3.1917 \(\int (-\frac{2}{x}+\frac{\sqrt{x}}{5}+x^{3/2}) \, dx\)

Optimal. Leaf size=23 \[ \frac{2 x^{5/2}}{5}+\frac{2 x^{3/2}}{15}-2 \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0025672, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 0, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \frac{2 x^{5/2}}{5}+\frac{2 x^{3/2}}{15}-2 \log (x) \]

Antiderivative was successfully verified.

[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*} \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)\\ \end{align*}

Mathematica [A]  time = 0.0074627, size = 23, normalized size = 1. \[ \frac{2 x^{5/2}}{5}+\frac{2 x^{3/2}}{15}-2 \log (x) \]

Antiderivative was successfully verified.

[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]

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Maple [A]  time = 0., size = 16, normalized size = 0.7 \begin{align*}{\frac{2}{15}{x}^{{\frac{3}{2}}}}+{\frac{2}{5}{x}^{{\frac{5}{2}}}}-2\,\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.961844, size = 20, normalized size = 0.87 \begin{align*} \frac{2}{5} \, x^{\frac{5}{2}} + \frac{2}{15} \, x^{\frac{3}{2}} - 2 \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 2.16926, size = 58, normalized size = 2.52 \begin{align*} \frac{2}{15} \,{\left (3 \, x^{2} + x\right )} \sqrt{x} - 4 \, \log \left (\sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Sympy [A]  time = 0.058089, size = 20, normalized size = 0.87 \begin{align*} \frac{2 x^{\frac{5}{2}}}{5} + \frac{2 x^{\frac{3}{2}}}{15} - 2 \log{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.06504, size = 22, normalized size = 0.96 \begin{align*} \frac{2}{5} \, x^{\frac{5}{2}} + \frac{2}{15} \, x^{\frac{3}{2}} - 2 \, \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))