3.578 \(\int \frac{\sqrt [3]{x}}{\sqrt [4]{x}+\sqrt{x}} \, dx\)

Optimal. Leaf size=76 \[ \frac{6 x^{5/6}}{5}-\frac{12 x^{7/12}}{7}+3 \sqrt [3]{x}-12 \sqrt [12]{x}+6 \log \left (\sqrt [12]{x}+1\right )-2 \log \left (\sqrt [4]{x}+1\right )-4 \sqrt{3} \tan ^{-1}\left (\frac{1-2 \sqrt [12]{x}}{\sqrt{3}}\right ) \]

[Out]

-12*x^(1/12) + 3*x^(1/3) - (12*x^(7/12))/7 + (6*x^(5/6))/5 - 4*Sqrt[3]*ArcTan[(1 - 2*x^(1/12))/Sqrt[3]] + 6*Lo
g[1 + x^(1/12)] - 2*Log[1 + x^(1/4)]

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Rubi [A]  time = 0.0344146, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {1584, 341, 50, 58, 618, 204, 31} \[ \frac{6 x^{5/6}}{5}-\frac{12 x^{7/12}}{7}+3 \sqrt [3]{x}-12 \sqrt [12]{x}+6 \log \left (\sqrt [12]{x}+1\right )-2 \log \left (\sqrt [4]{x}+1\right )-4 \sqrt{3} \tan ^{-1}\left (\frac{1-2 \sqrt [12]{x}}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^(1/3)/(x^(1/4) + Sqrt[x]),x]

[Out]

-12*x^(1/12) + 3*x^(1/3) - (12*x^(7/12))/7 + (6*x^(5/6))/5 - 4*Sqrt[3]*ArcTan[(1 - 2*x^(1/12))/Sqrt[3]] + 6*Lo
g[1 + x^(1/12)] - 2*Log[1 + x^(1/4)]

Rule 1584

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

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 58

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[-((b*c - a*d)/b), 3]}, -Sim
p[Log[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (Dist[3/(2*b*q), Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d
*x)^(1/3)], x] + Dist[3/(2*b*q^2), Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x
] && NegQ[(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}}{\sqrt [4]{x}+\sqrt{x}} \, dx &=\int \frac{\sqrt [12]{x}}{1+\sqrt [4]{x}} \, dx\\ &=4 \operatorname{Subst}\left (\int \frac{x^{10/3}}{1+x} \, dx,x,\sqrt [4]{x}\right )\\ &=\frac{6 x^{5/6}}{5}-4 \operatorname{Subst}\left (\int \frac{x^{7/3}}{1+x} \, dx,x,\sqrt [4]{x}\right )\\ &=-\frac{12 x^{7/12}}{7}+\frac{6 x^{5/6}}{5}+4 \operatorname{Subst}\left (\int \frac{x^{4/3}}{1+x} \, dx,x,\sqrt [4]{x}\right )\\ &=3 \sqrt [3]{x}-\frac{12 x^{7/12}}{7}+\frac{6 x^{5/6}}{5}-4 \operatorname{Subst}\left (\int \frac{\sqrt [3]{x}}{1+x} \, dx,x,\sqrt [4]{x}\right )\\ &=-12 \sqrt [12]{x}+3 \sqrt [3]{x}-\frac{12 x^{7/12}}{7}+\frac{6 x^{5/6}}{5}+4 \operatorname{Subst}\left (\int \frac{1}{x^{2/3} (1+x)} \, dx,x,\sqrt [4]{x}\right )\\ &=-12 \sqrt [12]{x}+3 \sqrt [3]{x}-\frac{12 x^{7/12}}{7}+\frac{6 x^{5/6}}{5}-2 \log \left (1+\sqrt [4]{x}\right )+6 \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,\sqrt [12]{x}\right )+6 \operatorname{Subst}\left (\int \frac{1}{1-x+x^2} \, dx,x,\sqrt [12]{x}\right )\\ &=-12 \sqrt [12]{x}+3 \sqrt [3]{x}-\frac{12 x^{7/12}}{7}+\frac{6 x^{5/6}}{5}+6 \log \left (1+\sqrt [12]{x}\right )-2 \log \left (1+\sqrt [4]{x}\right )-12 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 \sqrt [12]{x}\right )\\ &=-12 \sqrt [12]{x}+3 \sqrt [3]{x}-\frac{12 x^{7/12}}{7}+\frac{6 x^{5/6}}{5}-4 \sqrt{3} \tan ^{-1}\left (\frac{1-2 \sqrt [12]{x}}{\sqrt{3}}\right )+6 \log \left (1+\sqrt [12]{x}\right )-2 \log \left (1+\sqrt [4]{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0175064, size = 83, normalized size = 1.09 \[ \frac{6 x^{5/6}}{5}-\frac{12 x^{7/12}}{7}+3 \sqrt [3]{x}-12 \sqrt [12]{x}+4 \log \left (\sqrt [12]{x}+1\right )-2 \log \left (\sqrt [6]{x}-\sqrt [12]{x}+1\right )+4 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [12]{x}-1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^(1/3)/(x^(1/4) + Sqrt[x]),x]

