\(\int (-\frac {2}{x}+\frac {\sqrt {x}}{5}+x^{3/2}) \, dx\) [1917]

Optimal result

Integrand size = 20, antiderivative size = 23 \[ \int \left (-\frac {2}{x}+\frac {\sqrt {x}}{5}+x^{3/2}\right ) \, dx=\frac {2 x^{3/2}}{15}+\frac {2 x^{5/2}}{5}-2 \log (x) \]

[Out]

2/15*x^(3/2)+2/5*x^(5/2)-2*ln(x)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 23, 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 = {} \[ \int \left (-\frac {2}{x}+\frac {\sqrt {x}}{5}+x^{3/2}\right ) \, dx=\frac {2 x^{5/2}}{5}+\frac {2 x^{3/2}}{15}-2 \log (x) \]

[In]

Int[-2/x + Sqrt[x]/5 + x^(3/2),x]

[Out]

(2*x^(3/2))/15 + (2*x^(5/2))/5 - 2*Log[x]

Rubi steps \begin{align*} \text {integral}& = \frac {2 x^{3/2}}{15}+\frac {2 x^{5/2}}{5}-2 \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \left (-\frac {2}{x}+\frac {\sqrt {x}}{5}+x^{3/2}\right ) \, dx=\frac {2 x^{3/2}}{15}+\frac {2 x^{5/2}}{5}-2 \log (x) \]

[In]

Integrate[-2/x + Sqrt[x]/5 + x^(3/2),x]

[Out]

(2*x^(3/2))/15 + (2*x^(5/2))/5 - 2*Log[x]

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.70

method	result	size
derivativedivides	\(\frac {2 x^{\frac {3}{2}}}{15}+\frac {2 x^{\frac {5}{2}}}{5}-2 \ln \left (x \right )\)	\(16\)
default	\(\frac {2 x^{\frac {3}{2}}}{15}+\frac {2 x^{\frac {5}{2}}}{5}-2 \ln \left (x \right )\)	\(16\)
risch	\(\frac {2 x^{\frac {3}{2}}}{15}+\frac {2 x^{\frac {5}{2}}}{5}-2 \ln \left (x \right )\)	\(16\)
trager	\(\frac {2 x^{\frac {3}{2}} \left (3 x +1\right )}{15}+2 \ln \left (\frac {1}{x}\right )\)	\(18\)

[In]

int(-2/x+x^(3/2)+1/5*x^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/15*x^(3/2)+2/5*x^(5/2)-2*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \left (-\frac {2}{x}+\frac {\sqrt {x}}{5}+x^{3/2}\right ) \, dx=\frac {2}{15} \, {\left (3 \, x^{2} + x\right )} \sqrt {x} - 4 \, \log \left (\sqrt {x}\right ) \]

[In]

integrate(-2/x+x^(3/2)+1/5*x^(1/2),x, algorithm="fricas")

[Out]

2/15*(3*x^2 + x)*sqrt(x) - 4*log(sqrt(x))

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \left (-\frac {2}{x}+\frac {\sqrt {x}}{5}+x^{3/2}\right ) \, dx=\frac {2 x^{\frac {5}{2}}}{5} + \frac {2 x^{\frac {3}{2}}}{15} - 2 \log {\left (x \right )} \]

[In]

integrate(-2/x+x**(3/2)+1/5*x**(1/2),x)

[Out]

2*x**(5/2)/5 + 2*x**(3/2)/15 - 2*log(x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.65 \[ \int \left (-\frac {2}{x}+\frac {\sqrt {x}}{5}+x^{3/2}\right ) \, dx=\frac {2}{5} \, x^{\frac {5}{2}} + \frac {2}{15} \, x^{\frac {3}{2}} - 2 \, \log \left (x\right ) \]

[In]

integrate(-2/x+x^(3/2)+1/5*x^(1/2),x, algorithm="maxima")

[Out]

2/5*x^(5/2) + 2/15*x^(3/2) - 2*log(x)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.70 \[ \int \left (-\frac {2}{x}+\frac {\sqrt {x}}{5}+x^{3/2}\right ) \, dx=\frac {2}{5} \, x^{\frac {5}{2}} + \frac {2}{15} \, x^{\frac {3}{2}} - 2 \, \log \left ({\left | x \right |}\right ) \]

[In]

integrate(-2/x+x^(3/2)+1/5*x^(1/2),x, algorithm="giac")

[Out]

2/5*x^(5/2) + 2/15*x^(3/2) - 2*log(abs(x))

Mupad [B] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \left (-\frac {2}{x}+\frac {\sqrt {x}}{5}+x^{3/2}\right ) \, dx=\frac {2\,x^{3/2}}{15}-4\,\ln \left (\sqrt {x}\right )+\frac {2\,x^{5/2}}{5} \]

[In]

int(x^(1/2)/5 - 2/x + x^(3/2),x)

[Out]

(2*x^(3/2))/15 - 4*log(x^(1/2)) + (2*x^(5/2))/5