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

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

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Rubi [A]  time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {1}{2 x^2 (a+b)^3} \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} -\frac {1}{2 x^2 (a+b)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*x + b*x)^(-3),x]

[Out]

-1/2*1/((a + b)^3*x^2)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(a x+b x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a*x + b*x)^(-3),x]

[Out]

IntegrateAlgebraic[(a*x + b*x)^(-3), x]

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fricas [B]  time = 0.37, size = 26, normalized size = 2.17 \begin {gather*} -\frac {1}{2 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} x^{2}} \end {gather*}

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)

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giac [A]  time = 0.15, size = 16, normalized size = 1.33 \begin {gather*} -\frac {1}{2 \, {\left (a x + b x\right )}^{2} {\left (a + b\right )}} \end {gather*}

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))

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maple [A]  time = 0.05, size = 11, normalized size = 0.92 \begin {gather*} -\frac {1}{2 \left (a +b \right )^{3} x^{2}} \end {gather*}

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

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maxima [A]  time = 1.31, size = 16, normalized size = 1.33 \begin {gather*} -\frac {1}{2 \, {\left (a x + b x\right )}^{2} {\left (a + b\right )}} \end {gather*}

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))

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mupad [B]  time = 0.04, size = 26, normalized size = 2.17 \begin {gather*} -\frac {1}{2\,x^2\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x + b*x)^3,x)

[Out]

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

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sympy [B]  time = 0.09, size = 27, normalized size = 2.25 \begin {gather*} - \frac {1}{2 x^{2} \left (a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}\right )} \end {gather*}

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))

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