3.568 \(\int \frac {1}{\sqrt [3]{x}+\sqrt {x}} \, dx\)

Optimal. Leaf size=32 \[ 2 \sqrt {x}-3 \sqrt [3]{x}+6 \sqrt [6]{x}-6 \log \left (\sqrt [6]{x}+1\right ) \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1593, 266, 43} \[ 2 \sqrt {x}-3 \sqrt [3]{x}+6 \sqrt [6]{x}-6 \log \left (\sqrt [6]{x}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[(x^(1/3) + Sqrt[x])^(-1),x]

[Out]

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

Rule 43

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 266

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 1593

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

Rubi steps

\begin {align*} \int \frac {1}{\sqrt [3]{x}+\sqrt {x}} \, dx &=\int \frac {1}{\left (1+\sqrt [6]{x}\right ) \sqrt [3]{x}} \, dx\\ &=6 \operatorname {Subst}\left (\int \frac {x^3}{1+x} \, dx,x,\sqrt [6]{x}\right )\\ &=6 \operatorname {Subst}\left (\int \left (1+\frac {1}{-1-x}-x+x^2\right ) \, dx,x,\sqrt [6]{x}\right )\\ &=6 \sqrt [6]{x}-3 \sqrt [3]{x}+2 \sqrt {x}-6 \log \left (1+\sqrt [6]{x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 32, normalized size = 1.00 \[ 2 \sqrt {x}-3 \sqrt [3]{x}+6 \sqrt [6]{x}-6 \log \left (\sqrt [6]{x}+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(x^(1/3) + Sqrt[x])^(-1),x]

[Out]

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

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fricas [A]  time = 0.43, size = 24, normalized size = 0.75 \[ 2 \, \sqrt {x} - 3 \, x^{\frac {1}{3}} + 6 \, x^{\frac {1}{6}} - 6 \, \log \left (x^{\frac {1}{6}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.33, size = 24, normalized size = 0.75 \[ 2 \, \sqrt {x} - 3 \, x^{\frac {1}{3}} + 6 \, x^{\frac {1}{6}} - 6 \, \log \left (x^{\frac {1}{6}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.03, size = 92, normalized size = 2.88 \[ -\ln \left (x -1\right )+\ln \left (\sqrt {x}-1\right )-\ln \left (\sqrt {x}+1\right )-2 \ln \left (x^{\frac {1}{6}}+1\right )-2 \ln \left (x^{\frac {1}{3}}-1\right )+2 \ln \left (x^{\frac {1}{6}}-1\right )+\ln \left (x^{\frac {1}{3}}-x^{\frac {1}{6}}+1\right )-\ln \left (x^{\frac {1}{3}}+x^{\frac {1}{6}}+1\right )+\ln \left (x^{\frac {2}{3}}+x^{\frac {1}{3}}+1\right )+2 \sqrt {x}-3 x^{\frac {1}{3}}+6 x^{\frac {1}{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.70, size = 24, normalized size = 0.75 \[ 2 \, \sqrt {x} - 3 \, x^{\frac {1}{3}} + 6 \, x^{\frac {1}{6}} - 6 \, \log \left (x^{\frac {1}{6}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.03, size = 24, normalized size = 0.75 \[ 2\,\sqrt {x}-6\,\ln \left (x^{1/6}+1\right )-3\,x^{1/3}+6\,x^{1/6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(1/2) + x^(1/3)),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt [3]{x} + \sqrt {x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x**(1/3) + sqrt(x)), x)

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