3.1.3 \(\int \frac {x+x^2}{\sqrt {x}} \, dx\) [3]

Optimal. Leaf size=19 \[ \frac {2 x^{3/2}}{3}+\frac {2 x^{5/2}}{5} \]

[Out]

2/3*x^(3/2)+2/5*x^(5/2)

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Rubi [A]
time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {14} \begin {gather*} \frac {2 x^{5/2}}{5}+\frac {2 x^{3/2}}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x + x^2)/Sqrt[x],x]

[Out]

(2*x^(3/2))/3 + (2*x^(5/2))/5

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 14, normalized size = 0.74 \begin {gather*} \frac {2}{15} x^{3/2} (5+3 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x + x^2)/Sqrt[x],x]

[Out]

(2*x^(3/2)*(5 + 3*x))/15

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Mathics [A]
time = 1.71, size = 10, normalized size = 0.53 \begin {gather*} \frac {2 x^{\frac {3}{2}} \left (5+3 x\right )}{15} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2 x ^ (3 / 2) (5 + 3 x) / 15

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Maple [A]
time = 0.03, size = 12, normalized size = 0.63

method	result	size
gosper	\(\frac {2 x^{\frac {3}{2}} \left (5+3 x \right )}{15}\)	\(11\)
trager	\(\frac {2 x^{\frac {3}{2}} \left (5+3 x \right )}{15}\)	\(11\)
risch	\(\frac {2 x^{\frac {3}{2}} \left (5+3 x \right )}{15}\)	\(11\)
derivativedivides	\(\frac {2 x^{\frac {3}{2}}}{3}+\frac {2 x^{\frac {5}{2}}}{5}\)	\(12\)
default	\(\frac {2 x^{\frac {3}{2}}}{3}+\frac {2 x^{\frac {5}{2}}}{5}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*x^(3/2)+2/5*x^(5/2)

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Maxima [A]
time = 0.32, size = 11, normalized size = 0.58 \begin {gather*} \frac {2}{5} \, x^{\frac {5}{2}} + \frac {2}{3} \, x^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*x^(5/2) + 2/3*x^(3/2)

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Fricas [A]
time = 0.30, size = 14, normalized size = 0.74 \begin {gather*} \frac {2}{15} \, {\left (3 \, x^{2} + 5 \, x\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*x^2 + 5*x)*sqrt(x)

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Sympy [A]
time = 0.08, size = 15, normalized size = 0.79 \begin {gather*} \frac {2 x^{\frac {5}{2}}}{5} + \frac {2 x^{\frac {3}{2}}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x**(5/2)/5 + 2*x**(3/2)/3

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Giac [A]
time = 0.00, size = 23, normalized size = 1.21 \begin {gather*} \frac {2}{5} \sqrt {x} x^{2}+\frac {2}{3} \sqrt {x} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*x^(5/2) + 2/3*x^(3/2)

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Mupad [B]
time = 0.03, size = 10, normalized size = 0.53 \begin {gather*} \frac {2\,x^{3/2}\,\left (3\,x+5\right )}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + x^2)/x^(1/2),x)

[Out]

(2*x^(3/2)*(3*x + 5))/15

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