3.572 \(\int \frac{1}{\sqrt [4]{x}+\sqrt [3]{x}} \, dx\)

Optimal. Leaf size=73 \[ \frac{3 x^{2/3}}{2}-\frac{12 x^{7/12}}{7}-\frac{12 x^{5/12}}{5}+2 \sqrt{x}+3 \sqrt [3]{x}-4 \sqrt [4]{x}+6 \sqrt [6]{x}-12 \sqrt [12]{x}+12 \log \left (\sqrt [12]{x}+1\right ) \]

[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 + 12*Log[1 + x^(1/12)]

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Rubi [A]  time = 0.0282093, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {1593, 266, 43} \[ \frac{3 x^{2/3}}{2}-\frac{12 x^{7/12}}{7}-\frac{12 x^{5/12}}{5}+2 \sqrt{x}+3 \sqrt [3]{x}-4 \sqrt [4]{x}+6 \sqrt [6]{x}-12 \sqrt [12]{x}+12 \log \left (\sqrt [12]{x}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[(x^(1/4) + x^(1/3))^(-1),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 + 12*Log[1 + x^(1/12)]

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 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{1}{\sqrt [4]{x}+\sqrt [3]{x}} \, dx &=\int \frac{1}{\left (1+\sqrt [12]{x}\right ) \sqrt [4]{x}} \, dx\\ &=12 \operatorname{Subst}\left (\int \frac{x^8}{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+\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}+12 \log \left (1+\sqrt [12]{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0240054, size = 73, normalized size = 1. \[ \frac{3 x^{2/3}}{2}-\frac{12 x^{7/12}}{7}-\frac{12 x^{5/12}}{5}+2 \sqrt{x}+3 \sqrt [3]{x}-4 \sqrt [4]{x}+6 \sqrt [6]{x}-12 \sqrt [12]{x}+12 \log \left (\sqrt [12]{x}+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(x^(1/4) + x^(1/3))^(-1),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 + 12*Log[1 + x^(1/12)]

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Maple [B]  time = 0.087, size = 173, normalized size = 2.4 \begin{align*} -\ln \left ({x}^{{\frac{2}{3}}}+\sqrt [3]{x}+1 \right ) +2\,\ln \left ( \sqrt [3]{x}-1 \right ) +\ln \left ( -1+\sqrt{x} \right ) -\ln \left ( 1+\sqrt{x} \right ) -2\,\ln \left ( 1+\sqrt [6]{x} \right ) +\ln \left ( 1-\sqrt [6]{x}+\sqrt [3]{x} \right ) +2\,\ln \left ( \sqrt [6]{x}-1 \right ) -\ln \left ( \sqrt [3]{x}+\sqrt [6]{x}+1 \right ) -2\,\ln \left ( \sqrt [4]{x}-1 \right ) -{\frac{12}{7}{x}^{{\frac{7}{12}}}}-{\frac{12}{5}{x}^{{\frac{5}{12}}}}-12\,{x}^{1/12}+2\,\ln \left ( 1+\sqrt [4]{x} \right ) +\ln \left ( x-1 \right ) +{\frac{3}{2}{x}^{{\frac{2}{3}}}}+6\,\sqrt [6]{x}-4\,\sqrt [4]{x}+3\,\sqrt [3]{x}+2\,\sqrt{x}+4\,\ln \left ( 1+{x}^{1/12} \right ) -2\,\ln \left ( 1-{x}^{1/12}+\sqrt [6]{x} \right ) -4\,\ln \left ({x}^{1/12}-1 \right ) +2\,\ln \left ( \sqrt [6]{x}+{x}^{1/12}+1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.29995, size = 66, normalized size = 0.9 \begin{align*} \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(1/(x^(1/4)+x^(1/3)),x, algorithm="maxima")

[Out]

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 = 1.18504, size = 176, normalized size = 2.41 \begin{align*} \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(1/(x^(1/4)+x^(1/3)),x, algorithm="fricas")

[Out]

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt [4]{x} + \sqrt [3]{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x**(1/4) + x**(1/3)), x)

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Giac [A]  time = 1.1225, size = 66, normalized size = 0.9 \begin{align*} \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(1/(x^(1/4)+x^(1/3)),x, algorithm="giac")

[Out]

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)