3.432 \(\int \sqrt{x+x^{5/2}} \, dx\)

Optimal. Leaf size=20 \[ \frac{4 \left (x^{5/2}+x\right )^{3/2}}{9 x^{3/2}} \]

[Out]

(4*(x + x^(5/2))^(3/2))/(9*x^(3/2))

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Rubi [A]  time = 0.003314, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2000} \[ \frac{4 \left (x^{5/2}+x\right )^{3/2}}{9 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x + x^(5/2)],x]

[Out]

(4*(x + x^(5/2))^(3/2))/(9*x^(3/2))

Rule 2000

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + b*x^n)^(p + 1)/(b*(n - j)*(p + 1)*x
^(n - 1)), x] /; FreeQ[{a, b, j, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && EqQ[j*p - n + j + 1, 0]

Rubi steps

\begin{align*} \int \sqrt{x+x^{5/2}} \, dx &=\frac{4 \left (x+x^{5/2}\right )^{3/2}}{9 x^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0099119, size = 20, normalized size = 1. \[ \frac{4 \left (x^{5/2}+x\right )^{3/2}}{9 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x + x^(5/2)],x]

[Out]

(4*(x + x^(5/2))^(3/2))/(9*x^(3/2))

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Maple [A]  time = 0.009, size = 18, normalized size = 0.9 \begin{align*}{\frac{4}{9}\sqrt{x+{x}^{{\frac{5}{2}}}} \left ( 1+{x}^{{\frac{3}{2}}} \right ){\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x+x^(5/2))^(1/2),x)

[Out]

4/9*(x+x^(5/2))^(1/2)/x^(1/2)*(1+x^(3/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{x^{\frac{5}{2}} + x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(x^(5/2) + x), x)

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Fricas [A]  time = 1.1921, size = 55, normalized size = 2.75 \begin{align*} \frac{4 \, \sqrt{x^{\frac{5}{2}} + x}{\left (x^{2} + \sqrt{x}\right )}}{9 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/9*sqrt(x^(5/2) + x)*(x^2 + sqrt(x))/x

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{x^{\frac{5}{2}} + x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x**(5/2) + x), x)

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Giac [A]  time = 1.27574, size = 15, normalized size = 0.75 \begin{align*} \frac{4}{9} \,{\left (x^{\frac{3}{2}} + 1\right )}^{\frac{3}{2}} - \frac{4}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/9*(x^(3/2) + 1)^(3/2) - 4/9