\(\int \frac {1}{(c+d x)^{5/2}} \, dx\) [1439]

Optimal result

Integrand size = 9, antiderivative size = 16 \[ \int \frac {1}{(c+d x)^{5/2}} \, dx=-\frac {2}{3 d (c+d x)^{3/2}} \]

[Out]

-2/3/d/(d*x+c)^(3/2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, 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} \[ \int \frac {1}{(c+d x)^{5/2}} \, dx=-\frac {2}{3 d (c+d x)^{3/2}} \]

[In]

Int[(c + d*x)^(-5/2),x]

[Out]

-2/(3*d*(c + d*x)^(3/2))

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 {2}{3 d (c+d x)^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(c+d x)^{5/2}} \, dx=-\frac {2}{3 d (c+d x)^{3/2}} \]

[In]

Integrate[(c + d*x)^(-5/2),x]

[Out]

-2/(3*d*(c + d*x)^(3/2))

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81

method	result	size
gosper	\(-\frac {2}{3 d \left (d x +c \right )^{\frac {3}{2}}}\)	\(13\)
derivativedivides	\(-\frac {2}{3 d \left (d x +c \right )^{\frac {3}{2}}}\)	\(13\)
default	\(-\frac {2}{3 d \left (d x +c \right )^{\frac {3}{2}}}\)	\(13\)
trager	\(-\frac {2}{3 d \left (d x +c \right )^{\frac {3}{2}}}\)	\(13\)
pseudoelliptic	\(-\frac {2}{3 d \left (d x +c \right )^{\frac {3}{2}}}\)	\(13\)

[In]

int(1/(d*x+c)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-2/3/d/(d*x+c)^(3/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (12) = 24\).

Time = 0.22 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.94 \[ \int \frac {1}{(c+d x)^{5/2}} \, dx=-\frac {2 \, \sqrt {d x + c}}{3 \, {\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \]

[In]

integrate(1/(d*x+c)^(5/2),x, algorithm="fricas")

[Out]

-2/3*sqrt(d*x + c)/(d^3*x^2 + 2*c*d^2*x + c^2*d)

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(c+d x)^{5/2}} \, dx=- \frac {2}{3 d \left (c + d x\right )^{\frac {3}{2}}} \]

[In]

integrate(1/(d*x+c)**(5/2),x)

[Out]

-2/(3*d*(c + d*x)**(3/2))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{(c+d x)^{5/2}} \, dx=-\frac {2}{3 \, {\left (d x + c\right )}^{\frac {3}{2}} d} \]

[In]

integrate(1/(d*x+c)^(5/2),x, algorithm="maxima")

[Out]

-2/3/((d*x + c)^(3/2)*d)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{(c+d x)^{5/2}} \, dx=-\frac {2}{3 \, {\left (d x + c\right )}^{\frac {3}{2}} d} \]

[In]

integrate(1/(d*x+c)^(5/2),x, algorithm="giac")

[Out]

-2/3/((d*x + c)^(3/2)*d)

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{(c+d x)^{5/2}} \, dx=-\frac {2}{3\,d\,{\left (c+d\,x\right )}^{3/2}} \]

[In]

int(1/(c + d*x)^(5/2),x)

[Out]

-2/(3*d*(c + d*x)^(3/2))