Optimal. Leaf size=44 \[ -\frac {9}{5} \sqrt [3]{1-5 x}+\frac {3}{10} (1-5 x)^{2/3}+x+\frac {27}{5} \log \left (3+\sqrt [3]{1-5 x}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {442, 383, 78}
\begin {gather*} x+\frac {3}{10} (1-5 x)^{2/3}-\frac {9}{5} \sqrt [3]{1-5 x}+\frac {27}{5} \log \left (\sqrt [3]{1-5 x}+3\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 383
Rule 442
Rubi steps
\begin {align*} \int \frac {2+\sqrt [3]{1-5 x}}{3+\sqrt [3]{1-5 x}} \, dx &=-\left (\frac {1}{5} \text {Subst}\left (\int \frac {2+\sqrt [3]{x}}{3+\sqrt [3]{x}} \, dx,x,1-5 x\right )\right )\\ &=-\left (\frac {3}{5} \text {Subst}\left (\int \frac {x^2 (2+x)}{3+x} \, dx,x,\sqrt [3]{1-5 x}\right )\right )\\ &=-\left (\frac {3}{5} \text {Subst}\left (\int \left (3-x+x^2-\frac {9}{3+x}\right ) \, dx,x,\sqrt [3]{1-5 x}\right )\right )\\ &=-\frac {9}{5} \sqrt [3]{1-5 x}+\frac {3}{10} (1-5 x)^{2/3}+x+\frac {27}{5} \log \left (3+\sqrt [3]{1-5 x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 45, normalized size = 1.02 \begin {gather*} \frac {1}{10} \left (-2-18 \sqrt [3]{1-5 x}+3 (1-5 x)^{2/3}+10 x+54 \log \left (3+\sqrt [3]{1-5 x}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.32, size = 34, normalized size = 0.77
method | result | size |
derivativedivides | \(-\frac {1}{5}+x +\frac {3 \left (1-5 x \right )^{\frac {2}{3}}}{10}-\frac {9 \left (1-5 x \right )^{\frac {1}{3}}}{5}+\frac {27 \ln \left (3+\left (1-5 x \right )^{\frac {1}{3}}\right )}{5}\) | \(34\) |
default | \(-\frac {1}{5}+x +\frac {3 \left (1-5 x \right )^{\frac {2}{3}}}{10}-\frac {9 \left (1-5 x \right )^{\frac {1}{3}}}{5}+\frac {27 \ln \left (3+\left (1-5 x \right )^{\frac {1}{3}}\right )}{5}\) | \(34\) |
trager | \(x -\frac {9 \left (1-5 x \right )^{\frac {1}{3}}}{5}+\frac {3 \left (1-5 x \right )^{\frac {2}{3}}}{10}+\frac {9 \ln \left (9 \left (1-5 x \right )^{\frac {2}{3}}+27 \left (1-5 x \right )^{\frac {1}{3}}-5 x +28\right )}{5}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 33, normalized size = 0.75 \begin {gather*} x + \frac {3}{10} \, {\left (-5 \, x + 1\right )}^{\frac {2}{3}} - \frac {9}{5} \, {\left (-5 \, x + 1\right )}^{\frac {1}{3}} + \frac {27}{5} \, \log \left ({\left (-5 \, x + 1\right )}^{\frac {1}{3}} + 3\right ) - \frac {1}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 32, normalized size = 0.73 \begin {gather*} x + \frac {3}{10} \, {\left (-5 \, x + 1\right )}^{\frac {2}{3}} - \frac {9}{5} \, {\left (-5 \, x + 1\right )}^{\frac {1}{3}} + \frac {27}{5} \, \log \left ({\left (-5 \, x + 1\right )}^{\frac {1}{3}} + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 39, normalized size = 0.89 \begin {gather*} x + \frac {3 \left (1 - 5 x\right )^{\frac {2}{3}}}{10} - \frac {9 \sqrt [3]{1 - 5 x}}{5} + \frac {27 \log {\left (\sqrt [3]{1 - 5 x} + 3 \right )}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.36, size = 33, normalized size = 0.75 \begin {gather*} x + \frac {3}{10} \, {\left (-5 \, x + 1\right )}^{\frac {2}{3}} - \frac {9}{5} \, {\left (-5 \, x + 1\right )}^{\frac {1}{3}} + \frac {27}{5} \, \log \left ({\left (-5 \, x + 1\right )}^{\frac {1}{3}} + 3\right ) - \frac {1}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 32, normalized size = 0.73 \begin {gather*} x+\frac {27\,\ln \left ({\left (1-5\,x\right )}^{1/3}+3\right )}{5}-\frac {9\,{\left (1-5\,x\right )}^{1/3}}{5}+\frac {3\,{\left (1-5\,x\right )}^{2/3}}{10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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