Integrand size = 13, antiderivative size = 140 \[ \int \frac {1}{x^{2/3} (a+b x)^3} \, dx=\frac {\sqrt [3]{x}}{2 a (a+b x)^2}+\frac {5 \sqrt [3]{x}}{6 a^2 (a+b x)}-\frac {5 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{8/3} \sqrt [3]{b}}+\frac {5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{6 a^{8/3} \sqrt [3]{b}}-\frac {5 \log (a+b x)}{18 a^{8/3} \sqrt [3]{b}} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 140, 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 = {44, 60, 631, 210, 31} \[ \int \frac {1}{x^{2/3} (a+b x)^3} \, dx=-\frac {5 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{8/3} \sqrt [3]{b}}+\frac {5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{6 a^{8/3} \sqrt [3]{b}}-\frac {5 \log (a+b x)}{18 a^{8/3} \sqrt [3]{b}}+\frac {5 \sqrt [3]{x}}{6 a^2 (a+b x)}+\frac {\sqrt [3]{x}}{2 a (a+b x)^2} \]
[In]
[Out]
Rule 31
Rule 44
Rule 60
Rule 210
Rule 631
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [3]{x}}{2 a (a+b x)^2}+\frac {5 \int \frac {1}{x^{2/3} (a+b x)^2} \, dx}{6 a} \\ & = \frac {\sqrt [3]{x}}{2 a (a+b x)^2}+\frac {5 \sqrt [3]{x}}{6 a^2 (a+b x)}+\frac {5 \int \frac {1}{x^{2/3} (a+b x)} \, dx}{9 a^2} \\ & = \frac {\sqrt [3]{x}}{2 a (a+b x)^2}+\frac {5 \sqrt [3]{x}}{6 a^2 (a+b x)}-\frac {5 \log (a+b x)}{18 a^{8/3} \sqrt [3]{b}}+\frac {5 \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 )}{6 a^{7/3} b^{2/3}}+\frac {5 \text {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx,x,\sqrt [3]{x}\right )}{6 a^{8/3} \sqrt [3]{b}} \\ & = \frac {\sqrt [3]{x}}{2 a (a+b x)^2}+\frac {5 \sqrt [3]{x}}{6 a^2 (a+b x)}+\frac {5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{6 a^{8/3} \sqrt [3]{b}}-\frac {5 \log (a+b x)}{18 a^{8/3} \sqrt [3]{b}}+\frac {5 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}\right )}{3 a^{8/3} \sqrt [3]{b}} \\ & = \frac {\sqrt [3]{x}}{2 a (a+b x)^2}+\frac {5 \sqrt [3]{x}}{6 a^2 (a+b x)}-\frac {5 \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{3 \sqrt {3} a^{8/3} \sqrt [3]{b}}+\frac {5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{6 a^{8/3} \sqrt [3]{b}}-\frac {5 \log (a+b x)}{18 a^{8/3} \sqrt [3]{b}} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.01 \[ \int \frac {1}{x^{2/3} (a+b x)^3} \, dx=\frac {\frac {3 a^{2/3} \sqrt [3]{x} (8 a+5 b x)}{(a+b x)^2}-\frac {10 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}+\frac {10 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{\sqrt [3]{b}}-\frac {5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{\sqrt [3]{b}}}{18 a^{8/3}} \]
[In]
[Out]
Time = 0.11 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.99
method | result | size |
derivativedivides | \(\frac {x^{\frac {1}{3}}}{2 a \left (b x +a \right )^{2}}+\frac {\frac {5 x^{\frac {1}{3}}}{6 a \left (b x +a \right )}+\frac {5 \left (\frac {2 \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{2 a}}{a}\) | \(139\) |
default | \(\frac {x^{\frac {1}{3}}}{2 a \left (b x +a \right )^{2}}+\frac {\frac {5 x^{\frac {1}{3}}}{6 a \left (b x +a \right )}+\frac {5 \left (\frac {2 \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{2 a}}{a}\) | \(139\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (99) = 198\).
