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

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) + 2*Sqrt[x] - 6*Log[1 + x^(1/6)]

________________________________________________________________________________________

Rubi [A]  time = 0.0136464, antiderivative size = 32, normalized size of antiderivative = 1., 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 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]

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 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])

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*}

Mathematica [A]  time = 0.0125579, size = 32, normalized size = 1. \[ 2 \sqrt{x}-3 \sqrt [3]{x}+6 \sqrt [6]{x}-6 \log \left (\sqrt [6]{x}+1\right ) \]

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)]

________________________________________________________________________________________

Maple [B]  time = 0., size = 92, normalized size = 2.9 \begin{align*} 2\,\ln \left ( -1+\sqrt [6]{x} \right ) -\ln \left ( \sqrt [3]{x}+\sqrt [6]{x}+1 \right ) -2\,\ln \left ( 1+\sqrt [6]{x} \right ) +\ln \left ( \sqrt [3]{x}-\sqrt [6]{x}+1 \right ) +2\,\sqrt{x}+\ln \left ( \sqrt{x}-1 \right ) -\ln \left ( \sqrt{x}+1 \right ) +6\,\sqrt [6]{x}-\ln \left ( -1+x \right ) -2\,\ln \left ( -1+\sqrt [3]{x} \right ) +\ln \left ({x}^{{\frac{2}{3}}}+\sqrt [3]{x}+1 \right ) -3\,\sqrt [3]{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.930065, size = 32, normalized size = 1. \begin{align*} 2 \, \sqrt{x} - 3 \, x^{\frac{1}{3}} + 6 \, x^{\frac{1}{6}} - 6 \, \log \left (x^{\frac{1}{6}} + 1\right ) \end{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 1.73119, size = 76, normalized size = 2.38 \begin{align*} 2 \, \sqrt{x} - 3 \, x^{\frac{1}{3}} + 6 \, x^{\frac{1}{6}} - 6 \, \log \left (x^{\frac{1}{6}} + 1\right ) \end{align*}

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)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt [3]{x} + \sqrt{x}}\, dx \end{align*}

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)

________________________________________________________________________________________

Giac [A]  time = 1.07626, size = 32, normalized size = 1. \begin{align*} 2 \, \sqrt{x} - 3 \, x^{\frac{1}{3}} + 6 \, x^{\frac{1}{6}} - 6 \, \log \left (x^{\frac{1}{6}} + 1\right ) \end{align*}

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)