\(\int (\frac {1}{x^{3/2}}+x^{3/2}) \, dx\) [1914]

Optimal result

Integrand size = 11, antiderivative size = 17 \[ \int \left (\frac {1}{x^{3/2}}+x^{3/2}\right ) \, dx=-\frac {2}{\sqrt {x}}+\frac {2 x^{5/2}}{5} \]

[Out]

2/5*x^(5/2)-2/x^(1/2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \left (\frac {1}{x^{3/2}}+x^{3/2}\right ) \, dx=\frac {2 x^{5/2}}{5}-\frac {2}{\sqrt {x}} \]

[In]

Int[x^(-3/2) + x^(3/2),x]

[Out]

-2/Sqrt[x] + (2*x^(5/2))/5

Rubi steps \begin{align*} \text {integral}& = -\frac {2}{\sqrt {x}}+\frac {2 x^{5/2}}{5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \left (\frac {1}{x^{3/2}}+x^{3/2}\right ) \, dx=\frac {2 \left (-5+x^3\right )}{5 \sqrt {x}} \]

[In]

Integrate[x^(-3/2) + x^(3/2),x]

[Out]

(2*(-5 + x^3))/(5*Sqrt[x])

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.65

method	result	size
gosper	\(\frac {\frac {2 x^{3}}{5}-2}{\sqrt {x}}\)	\(11\)
trager	\(\frac {\frac {2 x^{3}}{5}-2}{\sqrt {x}}\)	\(11\)
derivativedivides	\(\frac {2 x^{\frac {5}{2}}}{5}-\frac {2}{\sqrt {x}}\)	\(12\)
default	\(\frac {2 x^{\frac {5}{2}}}{5}-\frac {2}{\sqrt {x}}\)	\(12\)
risch	\(\frac {2 x^{\frac {5}{2}}}{5}-\frac {2}{\sqrt {x}}\)	\(12\)

[In]

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

[Out]

2/5*(x^3-5)/x^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.59 \[ \int \left (\frac {1}{x^{3/2}}+x^{3/2}\right ) \, dx=\frac {2 \, {\left (x^{3} - 5\right )}}{5 \, \sqrt {x}} \]

[In]

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

[Out]

2/5*(x^3 - 5)/sqrt(x)

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \left (\frac {1}{x^{3/2}}+x^{3/2}\right ) \, dx=\frac {2 x^{\frac {5}{2}}}{5} - \frac {2}{\sqrt {x}} \]

[In]

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

[Out]

2*x**(5/2)/5 - 2/sqrt(x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.65 \[ \int \left (\frac {1}{x^{3/2}}+x^{3/2}\right ) \, dx=\frac {2}{5} \, x^{\frac {5}{2}} - \frac {2}{\sqrt {x}} \]

[In]

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

[Out]

2/5*x^(5/2) - 2/sqrt(x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.65 \[ \int \left (\frac {1}{x^{3/2}}+x^{3/2}\right ) \, dx=\frac {2}{5} \, x^{\frac {5}{2}} - \frac {2}{\sqrt {x}} \]

[In]

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

[Out]

2/5*x^(5/2) - 2/sqrt(x)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.71 \[ \int \left (\frac {1}{x^{3/2}}+x^{3/2}\right ) \, dx=\frac {2\,x^3-10}{5\,\sqrt {x}} \]

[In]

int(1/x^(3/2) + x^(3/2),x)

[Out]

(2*x^3 - 10)/(5*x^(1/2))