Time = 0.24 (sec) , antiderivative size = 499, normalized size of antiderivative = 3.56 \[ \int \frac {1}{x^{2/3} (a+b x)^3} \, dx=\left [\frac {15 \, \sqrt {\frac {1}{3}} {\left (a b^{3} x^{2} + 2 \, a^{2} b^{2} x + a^{3} b\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a b x - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{\frac {2}{3}} - \left (a^{2} b\right )^{\frac {1}{3}} a + \left (a^{2} b\right )^{\frac {2}{3}} x^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} - 3 \, \left (a^{2} b\right )^{\frac {1}{3}} a x^{\frac {1}{3}}}{b x + a}\right ) - 5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{\frac {2}{3}} + \left (a^{2} b\right )^{\frac {1}{3}} a - \left (a^{2} b\right )^{\frac {2}{3}} x^{\frac {1}{3}}\right ) + 10 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{\frac {1}{3}} + \left (a^{2} b\right )^{\frac {2}{3}}\right ) + 3 \, {\left (5 \, a^{2} b^{2} x + 8 \, a^{3} b\right )} x^{\frac {1}{3}}}{18 \, {\left (a^{4} b^{3} x^{2} + 2 \, a^{5} b^{2} x + a^{6} b\right )}}, \frac {30 \, \sqrt {\frac {1}{3}} {\left (a b^{3} x^{2} + 2 \, a^{2} b^{2} x + a^{3} b\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (-\frac {\sqrt {\frac {1}{3}} {\left (\left (a^{2} b\right )^{\frac {1}{3}} a - 2 \, \left (a^{2} b\right )^{\frac {2}{3}} x^{\frac {1}{3}}\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{a^{2}}\right ) - 5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{\frac {2}{3}} + \left (a^{2} b\right )^{\frac {1}{3}} a - \left (a^{2} b\right )^{\frac {2}{3}} x^{\frac {1}{3}}\right ) + 10 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{\frac {1}{3}} + \left (a^{2} b\right )^{\frac {2}{3}}\right ) + 3 \, {\left (5 \, a^{2} b^{2} x + 8 \, a^{3} b\right )} x^{\frac {1}{3}}}{18 \, {\left (a^{4} b^{3} x^{2} + 2 \, a^{5} b^{2} x + a^{6} b\right )}}\right ] \]
[In]
[Out]
Timed out. \[ \int \frac {1}{x^{2/3} (a+b x)^3} \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^{2/3} (a+b x)^3} \, dx=\frac {5 \, b x^{\frac {4}{3}} + 8 \, a x^{\frac {1}{3}}}{6 \, {\left (a^{2} b^{2} x^{2} + 2 \, a^{3} b x + a^{4}\right )}} + \frac {5 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {5 \, \log \left (x^{\frac {2}{3}} - x^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {5 \, \log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.02 \[ \int \frac {1}{x^{2/3} (a+b x)^3} \, dx=-\frac {5 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x^{\frac {1}{3}} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{3}} + \frac {5 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{3} b} + \frac {5 \, \left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{\frac {2}{3}} + x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{3} b} + \frac {5 \, b x^{\frac {4}{3}} + 8 \, a x^{\frac {1}{3}}}{6 \, {\left (b x + a\right )}^{2} a^{2}} \]
[In]
[Out]
Time = 0.25 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.12 \[ \int \frac {1}{x^{2/3} (a+b x)^3} \, dx=\frac {\frac {4\,x^{1/3}}{3\,a}+\frac {5\,b\,x^{4/3}}{6\,a^2}}{a^2+2\,a\,b\,x+b^2\,x^2}+\frac {5\,\ln \left (\frac {5\,b^{5/3}}{a^{5/3}}+\frac {5\,b^2\,x^{1/3}}{a^2}\right )}{9\,a^{8/3}\,b^{1/3}}+\frac {\ln \left (\frac {5\,b^2\,x^{1/3}}{a^2}+\frac {b^{5/3}\,\left (-5+\sqrt {3}\,5{}\mathrm {i}\right )}{2\,a^{5/3}}\right )\,\left (-5+\sqrt {3}\,5{}\mathrm {i}\right )}{18\,a^{8/3}\,b^{1/3}}-\frac {\ln \left (\frac {5\,b^2\,x^{1/3}}{a^2}-\frac {b^{5/3}\,\left (5+\sqrt {3}\,5{}\mathrm {i}\right )}{2\,a^{5/3}}\right )\,\left (5+\sqrt {3}\,5{}\mathrm {i}\right )}{18\,a^{8/3}\,b^{1/3}} \]
[In]
[Out]