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

Optimal. Leaf size=119 \[ \frac{6 x^{7/6}}{7}-\frac{12 x^{13/12}}{13}-\frac{12 x^{11/12}}{11}+\frac{6 x^{5/6}}{5}-\frac{4 x^{3/4}}{3}+\frac{3 x^{2/3}}{2}-\frac{12 x^{7/12}}{7}-\frac{12 x^{5/12}}{5}+x+2 \sqrt{x}+3 \sqrt [3]{x}-4 \sqrt [4]{x}+6 \sqrt [6]{x}-12 \sqrt [12]{x}+12 \log \left (\sqrt [12]{x}+1\right ) \]

[Out]

-12*x^(1/12) + 6*x^(1/6) - 4*x^(1/4) + 3*x^(1/3) - (12*x^(5/12))/5 + 2*Sqrt[x] - (12*x^(7/12))/7 + (3*x^(2/3))
/2 - (4*x^(3/4))/3 + (6*x^(5/6))/5 - (12*x^(11/12))/11 + x - (12*x^(13/12))/13 + (6*x^(7/6))/7 + 12*Log[1 + x^
(1/12)]

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Rubi [A]  time = 0.0493564, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {1584, 266, 43} \[ \frac{6 x^{7/6}}{7}-\frac{12 x^{13/12}}{13}-\frac{12 x^{11/12}}{11}+\frac{6 x^{5/6}}{5}-\frac{4 x^{3/4}}{3}+\frac{3 x^{2/3}}{2}-\frac{12 x^{7/12}}{7}-\frac{12 x^{5/12}}{5}+x+2 \sqrt{x}+3 \sqrt [3]{x}-4 \sqrt [4]{x}+6 \sqrt [6]{x}-12 \sqrt [12]{x}+12 \log \left (\sqrt [12]{x}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(x^(1/4) + x^(1/3)),x]

[Out]

-12*x^(1/12) + 6*x^(1/6) - 4*x^(1/4) + 3*x^(1/3) - (12*x^(5/12))/5 + 2*Sqrt[x] - (12*x^(7/12))/7 + (3*x^(2/3))
/2 - (4*x^(3/4))/3 + (6*x^(5/6))/5 - (12*x^(11/12))/11 + x - (12*x^(13/12))/13 + (6*x^(7/6))/7 + 12*Log[1 + x^
(1/12)]

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 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{\sqrt{x}}{\sqrt [4]{x}+\sqrt [3]{x}} \, dx &=\int \frac{\sqrt [4]{x}}{1+\sqrt [12]{x}} \, dx\\ &=12 \operatorname{Subst}\left (\int \frac{x^{14}}{1+x} \, dx,x,\sqrt [12]{x}\right )\\ &=12 \operatorname{Subst}\left (\int \left (-1+x-x^2+x^3-x^4+x^5-x^6+x^7-x^8+x^9-x^{10}+x^{11}-x^{12}+x^{13}+\frac{1}{1+x}\right ) \, dx,x,\sqrt [12]{x}\right )\\ &=-12 \sqrt [12]{x}+6 \sqrt [6]{x}-4 \sqrt [4]{x}+3 \sqrt [3]{x}-\frac{12 x^{5/12}}{5}+2 \sqrt{x}-\frac{12 x^{7/12}}{7}+\frac{3 x^{2/3}}{2}-\frac{4 x^{3/4}}{3}+\frac{6 x^{5/6}}{5}-\frac{12 x^{11/12}}{11}+x-\frac{12 x^{13/12}}{13}+\frac{6 x^{7/6}}{7}+12 \log \left (1+\sqrt [12]{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0329904, size = 119, normalized size = 1. \[ \frac{6 x^{7/6}}{7}-\frac{12 x^{13/12}}{13}-\frac{12 x^{11/12}}{11}+\frac{6 x^{5/6}}{5}-\frac{4 x^{3/4}}{3}+\frac{3 x^{2/3}}{2}-\frac{12 x^{7/12}}{7}-\frac{12 x^{5/12}}{5}+x+2 \sqrt{x}+3 \sqrt [3]{x}-4 \sqrt [4]{x}+6 \sqrt [6]{x}-12 \sqrt [12]{x}+12 \log \left (\sqrt [12]{x}+1\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-12*x^(1/12) + 6*x^(1/6) - 4*x^(1/4) + 3*x^(1/3) - (12*x^(5/12))/5 + 2*Sqrt[x] - (12*x^(7/12))/7 + (3*x^(2/3))
/2 - (4*x^(3/4))/3 + (6*x^(5/6))/5 - (12*x^(11/12))/11 + x - (12*x^(13/12))/13 + (6*x^(7/6))/7 + 12*Log[1 + x^
(1/12)]

