\(\int x^{3/2} \, dx\) [21]

Optimal result

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

[Out]

2/5*x^(5/2)

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} \[ \int x^{3/2} \, dx=\frac {2 x^{5/2}}{5} \]

[In]

Int[x^(3/2),x]

[Out]

(2*x^(5/2))/5

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 x^{5/2}}{5} \\ \end{align*}

Mathematica [A] (verified)

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

[In]

Integrate[x^(3/2),x]

[Out]

(2*x^(5/2))/5

Maple [A] (verified)

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

method	result	size
gosper	\(\frac {2 x^{\frac {5}{2}}}{5}\)	\(6\)
derivativedivides	\(\frac {2 x^{\frac {5}{2}}}{5}\)	\(6\)
default	\(\frac {2 x^{\frac {5}{2}}}{5}\)	\(6\)
trager	\(\frac {2 x^{\frac {5}{2}}}{5}\)	\(6\)
risch	\(\frac {2 x^{\frac {5}{2}}}{5}\)	\(6\)

[In]

int(x^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/5*x^(5/2)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.56 \[ \int x^{3/2} \, dx=\frac {2}{5} \, x^{\frac {5}{2}} \]

[In]

integrate(x^(3/2),x, algorithm="fricas")

[Out]

2/5*x^(5/2)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int x^{3/2} \, dx=\frac {2 x^{\frac {5}{2}}}{5} \]

[In]

integrate(x**(3/2),x)

[Out]

2*x**(5/2)/5

Maxima [A] (verification not implemented)

none

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

[In]

integrate(x^(3/2),x, algorithm="maxima")

[Out]

2/5*x^(5/2)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.56 \[ \int x^{3/2} \, dx=\frac {2}{5} \, x^{\frac {5}{2}} \]

[In]

integrate(x^(3/2),x, algorithm="giac")

[Out]

2/5*x^(5/2)

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.56 \[ \int x^{3/2} \, dx=\frac {2\,x^{5/2}}{5} \]

[In]

int(x^(3/2),x)

[Out]

(2*x^(5/2))/5