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

Optimal. Leaf size=17 \[ -\frac {2}{\sqrt {x}}+\frac {2 x^{5/2}}{5} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \frac {2 x^{5/2}}{5}-\frac {2}{\sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(-3/2) + x^(3/2),x]

[Out]

-2/Sqrt[x] + (2*x^(5/2))/5

Rubi steps

\begin {align*} \int \left (\frac {1}{x^{3/2}}+x^{3/2}\right ) \, dx &=-\frac {2}{\sqrt {x}}+\frac {2 x^{5/2}}{5}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 14, normalized size = 0.82 \begin {gather*} \frac {2 \left (-5+x^3\right )}{5 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(-3/2) + x^(3/2),x]

[Out]

(2*(-5 + x^3))/(5*Sqrt[x])

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Mathics [A]
time = 1.68, size = 10, normalized size = 0.59 \begin {gather*} \frac {2 \left (-5+x^3\right )}{5 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x^(-3/2) + x^(3/2),x]')

[Out]

2 (-5 + x ^ 3) / (5 Sqrt[x])

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Maple [A]
time = 0.03, size = 12, normalized size = 0.71

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	\(\frac {2 x^{\frac {5}{2}}}{5}-\frac {2}{\sqrt {x}}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 11, normalized size = 0.65 \begin {gather*} \frac {2}{5} \, x^{\frac {5}{2}} - \frac {2}{\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*x^(5/2) - 2/sqrt(x)

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Fricas [A]
time = 0.36, size = 10, normalized size = 0.59 \begin {gather*} \frac {2 \, {\left (x^{3} - 5\right )}}{5 \, \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(x^3 - 5)/sqrt(x)

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Sympy [A]
time = 0.03, size = 14, normalized size = 0.82 \begin {gather*} \frac {2 x^{\frac {5}{2}}}{5} - \frac {2}{\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x**(5/2)/5 - 2/sqrt(x)

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Giac [A]
time = 0.00, size = 22, normalized size = 1.29 \begin {gather*} -\frac {2}{\sqrt {x}}+\frac {2}{5} \sqrt {x} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*x^(5/2) - 2/sqrt(x)

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Mupad [B]
time = 0.03, size = 12, normalized size = 0.71 \begin {gather*} \frac {2\,x^3-10}{5\,\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(3/2) + x^(3/2),x)

[Out]

(2*x^3 - 10)/(5*x^(1/2))

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