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Maple [A]  time = 0.004, size = 76, normalized size = 0.6 \begin{align*} -12\,{x}^{1/12}+6\,\sqrt [6]{x}-4\,\sqrt [4]{x}+3\,\sqrt [3]{x}-{\frac{12}{5}{x}^{{\frac{5}{12}}}}-{\frac{12}{7}{x}^{{\frac{7}{12}}}}+{\frac{3}{2}{x}^{{\frac{2}{3}}}}-{\frac{4}{3}{x}^{{\frac{3}{4}}}}+{\frac{6}{5}{x}^{{\frac{5}{6}}}}-{\frac{12}{11}{x}^{{\frac{11}{12}}}}+x-{\frac{12}{13}{x}^{{\frac{13}{12}}}}+{\frac{6}{7}{x}^{{\frac{7}{6}}}}+12\,\ln \left ( 1+{x}^{1/12} \right ) +2\,\sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-12*x^(1/12)+6*x^(1/6)-4*x^(1/4)+3*x^(1/3)-12/5*x^(5/12)-12/7*x^(7/12)+3/2*x^(2/3)-4/3*x^(3/4)+6/5*x^(5/6)-12/
11*x^(11/12)+x-12/13*x^(13/12)+6/7*x^(7/6)+12*ln(1+x^(1/12))+2*x^(1/2)

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Maxima [A]  time = 1.17092, size = 101, normalized size = 0.85 \begin{align*} \frac{6}{7} \, x^{\frac{7}{6}} - \frac{12}{13} \, x^{\frac{13}{12}} + x - \frac{12}{11} \, x^{\frac{11}{12}} + \frac{6}{5} \, x^{\frac{5}{6}} - \frac{4}{3} \, x^{\frac{3}{4}} + \frac{3}{2} \, x^{\frac{2}{3}} - \frac{12}{7} \, x^{\frac{7}{12}} + 2 \, \sqrt{x} - \frac{12}{5} \, x^{\frac{5}{12}} + 3 \, x^{\frac{1}{3}} - 4 \, x^{\frac{1}{4}} + 6 \, x^{\frac{1}{6}} - 12 \, x^{\frac{1}{12}} + 12 \, \log \left (x^{\frac{1}{12}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6/7*x^(7/6) - 12/13*x^(13/12) + x - 12/11*x^(11/12) + 6/5*x^(5/6) - 4/3*x^(3/4) + 3/2*x^(2/3) - 12/7*x^(7/12)
+ 2*sqrt(x) - 12/5*x^(5/12) + 3*x^(1/3) - 4*x^(1/4) + 6*x^(1/6) - 12*x^(1/12) + 12*log(x^(1/12) + 1)

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Fricas [A]  time = 1.23181, size = 273, normalized size = 2.29 \begin{align*} \frac{6}{7} \,{\left (x + 7\right )} x^{\frac{1}{6}} - \frac{12}{13} \,{\left (x + 13\right )} x^{\frac{1}{12}} + x - \frac{12}{11} \, x^{\frac{11}{12}} + \frac{6}{5} \, x^{\frac{5}{6}} - \frac{4}{3} \, x^{\frac{3}{4}} + \frac{3}{2} \, x^{\frac{2}{3}} - \frac{12}{7} \, x^{\frac{7}{12}} + 2 \, \sqrt{x} - \frac{12}{5} \, x^{\frac{5}{12}} + 3 \, x^{\frac{1}{3}} - 4 \, x^{\frac{1}{4}} + 12 \, \log \left (x^{\frac{1}{12}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6/7*(x + 7)*x^(1/6) - 12/13*(x + 13)*x^(1/12) + x - 12/11*x^(11/12) + 6/5*x^(5/6) - 4/3*x^(3/4) + 3/2*x^(2/3)
- 12/7*x^(7/12) + 2*sqrt(x) - 12/5*x^(5/12) + 3*x^(1/3) - 4*x^(1/4) + 12*log(x^(1/12) + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.11659, size = 101, normalized size = 0.85 \begin{align*} \frac{6}{7} \, x^{\frac{7}{6}} - \frac{12}{13} \, x^{\frac{13}{12}} + x - \frac{12}{11} \, x^{\frac{11}{12}} + \frac{6}{5} \, x^{\frac{5}{6}} - \frac{4}{3} \, x^{\frac{3}{4}} + \frac{3}{2} \, x^{\frac{2}{3}} - \frac{12}{7} \, x^{\frac{7}{12}} + 2 \, \sqrt{x} - \frac{12}{5} \, x^{\frac{5}{12}} + 3 \, x^{\frac{1}{3}} - 4 \, x^{\frac{1}{4}} + 6 \, x^{\frac{1}{6}} - 12 \, x^{\frac{1}{12}} + 12 \, \log \left (x^{\frac{1}{12}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6/7*x^(7/6) - 12/13*x^(13/12) + x - 12/11*x^(11/12) + 6/5*x^(5/6) - 4/3*x^(3/4) + 3/2*x^(2/3) - 12/7*x^(7/12)
+ 2*sqrt(x) - 12/5*x^(5/12) + 3*x^(1/3) - 4*x^(1/4) + 6*x^(1/6) - 12*x^(1/12) + 12*log(x^(1/12) + 1)