Integrand size = 13, antiderivative size = 111 \[ \int \frac {x^{2/3}}{a+b x} \, dx=\frac {3 x^{2/3}}{2 b}+\frac {\sqrt {3} a^{2/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{b^{5/3}}+\frac {3 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{5/3}}-\frac {a^{2/3} \log (a+b x)}{2 b^{5/3}} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {52, 58, 631, 210, 31} \[ \int \frac {x^{2/3}}{a+b x} \, dx=\frac {\sqrt {3} a^{2/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{b^{5/3}}+\frac {3 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{5/3}}-\frac {a^{2/3} \log (a+b x)}{2 b^{5/3}}+\frac {3 x^{2/3}}{2 b} \]
[In]
[Out]
Rule 31
Rule 52
Rule 58
Rule 210
Rule 631
Rubi steps \begin{align*} \text {integral}& = \frac {3 x^{2/3}}{2 b}-\frac {a \int \frac {1}{\sqrt [3]{x} (a+b x)} \, dx}{b} \\ & = \frac {3 x^{2/3}}{2 b}-\frac {a^{2/3} \log (a+b x)}{2 b^{5/3}}-\frac {(3 a) \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^2}+\frac {\left (3 a^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx,x,\sqrt [3]{x}\right )}{2 b^{5/3}} \\ & = \frac {3 x^{2/3}}{2 b}+\frac {3 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{5/3}}-\frac {a^{2/3} \log (a+b x)}{2 b^{5/3}}-\frac {\left (3 a^{2/3}\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^{5/3}} \\ & = \frac {3 x^{2/3}}{2 b}+\frac {\sqrt {3} a^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{5/3}}+\frac {3 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{5/3}}-\frac {a^{2/3} \log (a+b x)}{2 b^{5/3}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.14 \[ \int \frac {x^{2/3}}{a+b x} \, dx=\frac {3 b^{2/3} x^{2/3}+2 \sqrt {3} a^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )-a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{2 b^{5/3}} \]
[In]
[Out]
Time = 0.10 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.96
method | result | size |
risch | \(\frac {3 x^{\frac {2}{3}}}{2 b}+\frac {a \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {a \ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{2 b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {a \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}}\) | \(107\) |
derivativedivides | \(\frac {3 x^{\frac {2}{3}}}{2 b}-\frac {3 \left (-\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a}{b}\) | \(112\) |
default | \(\frac {3 x^{\frac {2}{3}}}{2 b}-\frac {3 \left (-\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a}{b}\) | \(112\) |
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.15 \[ \int \frac {x^{2/3}}{a+b x} \, dx=-\frac {2 \, \sqrt {3} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x^{\frac {1}{3}} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} - \sqrt {3} a}{3 \, a}\right ) + \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (-b x^{\frac {1}{3}} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}} + a x^{\frac {2}{3}} + a \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}}\right ) - 2 \, \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (b \left (\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}} + a x^{\frac {1}{3}}\right ) - 3 \, x^{\frac {2}{3}}}{2 \, b} \]
[In]
[Out]
Time = 5.64 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.46 \[ \int \frac {x^{2/3}}{a+b x} \, dx=\begin {cases} \tilde {\infty } x^{\frac {2}{3}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {3 x^{\frac {5}{3}}}{5 a} & \text {for}\: b = 0 \\\frac {3 x^{\frac {2}{3}}}{2 b} & \text {for}\: a = 0 \\- \frac {a \log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{b^{2} \sqrt [3]{- \frac {a}{b}}} + \frac {a \log {\left (4 x^{\frac {2}{3}} + 4 \sqrt [3]{x} \sqrt [3]{- \frac {a}{b}} + 4 \left (- \frac {a}{b}\right )^{\frac {2}{3}} \right )}}{2 b^{2} \sqrt [3]{- \frac {a}{b}}} - \frac {\sqrt {3} a \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{b^{2} \sqrt [3]{- \frac {a}{b}}} + \frac {3 x^{\frac {2}{3}}}{2 b} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.03 \[ \int \frac {x^{2/3}}{a+b x} \, dx=-\frac {\sqrt {3} a \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {3 \, x^{\frac {2}{3}}}{2 \, b} - \frac {a \log \left (x^{\frac {2}{3}} - x^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{2 \, b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {a \log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.06 \[ \int \frac {x^{2/3}}{a+b x} \, dx=\frac {\left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x^{\frac {1}{3}} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{b} + \frac {3 \, x^{\frac {2}{3}}}{2 \, b} + \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{b^{3}} - \frac {\left (-a b^{2}\right )^{\frac {2}{3}} \log \left (x^{\frac {2}{3}} + x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{2 \, b^{3}} \]
[In]
[Out]
Time = 0.16 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.17 \[ \int \frac {x^{2/3}}{a+b x} \, dx=\frac {3\,x^{2/3}}{2\,b}+\frac {a^{2/3}\,\ln \left (\frac {9\,a^{7/3}}{b^{4/3}}+\frac {9\,a^2\,x^{1/3}}{b}\right )}{b^{5/3}}+\frac {a^{2/3}\,\ln \left (\frac {9\,a^{7/3}\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{b^{4/3}}+\frac {9\,a^2\,x^{1/3}}{b}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{b^{5/3}}-\frac {a^{2/3}\,\ln \left (\frac {9\,a^{7/3}\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{b^{4/3}}+\frac {9\,a^2\,x^{1/3}}{b}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{b^{5/3}} \]
[In]
[Out]