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

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Rubi [A]  time = 0.03, antiderivative size = 73, normalized size of antiderivative = 1.00, 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 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])

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 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]

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*}

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Mathematica [A]  time = 0.02, size = 73, normalized size = 1.00 \[ \frac {3 x^{2/3}}{2}-\frac {12 x^{7/12}}{7}-\frac {12 x^{5/12}}{5}+2 \sqrt {x}+3 \sqrt [3]{x}-4 \sqrt [4]{x}+6 \sqrt [6]{x}-12 \sqrt [12]{x}+12 \log \left (\sqrt [12]{x}+1\right ) \]

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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fricas [A]  time = 0.43, size = 49, normalized size = 0.67 \[ \frac {3}{2} \, x^{\frac {2}{3}} - \frac {12}{7} \, x^{\frac {7}{12}} + 2 \, \sqrt {x} - \frac {12}{5} \, x^{\frac {5}{12}} + 3 \, x^{\frac {1}{3}} - 4 \, x^{\frac {1}{4}} + 6 \, x^{\frac {1}{6}} - 12 \, x^{\frac {1}{12}} + 12 \, \log \left (x^{\frac {1}{12}} + 1\right ) \]

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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giac [A]  time = 0.32, size = 49, normalized size = 0.67 \[ \frac {3}{2} \, x^{\frac {2}{3}} - \frac {12}{7} \, x^{\frac {7}{12}} + 2 \, \sqrt {x} - \frac {12}{5} \, x^{\frac {5}{12}} + 3 \, x^{\frac {1}{3}} - 4 \, x^{\frac {1}{4}} + 6 \, x^{\frac {1}{6}} - 12 \, x^{\frac {1}{12}} + 12 \, \log \left (x^{\frac {1}{12}} + 1\right ) \]

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)

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maple [B]  time = 0.12, size = 173, normalized size = 2.37 \[ \ln \left (x -1\right )+\ln \left (\sqrt {x}-1\right )-\ln \left (\sqrt {x}+1\right )+2 \ln \left (x^{\frac {1}{4}}+1\right )-2 \ln \left (x^{\frac {1}{6}}+1\right )+4 \ln \left (x^{\frac {1}{12}}+1\right )+2 \ln \left (x^{\frac {1}{3}}-1\right )-2 \ln \left (x^{\frac {1}{4}}-1\right )+2 \ln \left (x^{\frac {1}{6}}-1\right )-4 \ln \left (x^{\frac {1}{12}}-1\right )+\ln \left (x^{\frac {1}{3}}-x^{\frac {1}{6}}+1\right )-2 \ln \left (x^{\frac {1}{6}}-x^{\frac {1}{12}}+1\right )-\ln \left (x^{\frac {1}{3}}+x^{\frac {1}{6}}+1\right )+2 \ln \left (x^{\frac {1}{6}}+x^{\frac {1}{12}}+1\right )-\ln \left (x^{\frac {2}{3}}+x^{\frac {1}{3}}+1\right )+\frac {3 x^{\frac {2}{3}}}{2}-\frac {12 x^{\frac {7}{12}}}{7}+2 \sqrt {x}-\frac {12 x^{\frac {5}{12}}}{5}+3 x^{\frac {1}{3}}-4 x^{\frac {1}{4}}+6 x^{\frac {1}{6}}-12 x^{\frac {1}{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.70, size = 49, normalized size = 0.67 \[ \frac {3}{2} \, x^{\frac {2}{3}} - \frac {12}{7} \, x^{\frac {7}{12}} + 2 \, \sqrt {x} - \frac {12}{5} \, x^{\frac {5}{12}} + 3 \, x^{\frac {1}{3}} - 4 \, x^{\frac {1}{4}} + 6 \, x^{\frac {1}{6}} - 12 \, x^{\frac {1}{12}} + 12 \, \log \left (x^{\frac {1}{12}} + 1\right ) \]

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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mupad [B]  time = 0.04, size = 49, normalized size = 0.67 \[ 12\,\ln \left (x^{1/12}+1\right )+2\,\sqrt {x}+3\,x^{1/3}-4\,x^{1/4}+\frac {3\,x^{2/3}}{2}+6\,x^{1/6}-12\,x^{1/12}-\frac {12\,x^{5/12}}{5}-\frac {12\,x^{7/12}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(1/3) + x^(1/4)),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt [4]{x} + \sqrt [3]{x}}\, dx \]

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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