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

Optimal. Leaf size=130 \[ \frac {4 x^{5/4}}{5}-\frac {6 x^{7/6}}{7}+\frac {12 x^{13/12}}{13}+\frac {12 x^{11/12}}{11}-\frac {6 x^{5/6}}{5}+\frac {4 x^{3/4}}{3}-\frac {3 x^{2/3}}{2}+\frac {12 x^{7/12}}{7}+\frac {12 x^{5/12}}{5}-x-2 \sqrt {x}-3 \sqrt [3]{x}+4 \sqrt [4]{x}-6 \sqrt [6]{x}+12 \sqrt [12]{x}-12 \log \left (\sqrt [12]{x}+1\right ) \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 130, 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} \[ \frac {4 x^{5/4}}{5}-\frac {6 x^{7/6}}{7}+\frac {12 x^{13/12}}{13}+\frac {12 x^{11/12}}{11}-\frac {6 x^{5/6}}{5}+\frac {4 x^{3/4}}{3}-\frac {3 x^{2/3}}{2}+\frac {12 x^{7/12}}{7}+\frac {12 x^{5/12}}{5}-x-2 \sqrt {x}-3 \sqrt [3]{x}+4 \sqrt [4]{x}-6 \sqrt [6]{x}+12 \sqrt [12]{x}-12 \log \left (\sqrt [12]{x}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[(x^(-1/3) + x^(-1/4))^(-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 + (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 + (4*x^(5/4))/5
- 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}{\frac {1}{\sqrt [3]{x}}+\frac {1}{\sqrt [4]{x}}} \, dx &=\int \frac {\sqrt [3]{x}}{1+\sqrt [12]{x}} \, dx\\ &=12 \operatorname {Subst}\left (\int \frac {x^{15}}{1+x} \, dx,x,\sqrt [12]{x}\right )\\ &=12 \operatorname {Subst}\left (\int \left (1+\frac {1}{-1-x}-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}+x^{14}\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}+\frac {4 x^{5/4}}{5}-12 \log \left (1+\sqrt [12]{x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 130, normalized size = 1.00 \[ \frac {4 x^{5/4}}{5}-\frac {6 x^{7/6}}{7}+\frac {12 x^{13/12}}{13}+\frac {12 x^{11/12}}{11}-\frac {6 x^{5/6}}{5}+\frac {4 x^{3/4}}{3}-\frac {3 x^{2/3}}{2}+\frac {12 x^{7/12}}{7}+\frac {12 x^{5/12}}{5}-x-2 \sqrt {x}-3 \sqrt [3]{x}+4 \sqrt [4]{x}-6 \sqrt [6]{x}+12 \sqrt [12]{x}-12 \log \left (\sqrt [12]{x}+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(x^(-1/3) + x^(-1/4))^(-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 + (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 + (4*x^(5/4))/5
- 12*Log[1 + x^(1/12)]

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fricas [A]  time = 0.44, size = 76, normalized size = 0.58 \[ \frac {4}{5} \, {\left (x + 5\right )} x^{\frac {1}{4}} - \frac {6}{7} \, {\left (x + 7\right )} x^{\frac {1}{6}} + \frac {12}{13} \, {\left (x + 13\right )} x^{\frac {1}{12}} - x + \frac {12}{11} \, x^{\frac {11}{12}} - \frac {6}{5} \, x^{\frac {5}{6}} + \frac {4}{3} \, x^{\frac {3}{4}} - \frac {3}{2} \, x^{\frac {2}{3}} + \frac {12}{7} \, x^{\frac {7}{12}} - 2 \, \sqrt {x} + \frac {12}{5} \, x^{\frac {5}{12}} - 3 \, x^{\frac {1}{3}} - 12 \, \log \left (x^{\frac {1}{12}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/5*(x + 5)*x^(1/4) - 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) - 12*log(x^(1/12) + 1)

