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

Optimal result

Integrand size = 7, antiderivative size = 12 \[ \int \frac {1}{(c+d x)^2} \, dx=-\frac {1}{d (c+d x)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {32} \[ \int \frac {1}{(c+d x)^2} \, dx=-\frac {1}{d (c+d x)} \]

[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*} \text {integral}& = -\frac {1}{d (c+d x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(c+d x)^2} \, dx=-\frac {1}{d (c+d x)} \]

[In]

Integrate[(c + d*x)^(-2),x]

[Out]

-(1/(d*(c + d*x)))

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 13, normalized size of antiderivative = 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	\(-\frac {1}{d \left (d x +c \right )}\)	\(13\)
parallelrisch	\(-\frac {1}{d \left (d x +c \right )}\)	\(13\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(c+d x)^2} \, dx=-\frac {1}{d^{2} x + c d} \]

[In]

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

[Out]

-1/(d^2*x + c*d)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {1}{(c+d x)^2} \, dx=- \frac {1}{c d + d^{2} x} \]

[In]

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

[Out]

-1/(c*d + d**2*x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(c+d x)^2} \, dx=-\frac {1}{{\left (d x + c\right )} d} \]

[In]

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

[Out]

-1/((d*x + c)*d)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(c+d x)^2} \, dx=-\frac {1}{{\left (d x + c\right )} d} \]

[In]

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

[Out]

-1/((d*x + c)*d)

Mupad [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(c+d x)^2} \, dx=-\frac {1}{d\,\left (c+d\,x\right )} \]

[In]

int(1/(c + d*x)^2,x)

[Out]

-1/(d*(c + d*x))