3.2256 \(\int \frac{\sqrt [3]{x}}{1+\sqrt{x}} \, dx\)

Optimal. Leaf size=58 \[ \frac{6 x^{5/6}}{5}-3 \sqrt [3]{x}-3 \log \left (\sqrt [6]{x}+1\right )+\log \left (\sqrt{x}+1\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{1-2 \sqrt [6]{x}}{\sqrt{3}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.02957, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {341, 50, 56, 618, 204, 31} \[ \frac{6 x^{5/6}}{5}-3 \sqrt [3]{x}-3 \log \left (\sqrt [6]{x}+1\right )+\log \left (\sqrt{x}+1\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{1-2 \sqrt [6]{x}}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^(1/3)/(1 + Sqrt[x]),x]

[Out]

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

Rule 341

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

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 56

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[-((b*c - a*d)/b), 3]}, Simp
[Log[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(
1/3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && Ne
gQ[(b*c - a*d)/b]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rule 31

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

Rubi steps

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

Mathematica [C]  time = 0.0078134, size = 35, normalized size = 0.6 \[ \frac{3}{5} \sqrt [3]{x} \left (5 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};-\sqrt{x}\right )+2 \sqrt{x}-5\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^(1/3)/(1 + Sqrt[x]),x]

[Out]

(3*x^(1/3)*(-5 + 2*Sqrt[x] + 5*Hypergeometric2F1[2/3, 1, 5/3, -Sqrt[x]]))/5

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Maple [A]  time = 0.007, size = 49, normalized size = 0.8 \begin{align*}{\frac{6}{5}{x}^{{\frac{5}{6}}}}-3\,\sqrt [3]{x}+\ln \left ( \sqrt [3]{x}-\sqrt [6]{x}+1 \right ) +2\,\sqrt{3}\arctan \left ( 1/3\, \left ( 2\,\sqrt [6]{x}-1 \right ) \sqrt{3} \right ) -2\,\ln \left ( 1+\sqrt [6]{x} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.51049, size = 65, normalized size = 1.12 \begin{align*} 2 \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{\frac{1}{6}} - 1\right )}\right ) + \frac{6}{5} \, x^{\frac{5}{6}} - 3 \, x^{\frac{1}{3}} + \log \left (x^{\frac{1}{3}} - x^{\frac{1}{6}} + 1\right ) - 2 \, \log \left (x^{\frac{1}{6}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.3426, size = 174, normalized size = 3. \begin{align*} 2 \, \sqrt{3} \arctan \left (\frac{2}{3} \, \sqrt{3} x^{\frac{1}{6}} - \frac{1}{3} \, \sqrt{3}\right ) + \frac{6}{5} \, x^{\frac{5}{6}} - 3 \, x^{\frac{1}{3}} + \log \left (x^{\frac{1}{3}} - x^{\frac{1}{6}} + 1\right ) - 2 \, \log \left (x^{\frac{1}{6}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 1.40805, size = 138, normalized size = 2.38 \begin{align*} \frac{16 x^{\frac{5}{6}} \Gamma \left (\frac{8}{3}\right )}{5 \Gamma \left (\frac{11}{3}\right )} - \frac{8 \sqrt [3]{x} \Gamma \left (\frac{8}{3}\right )}{\Gamma \left (\frac{11}{3}\right )} - \frac{16 e^{- \frac{2 i \pi }{3}} \log{\left (- \sqrt [6]{x} e^{\frac{i \pi }{3}} + 1 \right )} \Gamma \left (\frac{8}{3}\right )}{3 \Gamma \left (\frac{11}{3}\right )} - \frac{16 \log{\left (- \sqrt [6]{x} e^{i \pi } + 1 \right )} \Gamma \left (\frac{8}{3}\right )}{3 \Gamma \left (\frac{11}{3}\right )} - \frac{16 e^{\frac{2 i \pi }{3}} \log{\left (- \sqrt [6]{x} e^{\frac{5 i \pi }{3}} + 1 \right )} \Gamma \left (\frac{8}{3}\right )}{3 \Gamma \left (\frac{11}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

16*x**(5/6)*gamma(8/3)/(5*gamma(11/3)) - 8*x**(1/3)*gamma(8/3)/gamma(11/3) - 16*exp(-2*I*pi/3)*log(-x**(1/6)*e
xp_polar(I*pi/3) + 1)*gamma(8/3)/(3*gamma(11/3)) - 16*log(-x**(1/6)*exp_polar(I*pi) + 1)*gamma(8/3)/(3*gamma(1
1/3)) - 16*exp(2*I*pi/3)*log(-x**(1/6)*exp_polar(5*I*pi/3) + 1)*gamma(8/3)/(3*gamma(11/3))

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Giac [A]  time = 1.0984, size = 65, normalized size = 1.12 \begin{align*} 2 \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{\frac{1}{6}} - 1\right )}\right ) + \frac{6}{5} \, x^{\frac{5}{6}} - 3 \, x^{\frac{1}{3}} + \log \left (x^{\frac{1}{3}} - x^{\frac{1}{6}} + 1\right ) - 2 \, \log \left (x^{\frac{1}{6}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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