Optimal. Leaf size=119 \[ -12 \sqrt [12]{x}+6 \sqrt [6]{x}-4 \sqrt [4]{x}+3 \sqrt [3]{x}-\frac {12 x^{5/12}}{5}+2 \sqrt {x}-\frac {12 x^{7/12}}{7}+\frac {3 x^{2/3}}{2}-\frac {4 x^{3/4}}{3}+\frac {6 x^{5/6}}{5}-\frac {12 x^{11/12}}{11}+x-\frac {12 x^{13/12}}{13}+\frac {6 x^{7/6}}{7}+12 \log \left (1+\sqrt [12]{x}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 119, normalized size of antiderivative = 1.00, 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 = {1598, 272, 45}
\begin {gather*} \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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 1598
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 \text {Subst}\left (\int \frac {x^{14}}{1+x} \, dx,x,\sqrt [12]{x}\right )\\ &=12 \text {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*}
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Mathematica [A]
time = 0.03, size = 110, normalized size = 0.92 \begin {gather*} \frac {-360360 \sqrt [12]{x}+180180 \sqrt [6]{x}-120120 \sqrt [4]{x}+90090 \sqrt [3]{x}-72072 x^{5/12}+60060 \sqrt {x}-51480 x^{7/12}+45045 x^{2/3}-40040 x^{3/4}+36036 x^{5/6}-32760 x^{11/12}+30030 x-27720 x^{13/12}+25740 x^{7/6}}{30030}+12 \log \left (1+\sqrt [12]{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.30, size = 76, normalized size = 0.64
method | result | size |
derivativedivides | \(-12 x^{\frac {1}{12}}+6 x^{\frac {1}{6}}-4 x^{\frac {1}{4}}+3 x^{\frac {1}{3}}-\frac {12 x^{\frac {5}{12}}}{5}-\frac {12 x^{\frac {7}{12}}}{7}+\frac {3 x^{\frac {2}{3}}}{2}-\frac {4 x^{\frac {3}{4}}}{3}+\frac {6 x^{\frac {5}{6}}}{5}-\frac {12 x^{\frac {11}{12}}}{11}+x -\frac {12 x^{\frac {13}{12}}}{13}+\frac {6 x^{\frac {7}{6}}}{7}+12 \ln \left (1+x^{\frac {1}{12}}\right )+2 \sqrt {x}\) | \(76\) |
default | \(-12 x^{\frac {1}{12}}+6 x^{\frac {1}{6}}-4 x^{\frac {1}{4}}+3 x^{\frac {1}{3}}-\frac {12 x^{\frac {5}{12}}}{5}-\frac {12 x^{\frac {7}{12}}}{7}+\frac {3 x^{\frac {2}{3}}}{2}-\frac {4 x^{\frac {3}{4}}}{3}+\frac {6 x^{\frac {5}{6}}}{5}-\frac {12 x^{\frac {11}{12}}}{11}+x -\frac {12 x^{\frac {13}{12}}}{13}+\frac {6 x^{\frac {7}{6}}}{7}+12 \ln \left (1+x^{\frac {1}{12}}\right )+2 \sqrt {x}\) | \(76\) |
meijerg | \(-\frac {x^{\frac {1}{12}} \left (-25740 x^{\frac {13}{12}}+27720 x -30030 x^{\frac {11}{12}}+32760 x^{\frac {5}{6}}-36036 x^{\frac {3}{4}}+40040 x^{\frac {2}{3}}-45045 x^{\frac {7}{12}}+51480 \sqrt {x}-60060 x^{\frac {5}{12}}+72072 x^{\frac {1}{3}}-90090 x^{\frac {1}{4}}+120120 x^{\frac {1}{6}}-180180 x^{\frac {1}{12}}+360360\right )}{30030}+12 \ln \left (1+x^{\frac {1}{12}}\right )\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 75, normalized size = 0.63 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 71, normalized size = 0.60 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x}}{\sqrt [4]{x} + \sqrt [3]{x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.17, size = 75, normalized size = 0.63 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.15, size = 75, normalized size = 0.63 \begin {gather*} x+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}-\frac {4\,x^{3/4}}{3}+\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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