3.1358 \(\int \frac{1}{(c+d x)^3} \, dx\)

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

[Out]

-1/(2*d*(c + d*x)^2)

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Rubi [A]  time = 0.0017357, 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} \[ -\frac{1}{2 d (c+d x)^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(-3),x]

[Out]

-1/(2*d*(c + d*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}{(c+d x)^3} \, dx &=-\frac{1}{2 d (c+d x)^2}\\ \end{align*}

Mathematica [A]  time = 0.0024733, size = 14, normalized size = 1. \[ -\frac{1}{2 d (c+d x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^(-3),x]

[Out]

-1/(2*d*(c + d*x)^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*x+c)^3,x)

[Out]

-1/2/d/(d*x+c)^2

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Maxima [A]  time = 0.957684, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{2 \,{\left (d x + c\right )}^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^3,x, algorithm="maxima")

[Out]

-1/2/((d*x + c)^2*d)

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Fricas [A]  time = 1.62047, size = 49, normalized size = 3.5 \begin{align*} -\frac{1}{2 \,{\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^3,x, algorithm="fricas")

[Out]

-1/2/(d^3*x^2 + 2*c*d^2*x + c^2*d)

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Sympy [B]  time = 0.30866, size = 26, normalized size = 1.86 \begin{align*} - \frac{1}{2 c^{2} d + 4 c d^{2} x + 2 d^{3} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)**3,x)

[Out]

-1/(2*c**2*d + 4*c*d**2*x + 2*d**3*x**2)

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Giac [A]  time = 1.07505, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{2 \,{\left (d x + c\right )}^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^3,x, algorithm="giac")

[Out]

-1/2/((d*x + c)^2*d)