[Out]

-1/2/((a*x + b*x)^2*(a + b))

________________________________________________________________________________________

Fricas [B]  time = 0.820803, size = 59, normalized size = 4.92 \begin{align*} -\frac{1}{2 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+b*x)^3,x, algorithm="fricas")

[Out]

-1/2/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*x^2)

________________________________________________________________________________________

Sympy [B]  time = 0.107827, size = 27, normalized size = 2.25 \begin{align*} - \frac{1}{2 x^{2} \left (a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+b*x)**3,x)

[Out]

-1/(2*x**2*(a**3 + 3*a**2*b + 3*a*b**2 + b**3))

________________________________________________________________________________________

Giac [A]  time = 1.15199, size = 22, normalized size = 1.83 \begin{align*} -\frac{1}{2 \,{\left (a x + b x\right )}^{2}{\left (a + b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+b*x)^3,x, algorithm="giac")

[Out]

-1/2/((a*x + b*x)^2*(a + b))