\(\int \frac {1}{(c+d x)^{3/2}} \, dx\) [1429]

Optimal result

Integrand size = 9, antiderivative size = 14 \[ \int \frac {1}{(c+d x)^{3/2}} \, dx=-\frac {2}{d \sqrt {c+d x}} \]

[Out]

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

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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {32} \[ \int \frac {1}{(c+d x)^{3/2}} \, dx=-\frac {2}{d \sqrt {c+d x}} \]

[In]

Int[(c + d*x)^(-3/2),x]

[Out]

-2/(d*Sqrt[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 {2}{d \sqrt {c+d x}} \\ \end{align*}

Mathematica [A] (verified)

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

[In]

Integrate[(c + d*x)^(-3/2),x]

[Out]

-2/(d*Sqrt[c + d*x])

Maple [A] (verified)

Time = 0.70 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93

method	result	size
gosper	\(-\frac {2}{d \sqrt {d x +c}}\)	\(13\)
derivativedivides	\(-\frac {2}{d \sqrt {d x +c}}\)	\(13\)
default	\(-\frac {2}{d \sqrt {d x +c}}\)	\(13\)
trager	\(-\frac {2}{d \sqrt {d x +c}}\)	\(13\)
pseudoelliptic	\(-\frac {2}{d \sqrt {d x +c}}\)	\(13\)

[In]

int(1/(d*x+c)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.43 \[ \int \frac {1}{(c+d x)^{3/2}} \, dx=-\frac {2 \, \sqrt {d x + c}}{d^{2} x + c d} \]

[In]

integrate(1/(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

-2*sqrt(d*x + c)/(d^2*x + c*d)

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(c+d x)^{3/2}} \, dx=- \frac {2}{d \sqrt {c + d x}} \]

[In]

integrate(1/(d*x+c)**(3/2),x)

[Out]

-2/(d*sqrt(c + d*x))

Maxima [A] (verification not implemented)

none

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

[In]

integrate(1/(d*x+c)^(3/2),x, algorithm="maxima")

[Out]

-2/(sqrt(d*x + c)*d)

Giac [A] (verification not implemented)

none

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

[In]

integrate(1/(d*x+c)^(3/2),x, algorithm="giac")

[Out]

-2/(sqrt(d*x + c)*d)

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(c+d x)^{3/2}} \, dx=-\frac {2}{d\,\sqrt {c+d\,x}} \]

[In]

int(1/(c + d*x)^(3/2),x)

[Out]

-2/(d*(c + d*x)^(1/2))