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giac [A]  time = 0.36, size = 82, normalized size = 0.63 \[ \frac {4}{5} \, x^{\frac {5}{4}} - \frac {6}{7} \, x^{\frac {7}{6}} + \frac {12}{13} \, x^{\frac {13}{12}} - x + \frac {12}{11} \, x^{\frac {11}{12}} - \frac {6}{5} \, x^{\frac {5}{6}} + \frac {4}{3} \, x^{\frac {3}{4}} - \frac {3}{2} \, x^{\frac {2}{3}} + \frac {12}{7} \, x^{\frac {7}{12}} - 2 \, \sqrt {x} + \frac {12}{5} \, x^{\frac {5}{12}} - 3 \, x^{\frac {1}{3}} + 4 \, x^{\frac {1}{4}} - 6 \, x^{\frac {1}{6}} + 12 \, x^{\frac {1}{12}} - 12 \, \log \left (x^{\frac {1}{12}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/5*x^(5/4) - 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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maple [A]  time = 0.00, size = 83, normalized size = 0.64 \[ \frac {4 x^{\frac {5}{4}}}{5}-\frac {6 x^{\frac {7}{6}}}{7}+\frac {12 x^{\frac {13}{12}}}{13}-x -12 \ln \left (x^{\frac {1}{12}}+1\right )+\frac {12 x^{\frac {11}{12}}}{11}-\frac {6 x^{\frac {5}{6}}}{5}+\frac {4 x^{\frac {3}{4}}}{3}-\frac {3 x^{\frac {2}{3}}}{2}+\frac {12 x^{\frac {7}{12}}}{7}-2 \sqrt {x}+\frac {12 x^{\frac {5}{12}}}{5}-3 x^{\frac {1}{3}}+4 x^{\frac {1}{4}}-6 x^{\frac {1}{6}}+12 x^{\frac {1}{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maxima [A]  time = 0.73, size = 82, normalized size = 0.63 \[ \frac {4}{5} \, x^{\frac {5}{4}} - \frac {6}{7} \, x^{\frac {7}{6}} + \frac {12}{13} \, x^{\frac {13}{12}} - x + \frac {12}{11} \, x^{\frac {11}{12}} - \frac {6}{5} \, x^{\frac {5}{6}} + \frac {4}{3} \, x^{\frac {3}{4}} - \frac {3}{2} \, x^{\frac {2}{3}} + \frac {12}{7} \, x^{\frac {7}{12}} - 2 \, \sqrt {x} + \frac {12}{5} \, x^{\frac {5}{12}} - 3 \, x^{\frac {1}{3}} + 4 \, x^{\frac {1}{4}} - 6 \, x^{\frac {1}{6}} + 12 \, x^{\frac {1}{12}} - 12 \, \log \left (x^{\frac {1}{12}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/5*x^(5/4) - 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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mupad [B]  time = 0.15, size = 82, normalized size = 0.63 \[ 4\,x^{1/4}-12\,\ln \left (x^{1/12}+1\right )-2\,\sqrt {x}-3\,x^{1/3}-x-\frac {3\,x^{2/3}}{2}-6\,x^{1/6}+\frac {4\,x^{3/4}}{3}+\frac {4\,x^{5/4}}{5}-\frac {6\,x^{5/6}}{5}+12\,x^{1/12}-\frac {6\,x^{7/6}}{7}+\frac {12\,x^{5/12}}{5}+\frac {12\,x^{7/12}}{7}+\frac {12\,x^{11/12}}{11}+\frac {12\,x^{13/12}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1/x^(1/3) + 1/x^(1/4)),x)

[Out]

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

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sympy [A]  time = 3.08, size = 121, normalized size = 0.93 \[ \frac {12 x^{\frac {13}{12}}}{13} + \frac {12 x^{\frac {11}{12}}}{11} + \frac {12 x^{\frac {7}{12}}}{7} + \frac {12 x^{\frac {5}{12}}}{5} + 12 \sqrt [12]{x} - \frac {6 x^{\frac {7}{6}}}{7} - \frac {6 x^{\frac {5}{6}}}{5} - 6 \sqrt [6]{x} + \frac {4 x^{\frac {5}{4}}}{5} + \frac {4 x^{\frac {3}{4}}}{3} + 4 \sqrt [4]{x} - \frac {3 x^{\frac {2}{3}}}{2} - 3 \sqrt [3]{x} - 2 \sqrt {x} - x - 12 \log {\left (\sqrt [12]{x} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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