\(\int \frac {1}{(c+d x)^3} \, dx\) [1358]

Optimal result

Integrand size = 7, antiderivative size = 14 \[ \int \frac {1}{(c+d x)^3} \, dx=-\frac {1}{2 d (c+d x)^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, 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} \[ \int \frac {1}{(c+d x)^3} \, dx=-\frac {1}{2 d (c+d x)^2} \]

[In]

Int[(c + d*x)^(-3),x]

[Out]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(c+d x)^3} \, dx=-\frac {1}{2 d (c+d x)^2} \]

[In]

Integrate[(c + d*x)^(-3),x]

[Out]

-1/2*1/(d*(c + d*x)^2)

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93

method	result	size
gosper	\(-\frac {1}{2 d \left (d x +c \right )^{2}}\)	\(13\)
default	\(-\frac {1}{2 d \left (d x +c \right )^{2}}\)	\(13\)
norman	\(-\frac {1}{2 d \left (d x +c \right )^{2}}\)	\(13\)
risch	\(-\frac {1}{2 d \left (d x +c \right )^{2}}\)	\(13\)
parallelrisch	\(-\frac {1}{2 d \left (d x +c \right )^{2}}\)	\(13\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.71 \[ \int \frac {1}{(c+d x)^3} \, dx=-\frac {1}{2 \, {\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (12) = 24\).

Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.86 \[ \int \frac {1}{(c+d x)^3} \, dx=- \frac {1}{2 c^{2} d + 4 c d^{2} x + 2 d^{3} x^{2}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(c+d x)^3} \, dx=-\frac {1}{2 \, {\left (d x + c\right )}^{2} d} \]

[In]

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

[Out]

-1/2/((d*x + c)^2*d)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(c+d x)^3} \, dx=-\frac {1}{2 \, {\left (d x + c\right )}^{2} d} \]

[In]

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

[Out]

-1/2/((d*x + c)^2*d)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.86 \[ \int \frac {1}{(c+d x)^3} \, dx=-\frac {1}{2\,c^2\,d+4\,c\,d^2\,x+2\,d^3\,x^2} \]

[In]

int(1/(c + d*x)^3,x)

[Out]

-1/(2*c^2*d + 2*d^3*x^2 + 4*c*d^2*x)