3.1.12 \(\int \frac {\sqrt {x}}{-\sqrt [3]{x}+\sqrt {x}} \, dx\) [12]

Optimal. Leaf size=53 \[ 6 \sqrt [6]{x}+3 \sqrt [3]{x}+2 \sqrt {x}+\frac {3 x^{2/3}}{2}+\frac {6 x^{5/6}}{5}+x+6 \log \left (1-\sqrt [6]{x}\right ) \]

[Out]

6*x^(1/6)+3*x^(1/3)+2*x^(1/2)+3/2*x^(2/3)+6/5*x^(5/6)+x+6*ln(1-x^(1/6))

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Rubi [A]
time = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1598, 272, 45} \begin {gather*} \frac {6 x^{5/6}}{5}+\frac {3 x^{2/3}}{2}+x+2 \sqrt {x}+3 \sqrt [3]{x}+6 \sqrt [6]{x}+6 \log \left (1-\sqrt [6]{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(-x^(1/3) + Sqrt[x]),x]

[Out]

6*x^(1/6) + 3*x^(1/3) + 2*Sqrt[x] + (3*x^(2/3))/2 + (6*x^(5/6))/5 + x + 6*Log[1 - x^(1/6)]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 1598

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\int \frac {\sqrt [6]{x}}{-1+\sqrt [6]{x}} \, dx\\ &=6 \text {Subst}\left (\int \frac {x^6}{-1+x} \, dx,x,\sqrt [6]{x}\right )\\ &=6 \text {Subst}\left (\int \left (1+\frac {1}{-1+x}+x+x^2+x^3+x^4+x^5\right ) \, dx,x,\sqrt [6]{x}\right )\\ &=6 \sqrt [6]{x}+3 \sqrt [3]{x}+2 \sqrt {x}+\frac {3 x^{2/3}}{2}+\frac {6 x^{5/6}}{5}+x+6 \log \left (1-\sqrt [6]{x}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.01, size = 51, normalized size = 0.96 \begin {gather*} 6 \sqrt [6]{x}+3 \sqrt [3]{x}+2 \sqrt {x}+\frac {3 x^{2/3}}{2}+\frac {6 x^{5/6}}{5}+x+6 \log \left (-1+\sqrt [6]{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/(-x^(1/3) + Sqrt[x]),x]

[Out]

6*x^(1/6) + 3*x^(1/3) + 2*Sqrt[x] + (3*x^(2/3))/2 + (6*x^(5/6))/5 + x + 6*Log[-1 + x^(1/6)]

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Maple [A]
time = 0.07, size = 36, normalized size = 0.68

method	result	size
derivativedivides	\(x +\frac {6 x^{\frac {5}{6}}}{5}+\frac {3 x^{\frac {2}{3}}}{2}+2 \sqrt {x}+3 x^{\frac {1}{3}}+6 x^{\frac {1}{6}}+6 \ln \left (x^{\frac {1}{6}}-1\right )\)	\(36\)
default	\(x +\frac {6 x^{\frac {5}{6}}}{5}+\frac {3 x^{\frac {2}{3}}}{2}+2 \sqrt {x}+3 x^{\frac {1}{3}}+6 x^{\frac {1}{6}}+6 \ln \left (x^{\frac {1}{6}}-1\right )\)	\(36\)
meijerg	\(\frac {x^{\frac {1}{6}} \left (70 x^{\frac {5}{6}}+84 x^{\frac {2}{3}}+105 \sqrt {x}+140 x^{\frac {1}{3}}+210 x^{\frac {1}{6}}+420\right )}{70}+6 \ln \left (1-x^{\frac {1}{6}}\right )\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x+6/5*x^(5/6)+3/2*x^(2/3)+2*x^(1/2)+3*x^(1/3)+6*x^(1/6)+6*ln(x^(1/6)-1)

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Maxima [A]
time = 0.44, size = 35, normalized size = 0.66 \begin {gather*} x + \frac {6}{5} \, x^{\frac {5}{6}} + \frac {3}{2} \, x^{\frac {2}{3}} + 2 \, \sqrt {x} + 3 \, x^{\frac {1}{3}} + 6 \, x^{\frac {1}{6}} + 6 \, \log \left (x^{\frac {1}{6}} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 6/5*x^(5/6) + 3/2*x^(2/3) + 2*sqrt(x) + 3*x^(1/3) + 6*x^(1/6) + 6*log(x^(1/6) - 1)

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Fricas [A]
time = 0.57, size = 35, normalized size = 0.66 \begin {gather*} x + \frac {6}{5} \, x^{\frac {5}{6}} + \frac {3}{2} \, x^{\frac {2}{3}} + 2 \, \sqrt {x} + 3 \, x^{\frac {1}{3}} + 6 \, x^{\frac {1}{6}} + 6 \, \log \left (x^{\frac {1}{6}} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 6/5*x^(5/6) + 3/2*x^(2/3) + 2*sqrt(x) + 3*x^(1/3) + 6*x^(1/6) + 6*log(x^(1/6) - 1)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x}}{- \sqrt [3]{x} + \sqrt {x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x)/(-x**(1/3) + sqrt(x)), x)

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Giac [A]
time = 0.42, size = 36, normalized size = 0.68 \begin {gather*} x + \frac {6}{5} \, x^{\frac {5}{6}} + \frac {3}{2} \, x^{\frac {2}{3}} + 2 \, \sqrt {x} + 3 \, x^{\frac {1}{3}} + 6 \, x^{\frac {1}{6}} + 6 \, \log \left ({\left | x^{\frac {1}{6}} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 6/5*x^(5/6) + 3/2*x^(2/3) + 2*sqrt(x) + 3*x^(1/3) + 6*x^(1/6) + 6*log(abs(x^(1/6) - 1))

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Mupad [B]
time = 0.10, size = 35, normalized size = 0.66 \begin {gather*} x+6\,\ln \left (x^{1/6}-1\right )+2\,\sqrt {x}+3\,x^{1/3}+\frac {3\,x^{2/3}}{2}+6\,x^{1/6}+\frac {6\,x^{5/6}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)/(x^(1/2) - x^(1/3)),x)

[Out]

x + 6*log(x^(1/6) - 1) + 2*x^(1/2) + 3*x^(1/3) + (3*x^(2/3))/2 + 6*x^(1/6) + (6*x^(5/6))/5

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Chatgpt [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {not solved} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

not solved

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