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

Optimal. Leaf size=42 \[ \frac{3 x^{2/3}}{2}-3 \sqrt [3]{x}+6 \sqrt [6]{x}+3 \log \left (\sqrt [3]{x}+1\right )-6 \tan ^{-1}\left (\sqrt [6]{x}\right ) \]

[Out]

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

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Rubi [A]  time = 0.153555, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {6688, 1593, 1802, 635, 203, 260} \[ \frac{3 x^{2/3}}{2}-3 \sqrt [3]{x}+6 \sqrt [6]{x}+3 \log \left (\sqrt [3]{x}+1\right )-6 \tan ^{-1}\left (\sqrt [6]{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 + Sqrt[x])/((1 + x^(1/3))*Sqrt[x]),x]

[Out]

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

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

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 1802

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int \frac{1+\sqrt{x}}{\left (1+\sqrt [3]{x}\right ) \sqrt{x}} \, dx &=\int \frac{1+\frac{1}{\sqrt{x}}}{1+\sqrt [3]{x}} \, dx\\ &=6 \operatorname{Subst}\left (\int \frac{x^2+x^5}{1+x^2} \, dx,x,\sqrt [6]{x}\right )\\ &=6 \operatorname{Subst}\left (\int \frac{x^2 \left (1+x^3\right )}{1+x^2} \, dx,x,\sqrt [6]{x}\right )\\ &=6 \operatorname{Subst}\left (\int \left (1-x+x^3-\frac{1-x}{1+x^2}\right ) \, dx,x,\sqrt [6]{x}\right )\\ &=6 \sqrt [6]{x}-3 \sqrt [3]{x}+\frac{3 x^{2/3}}{2}-6 \operatorname{Subst}\left (\int \frac{1-x}{1+x^2} \, dx,x,\sqrt [6]{x}\right )\\ &=6 \sqrt [6]{x}-3 \sqrt [3]{x}+\frac{3 x^{2/3}}{2}-6 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt [6]{x}\right )+6 \operatorname{Subst}\left (\int \frac{x}{1+x^2} \, dx,x,\sqrt [6]{x}\right )\\ &=6 \sqrt [6]{x}-3 \sqrt [3]{x}+\frac{3 x^{2/3}}{2}-6 \tan ^{-1}\left (\sqrt [6]{x}\right )+3 \log \left (1+\sqrt [3]{x}\right )\\ \end{align*}

Mathematica [C]  time = 0.0222615, size = 54, normalized size = 1.29 \[ \frac{3 x^{2/3}}{2}-3 \sqrt [3]{x}+6 \sqrt [6]{x}+(3+3 i) \log \left (-\sqrt [6]{x}+i\right )+(3-3 i) \log \left (\sqrt [6]{x}+i\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.012, size = 48, normalized size = 1.1 \begin{align*} \ln \left ( 1+x \right ) +{\frac{3}{2}{x}^{{\frac{2}{3}}}}-\ln \left ({x}^{{\frac{2}{3}}}-\sqrt [3]{x}+1 \right ) +2\,\ln \left ( 1+\sqrt [3]{x} \right ) -3\,\sqrt [3]{x}+6\,\sqrt [6]{x}-6\,\arctan \left ( \sqrt [6]{x} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.65704, size = 41, normalized size = 0.98 \begin{align*} \frac{3}{2} \, x^{\frac{2}{3}} - 3 \, x^{\frac{1}{3}} + 6 \, x^{\frac{1}{6}} - 6 \, \arctan \left (x^{\frac{1}{6}}\right ) + 3 \, \log \left (x^{\frac{1}{3}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.65724, size = 105, normalized size = 2.5 \begin{align*} \frac{3}{2} \, x^{\frac{2}{3}} - 3 \, x^{\frac{1}{3}} + 6 \, x^{\frac{1}{6}} - 6 \, \arctan \left (x^{\frac{1}{6}}\right ) + 3 \, \log \left (x^{\frac{1}{3}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 16.6806, size = 39, normalized size = 0.93 \begin{align*} 6 \sqrt [6]{x} + \frac{3 x^{\frac{2}{3}}}{2} - 3 \sqrt [3]{x} + 3 \log{\left (\sqrt [3]{x} + 1 \right )} - 6 \operatorname{atan}{\left (\sqrt [6]{x} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.07974, size = 41, normalized size = 0.98 \begin{align*} \frac{3}{2} \, x^{\frac{2}{3}} - 3 \, x^{\frac{1}{3}} + 6 \, x^{\frac{1}{6}} - 6 \, \arctan \left (x^{\frac{1}{6}}\right ) + 3 \, \log \left (x^{\frac{1}{3}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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