Integrand size = 13, antiderivative size = 125 \[ \int \frac {x^{5/3}}{a+b x} \, dx=-\frac {3 a x^{2/3}}{2 b^2}+\frac {3 x^{5/3}}{5 b}-\frac {\sqrt {3} a^{5/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{b^{8/3}}-\frac {3 a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{8/3}}+\frac {a^{5/3} \log (a+b x)}{2 b^{8/3}} \]
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Time = 0.05 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {52, 58, 631, 210, 31} \[ \int \frac {x^{5/3}}{a+b x} \, dx=-\frac {\sqrt {3} a^{5/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{b^{8/3}}-\frac {3 a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{8/3}}+\frac {a^{5/3} \log (a+b x)}{2 b^{8/3}}-\frac {3 a x^{2/3}}{2 b^2}+\frac {3 x^{5/3}}{5 b} \]
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Rule 31
Rule 52
Rule 58
Rule 210
Rule 631
Rubi steps \begin{align*} \text {integral}& = \frac {3 x^{5/3}}{5 b}-\frac {a \int \frac {x^{2/3}}{a+b x} \, dx}{b} \\ & = -\frac {3 a x^{2/3}}{2 b^2}+\frac {3 x^{5/3}}{5 b}+\frac {a^2 \int \frac {1}{\sqrt [3]{x} (a+b x)} \, dx}{b^2} \\ & = -\frac {3 a x^{2/3}}{2 b^2}+\frac {3 x^{5/3}}{5 b}+\frac {a^{5/3} \log (a+b x)}{2 b^{8/3}}+\frac {\left (3 a^2\right ) \text {Subst}\left (\int \frac {1}{\frac {a^{2/3}}{b^{2/3}}-\frac {\sqrt [3]{a} x}{\sqrt [3]{b}}+x^2} \, dx,x,\sqrt [3]{x}\right )}{2 b^3}-\frac {\left (3 a^{5/3}\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx,x,\sqrt [3]{x}\right )}{2 b^{8/3}} \\ & = -\frac {3 a x^{2/3}}{2 b^2}+\frac {3 x^{5/3}}{5 b}-\frac {3 a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{8/3}}+\frac {a^{5/3} \log (a+b x)}{2 b^{8/3}}+\frac {\left (3 a^{5/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}\right )}{b^{8/3}} \\ & = -\frac {3 a x^{2/3}}{2 b^2}+\frac {3 x^{5/3}}{5 b}-\frac {\sqrt {3} a^{5/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{8/3}}-\frac {3 a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{8/3}}+\frac {a^{5/3} \log (a+b x)}{2 b^{8/3}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.12 \[ \int \frac {x^{5/3}}{a+b x} \, dx=\frac {-15 a b^{2/3} x^{2/3}+6 b^{5/3} x^{5/3}-10 \sqrt {3} a^{5/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-10 a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )+5 a^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{10 b^{8/3}} \]
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Time = 0.11 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.97
method | result | size |
risch | \(-\frac {3 \left (-2 b x +5 a \right ) x^{\frac {2}{3}}}{10 b^{2}}-\frac {a^{2} \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {a^{2} \ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{2 b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {a^{2} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}}\) | \(121\) |
derivativedivides | \(-\frac {3 \left (-\frac {b \,x^{\frac {5}{3}}}{5}+\frac {a \,x^{\frac {2}{3}}}{2}\right )}{b^{2}}+\frac {3 \left (-\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a^{2}}{b^{2}}\) | \(124\) |
default | \(-\frac {3 \left (-\frac {b \,x^{\frac {5}{3}}}{5}+\frac {a \,x^{\frac {2}{3}}}{2}\right )}{b^{2}}+\frac {3 \left (-\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a^{2}}{b^{2}}\) | \(124\) |
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Time = 0.22 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.18 \[ \int \frac {x^{5/3}}{a+b x} \, dx=\frac {10 \, \sqrt {3} a \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x^{\frac {1}{3}} \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} + \sqrt {3} a}{3 \, a}\right ) - 5 \, a \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (-b x^{\frac {1}{3}} \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}} + a x^{\frac {2}{3}} - a \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}}\right ) + 10 \, a \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (b \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}} + a x^{\frac {1}{3}}\right ) + 3 \, {\left (2 \, b x - 5 \, a\right )} x^{\frac {2}{3}}}{10 \, b^{2}} \]
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Time = 58.03 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.44 \[ \int \frac {x^{5/3}}{a+b x} \, dx=\begin {cases} \tilde {\infty } x^{\frac {5}{3}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {3 x^{\frac {8}{3}}}{8 a} & \text {for}\: b = 0 \\\frac {3 x^{\frac {5}{3}}}{5 b} & \text {for}\: a = 0 \\\frac {a^{2} \log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{b^{3} \sqrt [3]{- \frac {a}{b}}} - \frac {a^{2} \log {\left (4 x^{\frac {2}{3}} + 4 \sqrt [3]{x} \sqrt [3]{- \frac {a}{b}} + 4 \left (- \frac {a}{b}\right )^{\frac {2}{3}} \right )}}{2 b^{3} \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3} a^{2} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{b^{3} \sqrt [3]{- \frac {a}{b}}} - \frac {3 a x^{\frac {2}{3}}}{2 b^{2}} + \frac {3 x^{\frac {5}{3}}}{5 b} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.04 \[ \int \frac {x^{5/3}}{a+b x} \, dx=\frac {\sqrt {3} a^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {a^{2} \log \left (x^{\frac {2}{3}} - x^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{2 \, b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {a^{2} \log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {3 \, {\left (2 \, b x^{\frac {5}{3}} - 5 \, a x^{\frac {2}{3}}\right )}}{10 \, b^{2}} \]
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Time = 0.30 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.10 \[ \int \frac {x^{5/3}}{a+b x} \, dx=-\frac {a \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x^{\frac {1}{3}} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{b^{2}} - \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} a \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{b^{4}} + \frac {\left (-a b^{2}\right )^{\frac {2}{3}} a \log \left (x^{\frac {2}{3}} + x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{2 \, b^{4}} + \frac {3 \, {\left (2 \, b^{4} x^{\frac {5}{3}} - 5 \, a b^{3} x^{\frac {2}{3}}\right )}}{10 \, b^{5}} \]
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Time = 0.26 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.21 \[ \int \frac {x^{5/3}}{a+b x} \, dx=\frac {3\,x^{5/3}}{5\,b}+\frac {{\left (-a\right )}^{5/3}\,\ln \left (\frac {9\,a^4\,x^{1/3}}{b^3}-\frac {9\,{\left (-a\right )}^{13/3}}{b^{10/3}}\right )}{b^{8/3}}-\frac {3\,a\,x^{2/3}}{2\,b^2}+\frac {{\left (-a\right )}^{5/3}\,\ln \left (\frac {9\,a^4\,x^{1/3}}{b^3}-\frac {9\,{\left (-a\right )}^{13/3}\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{b^{10/3}}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{b^{8/3}}-\frac {{\left (-a\right )}^{5/3}\,\ln \left (\frac {9\,a^4\,x^{1/3}}{b^3}-\frac {9\,{\left (-a\right )}^{13/3}\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{b^{10/3}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{b^{8/3}} \]
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