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

Optimal. Leaf size=41 \[ \frac{6 x^{7/6}}{7}-\frac{6 x^{5/6}}{5}+2 \sqrt{x}-6 \sqrt [6]{x}+6 \tan ^{-1}\left (\sqrt [6]{x}\right ) \]

[Out]

-6*x^(1/6) + 2*Sqrt[x] - (6*x^(5/6))/5 + (6*x^(7/6))/7 + 6*ArcTan[x^(1/6)]

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Rubi [A]  time = 0.010337, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {341, 50, 63, 203} \[ \frac{6 x^{7/6}}{7}-\frac{6 x^{5/6}}{5}+2 \sqrt{x}-6 \sqrt [6]{x}+6 \tan ^{-1}\left (\sqrt [6]{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(1 + x^(1/3)),x]

[Out]

-6*x^(1/6) + 2*Sqrt[x] - (6*x^(5/6))/5 + (6*x^(7/6))/7 + 6*ArcTan[x^(1/6)]

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 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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])

Rubi steps

\begin{align*} \int \frac{\sqrt{x}}{1+\sqrt [3]{x}} \, dx &=3 \operatorname{Subst}\left (\int \frac{x^{7/2}}{1+x} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{6 x^{7/6}}{7}-3 \operatorname{Subst}\left (\int \frac{x^{5/2}}{1+x} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{6 x^{5/6}}{5}+\frac{6 x^{7/6}}{7}+3 \operatorname{Subst}\left (\int \frac{x^{3/2}}{1+x} \, dx,x,\sqrt [3]{x}\right )\\ &=2 \sqrt{x}-\frac{6 x^{5/6}}{5}+\frac{6 x^{7/6}}{7}-3 \operatorname{Subst}\left (\int \frac{\sqrt{x}}{1+x} \, dx,x,\sqrt [3]{x}\right )\\ &=-6 \sqrt [6]{x}+2 \sqrt{x}-\frac{6 x^{5/6}}{5}+\frac{6 x^{7/6}}{7}+3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} (1+x)} \, dx,x,\sqrt [3]{x}\right )\\ &=-6 \sqrt [6]{x}+2 \sqrt{x}-\frac{6 x^{5/6}}{5}+\frac{6 x^{7/6}}{7}+6 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt [6]{x}\right )\\ &=-6 \sqrt [6]{x}+2 \sqrt{x}-\frac{6 x^{5/6}}{5}+\frac{6 x^{7/6}}{7}+6 \tan ^{-1}\left (\sqrt [6]{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0074931, size = 41, normalized size = 1. \[ \frac{6 x^{7/6}}{7}-\frac{6 x^{5/6}}{5}+2 \sqrt{x}-6 \sqrt [6]{x}+6 \tan ^{-1}\left (\sqrt [6]{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/(1 + x^(1/3)),x]

[Out]

-6*x^(1/6) + 2*Sqrt[x] - (6*x^(5/6))/5 + (6*x^(7/6))/7 + 6*ArcTan[x^(1/6)]

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Maple [A]  time = 0.001, size = 28, normalized size = 0.7 \begin{align*} -6\,\sqrt [6]{x}-{\frac{6}{5}{x}^{{\frac{5}{6}}}}+{\frac{6}{7}{x}^{{\frac{7}{6}}}}+6\,\arctan \left ( \sqrt [6]{x} \right ) +2\,\sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6*x^(1/6)-6/5*x^(5/6)+6/7*x^(7/6)+6*arctan(x^(1/6))+2*x^(1/2)

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Maxima [A]  time = 1.49019, size = 36, normalized size = 0.88 \begin{align*} \frac{6}{7} \, x^{\frac{7}{6}} - \frac{6}{5} \, x^{\frac{5}{6}} + 2 \, \sqrt{x} - 6 \, x^{\frac{1}{6}} + 6 \, \arctan \left (x^{\frac{1}{6}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6/7*x^(7/6) - 6/5*x^(5/6) + 2*sqrt(x) - 6*x^(1/6) + 6*arctan(x^(1/6))

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Fricas [A]  time = 1.48156, size = 90, normalized size = 2.2 \begin{align*} \frac{6}{7} \,{\left (x - 7\right )} x^{\frac{1}{6}} - \frac{6}{5} \, x^{\frac{5}{6}} + 2 \, \sqrt{x} + 6 \, \arctan \left (x^{\frac{1}{6}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6/7*(x - 7)*x^(1/6) - 6/5*x^(5/6) + 2*sqrt(x) + 6*arctan(x^(1/6))

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Sympy [A]  time = 2.89894, size = 37, normalized size = 0.9 \begin{align*} \frac{6 x^{\frac{7}{6}}}{7} - \frac{6 x^{\frac{5}{6}}}{5} - 6 \sqrt [6]{x} + 2 \sqrt{x} + 6 \operatorname{atan}{\left (\sqrt [6]{x} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6*x**(7/6)/7 - 6*x**(5/6)/5 - 6*x**(1/6) + 2*sqrt(x) + 6*atan(x**(1/6))

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Giac [A]  time = 1.12019, size = 36, normalized size = 0.88 \begin{align*} \frac{6}{7} \, x^{\frac{7}{6}} - \frac{6}{5} \, x^{\frac{5}{6}} + 2 \, \sqrt{x} - 6 \, x^{\frac{1}{6}} + 6 \, \arctan \left (x^{\frac{1}{6}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6/7*x^(7/6) - 6/5*x^(5/6) + 2*sqrt(x) - 6*x^(1/6) + 6*arctan(x^(1/6))