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

Optimal. Leaf size=14 \[ -\frac{1}{2 b (a+b x)^2} \]

[Out]

-1/(2*b*(a + b*x)^2)

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Rubi [A]  time = 0.0014833, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {32} \[ -\frac{1}{2 b (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(-3),x]

[Out]

-1/(2*b*(a + b*x)^2)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.003185, size = 14, normalized size = 1. \[ -\frac{1}{2 b (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(-3),x]

[Out]

-1/(2*b*(a + b*x)^2)

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Maple [A]  time = 0., size = 13, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,b \left ( bx+a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02536, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{2 \,{\left (b x + a\right )}^{2} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/((b*x + a)^2*b)

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Fricas [A]  time = 1.42328, size = 49, normalized size = 3.5 \begin{align*} -\frac{1}{2 \,{\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(b^3*x^2 + 2*a*b^2*x + a^2*b)

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Sympy [B]  time = 0.457129, size = 26, normalized size = 1.86 \begin{align*} - \frac{1}{2 a^{2} b + 4 a b^{2} x + 2 b^{3} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(2*a**2*b + 4*a*b**2*x + 2*b**3*x**2)

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Giac [A]  time = 1.20683, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{2 \,{\left (b x + a\right )}^{2} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/((b*x + a)^2*b)