3.1.89 \(\int \frac {1+x^2}{\sqrt {x}} \, dx\) [89]

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

[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 = 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}+2 \sqrt {x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x^2)/Sqrt[x],x]

[Out]

2*Sqrt[x] + (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 {1+x^2}{\sqrt {x}} \, dx &=\int \left (\frac {1}{\sqrt {x}}+x^{3/2}\right ) \, dx\\ &=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}{5} \sqrt {x} \left (5+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^2)/Sqrt[x],x]

[Out]

(2*Sqrt[x]*(5 + x^2))/5

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Mathics [A]
time = 1.77, size = 10, normalized size = 0.59 \begin {gather*} \frac {2 \sqrt {x} \left (5+x^2\right )}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(x^2 + 1)/Sqrt[x],x]')

[Out]

2 Sqrt[x] (5 + x ^ 2) / 5

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Maple [A]
time = 0.01, size = 12, normalized size = 0.71

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 11, normalized size = 0.65 \begin {gather*} \frac {2}{5} \, x^{\frac {5}{2}} + 2 \, \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*x^(5/2) + 2*sqrt(x)

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Fricas [A]
time = 0.31, size = 10, normalized size = 0.59 \begin {gather*} \frac {2}{5} \, {\left (x^{2} + 5\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(x^2 + 5)*sqrt(x)

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Sympy [A]
time = 0.08, size = 14, normalized size = 0.82 \begin {gather*} \frac {2 x^{\frac {5}{2}}}{5} + 2 \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x**(5/2)/5 + 2*sqrt(x)

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Giac [A]
time = 0.00, size = 20, normalized size = 1.18 \begin {gather*} \frac {2}{5} \sqrt {x} x^{2}+2 \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*x^(5/2) + 2*sqrt(x)

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Mupad [B]
time = 0.02, size = 10, normalized size = 0.59 \begin {gather*} \frac {2\,\sqrt {x}\,\left (x^2+5\right )}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 + 1)/x^(1/2),x)

[Out]

(2*x^(1/2)*(x^2 + 5))/5

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