\(\int \frac {1}{x^{5/2}} \, dx\) [25]

Optimal result

Integrand size = 5, antiderivative size = 9 \[ \int \frac {1}{x^{5/2}} \, dx=-\frac {2}{3 x^{3/2}} \]

[Out]

-2/3/x^(3/2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {30} \[ \int \frac {1}{x^{5/2}} \, dx=-\frac {2}{3 x^{3/2}} \]

[In]

Int[x^(-5/2),x]

[Out]

-2/(3*x^(3/2))

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {2}{3 x^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^{5/2}} \, dx=-\frac {2}{3 x^{3/2}} \]

[In]

Integrate[x^(-5/2),x]

[Out]

-2/(3*x^(3/2))

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.67

method	result	size
gosper	\(-\frac {2}{3 x^{\frac {3}{2}}}\)	\(6\)
derivativedivides	\(-\frac {2}{3 x^{\frac {3}{2}}}\)	\(6\)
default	\(-\frac {2}{3 x^{\frac {3}{2}}}\)	\(6\)
trager	\(-\frac {2}{3 x^{\frac {3}{2}}}\)	\(6\)
risch	\(-\frac {2}{3 x^{\frac {3}{2}}}\)	\(6\)

[In]

int(1/x^(5/2),x,method=_RETURNVERBOSE)

[Out]

-2/3/x^(3/2)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.56 \[ \int \frac {1}{x^{5/2}} \, dx=-\frac {2}{3 \, x^{\frac {3}{2}}} \]

[In]

integrate(1/x^(5/2),x, algorithm="fricas")

[Out]

-2/3/x^(3/2)

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^{5/2}} \, dx=- \frac {2}{3 x^{\frac {3}{2}}} \]

[In]

integrate(1/x**(5/2),x)

[Out]

-2/(3*x**(3/2))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.56 \[ \int \frac {1}{x^{5/2}} \, dx=-\frac {2}{3 \, x^{\frac {3}{2}}} \]

[In]

integrate(1/x^(5/2),x, algorithm="maxima")

[Out]

-2/3/x^(3/2)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.56 \[ \int \frac {1}{x^{5/2}} \, dx=-\frac {2}{3 \, x^{\frac {3}{2}}} \]

[In]

integrate(1/x^(5/2),x, algorithm="giac")

[Out]

-2/3/x^(3/2)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.56 \[ \int \frac {1}{x^{5/2}} \, dx=-\frac {2}{3\,x^{3/2}} \]

[In]

int(1/x^(5/2),x)

[Out]

-2/(3*x^(3/2))