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

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \frac {2 x^{5/2}}{5}+\frac {2 x^{3/2}}{15}-2 \log (x) \end {gather*}

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*}

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Mathematica [A]
time = 0.01, size = 23, normalized size = 1.00 \begin {gather*} \frac {2 x^{3/2}}{15}+\frac {2 x^{5/2}}{5}-2 \log (x) \end {gather*}

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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Mathics [A]
time = 1.63, size = 15, normalized size = 0.65 \begin {gather*} \frac {2 x^{\frac {3}{2}}}{15}+\frac {2 x^{\frac {5}{2}}}{5}-2 \text {Log}\left [x\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('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 = 0.04, size = 16, normalized size = 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	\(\frac {2 x^{\frac {3}{2}}}{15}+\frac {2 x^{\frac {5}{2}}}{5}-2 \ln \left (x \right )\)	\(16\)
trager	\(\frac {2 x^{\frac {3}{2}} \left (1+3 x \right )}{15}-2 \ln \left (x \right )\)	\(16\)
risch	\(\frac {2 x^{\frac {3}{2}}}{15}+\frac {2 x^{\frac {5}{2}}}{5}-2 \ln \left (x \right )\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [A]
time = 0.25, size = 15, normalized size = 0.65 \begin {gather*} \frac {2}{5} \, x^{\frac {5}{2}} + \frac {2}{15} \, x^{\frac {3}{2}} - 2 \, \log \left (x\right ) \end {gather*}

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 = 0.30, size = 19, normalized size = 0.83 \begin {gather*} \frac {2}{15} \, {\left (3 \, x^{2} + x\right )} \sqrt {x} - 4 \, \log \left (\sqrt {x}\right ) \end {gather*}

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 = 0.03, size = 20, normalized size = 0.87 \begin {gather*} \frac {2 x^{\frac {5}{2}}}{5} + \frac {2 x^{\frac {3}{2}}}{15} - 2 \log {\left (x \right )} \end {gather*}

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 = 0.00, size = 31, normalized size = 1.35 \begin {gather*} -2 \ln \left |x\right |+\frac {2}{5} \sqrt {x} x^{2}+\frac {2 \sqrt {x} x}{5\cdot 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.28, size = 17, normalized size = 0.74 \begin {gather*} \frac {2\,x^{3/2}}{15}-4\,\ln \left (\sqrt {x}\right )+\frac {2\,x^{5/2}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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