3.14.48 \(\int \frac {1}{(c+d x)^2} \, dx\) [1348]

Optimal. Leaf size=12 \[ -\frac {1}{d (c+d x)} \]

[Out]

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

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

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(-2),x]

[Out]

-(1/(d*(c + d*x)))

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

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Mathematica [A]
time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} -\frac {1}{d (c+d x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^(-2),x]

[Out]

-(1/(d*(c + d*x)))

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Maple [A]
time = 0.13, size = 13, normalized size = 1.08

method	result	size
gosper	\(-\frac {1}{d \left (d x +c \right )}\)	\(13\)
default	\(-\frac {1}{d \left (d x +c \right )}\)	\(13\)
norman	\(\frac {x}{c \left (d x +c \right )}\)	\(13\)
risch	\(-\frac {1}{d \left (d x +c \right )}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*x+c)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.31, size = 12, normalized size = 1.00 \begin {gather*} -\frac {1}{{\left (d x + c\right )} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^2,x, algorithm="maxima")

[Out]

-1/((d*x + c)*d)

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Fricas [A]
time = 1.33, size = 13, normalized size = 1.08 \begin {gather*} -\frac {1}{d^{2} x + c d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^2,x, algorithm="fricas")

[Out]

-1/(d^2*x + c*d)

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Sympy [A]
time = 0.04, size = 10, normalized size = 0.83 \begin {gather*} - \frac {1}{c d + d^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)**2,x)

[Out]

-1/(c*d + d**2*x)

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Giac [A]
time = 1.22, size = 12, normalized size = 1.00 \begin {gather*} -\frac {1}{{\left (d x + c\right )} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^2,x, algorithm="giac")

[Out]

-1/((d*x + c)*d)

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Mupad [B]
time = 0.19, size = 12, normalized size = 1.00 \begin {gather*} -\frac {1}{d\,\left (c+d\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c + d*x)^2,x)

[Out]

-1/(d*(c + d*x))

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