3.1914 \(\int (\frac{1}{x^{3/2}}+x^{3/2}) \, dx\)

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

[Out]

-2/Sqrt[x] + (2*x^(5/2))/5

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Rubi [A]  time = 0.0018102, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 0, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \frac{2 x^{5/2}}{5}-\frac{2}{\sqrt{x}} \]

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*}

Mathematica [A]  time = 0.0041351, size = 14, normalized size = 0.82 \[ \frac{2 \left (x^3-5\right )}{5 \sqrt{x}} \]

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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Maple [A]  time = 0.003, size = 11, normalized size = 0.7 \begin{align*}{\frac{2\,{x}^{3}-10}{5}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.969533, size = 15, normalized size = 0.88 \begin{align*} \frac{2}{5} \, x^{\frac{5}{2}} - \frac{2}{\sqrt{x}} \end{align*}

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 = 2.18133, size = 31, normalized size = 1.82 \begin{align*} \frac{2 \,{\left (x^{3} - 5\right )}}{5 \, \sqrt{x}} \end{align*}

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 = 0.053506, size = 14, normalized size = 0.82 \begin{align*} \frac{2 x^{\frac{5}{2}}}{5} - \frac{2}{\sqrt{x}} \end{align*}

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 = 1.04802, size = 15, normalized size = 0.88 \begin{align*} \frac{2}{5} \, x^{\frac{5}{2}} - \frac{2}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*x^(5/2) - 2/sqrt(x)