[Out]

-12*x^(1/12) + 3*x^(1/3) - (12*x^(7/12))/7 + (6*x^(5/6))/5 + 4*Sqrt[3]*ArcTan[(-1 + 2*x^(1/12))/Sqrt[3]] + 4*L
og[1 + x^(1/12)] - 2*Log[1 - x^(1/12) + x^(1/6)]

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Maple [A]  time = 0.003, size = 61, normalized size = 0.8 \begin{align*}{\frac{6}{5}{x}^{{\frac{5}{6}}}}-{\frac{12}{7}{x}^{{\frac{7}{12}}}}+3\,\sqrt [3]{x}-12\,{x}^{1/12}-2\,\ln \left ( 1-{x}^{1/12}+\sqrt [6]{x} \right ) +4\,\sqrt{3}\arctan \left ( 1/3\, \left ( 2\,{x}^{1/12}-1 \right ) \sqrt{3} \right ) +4\,\ln \left ( 1+{x}^{1/12} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/3)/(x^(1/4)+x^(1/2)),x)

[Out]

6/5*x^(5/6)-12/7*x^(7/12)+3*x^(1/3)-12*x^(1/12)-2*ln(1-x^(1/12)+x^(1/6))+4*3^(1/2)*arctan(1/3*(2*x^(1/12)-1)*3
^(1/2))+4*ln(1+x^(1/12))

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Maxima [A]  time = 1.89103, size = 81, normalized size = 1.07 \begin{align*} 4 \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{\frac{1}{12}} - 1\right )}\right ) + \frac{6}{5} \, x^{\frac{5}{6}} - \frac{12}{7} \, x^{\frac{7}{12}} + 3 \, x^{\frac{1}{3}} - 12 \, x^{\frac{1}{12}} - 2 \, \log \left (x^{\frac{1}{6}} - x^{\frac{1}{12}} + 1\right ) + 4 \, \log \left (x^{\frac{1}{12}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/3)/(x^(1/4)+x^(1/2)),x, algorithm="maxima")

[Out]

4*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^(1/12) - 1)) + 6/5*x^(5/6) - 12/7*x^(7/12) + 3*x^(1/3) - 12*x^(1/12) - 2*log
(x^(1/6) - x^(1/12) + 1) + 4*log(x^(1/12) + 1)

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Fricas [A]  time = 1.32073, size = 221, normalized size = 2.91 \begin{align*} 4 \, \sqrt{3} \arctan \left (\frac{2}{3} \, \sqrt{3} x^{\frac{1}{12}} - \frac{1}{3} \, \sqrt{3}\right ) + \frac{6}{5} \, x^{\frac{5}{6}} - \frac{12}{7} \, x^{\frac{7}{12}} + 3 \, x^{\frac{1}{3}} - 12 \, x^{\frac{1}{12}} - 2 \, \log \left (x^{\frac{1}{6}} - x^{\frac{1}{12}} + 1\right ) + 4 \, \log \left (x^{\frac{1}{12}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/3)/(x^(1/4)+x^(1/2)),x, algorithm="fricas")

[Out]

4*sqrt(3)*arctan(2/3*sqrt(3)*x^(1/12) - 1/3*sqrt(3)) + 6/5*x^(5/6) - 12/7*x^(7/12) + 3*x^(1/3) - 12*x^(1/12) -
2*log(x^(1/6) - x^(1/12) + 1) + 4*log(x^(1/12) + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt [3]{x}}{\sqrt [4]{x} + \sqrt{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(1/3)/(x**(1/4)+x**(1/2)),x)

[Out]

Integral(x**(1/3)/(x**(1/4) + sqrt(x)), x)

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Giac [A]  time = 1.12424, size = 81, normalized size = 1.07 \begin{align*} 4 \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{\frac{1}{12}} - 1\right )}\right ) + \frac{6}{5} \, x^{\frac{5}{6}} - \frac{12}{7} \, x^{\frac{7}{12}} + 3 \, x^{\frac{1}{3}} - 12 \, x^{\frac{1}{12}} - 2 \, \log \left (x^{\frac{1}{6}} - x^{\frac{1}{12}} + 1\right ) + 4 \, \log \left (x^{\frac{1}{12}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/3)/(x^(1/4)+x^(1/2)),x, algorithm="giac")

[Out]

4*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^(1/12) - 1)) + 6/5*x^(5/6) - 12/7*x^(7/12) + 3*x^(1/3) - 12*x^(1/12) - 2*log
(x^(1/6) - x^(1/12) + 1) + 4*log(x^(1/12) + 1)