Integrand size = 17, antiderivative size = 140 \[ \int \frac {1}{(a+b x) (c+d x)^{2/3}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt [3]{b} (b c-a d)^{2/3}}-\frac {\log (a+b x)}{2 \sqrt [3]{b} (b c-a d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{2 \sqrt [3]{b} (b c-a d)^{2/3}} \]
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Time = 0.07 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {59, 631, 210, 31} \[ \int \frac {1}{(a+b x) (c+d x)^{2/3}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}}+1}{\sqrt {3}}\right )}{\sqrt [3]{b} (b c-a d)^{2/3}}-\frac {\log (a+b x)}{2 \sqrt [3]{b} (b c-a d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{2 \sqrt [3]{b} (b c-a d)^{2/3}} \]
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Rule 31
Rule 59
Rule 210
Rule 631
Rubi steps \begin{align*} \text {integral}& = -\frac {\log (a+b x)}{2 \sqrt [3]{b} (b c-a d)^{2/3}}-\frac {3 \text {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{b c-a d}}{\sqrt [3]{b}}-x} \, dx,x,\sqrt [3]{c+d x}\right )}{2 \sqrt [3]{b} (b c-a d)^{2/3}}-\frac {3 \text {Subst}\left (\int \frac {1}{\frac {(b c-a d)^{2/3}}{b^{2/3}}+\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{b}}+x^2} \, dx,x,\sqrt [3]{c+d x}\right )}{2 b^{2/3} \sqrt [3]{b c-a d}} \\ & = -\frac {\log (a+b x)}{2 \sqrt [3]{b} (b c-a d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{2 \sqrt [3]{b} (b c-a d)^{2/3}}+\frac {3 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}}\right )}{\sqrt [3]{b} (b c-a d)^{2/3}} \\ & = -\frac {\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt [3]{b} (b c-a d)^{2/3}}-\frac {\log (a+b x)}{2 \sqrt [3]{b} (b c-a d)^{2/3}}+\frac {3 \log \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{2 \sqrt [3]{b} (b c-a d)^{2/3}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(a+b x) (c+d x)^{2/3}} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{-b c+a d}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{-b c+a d}+\sqrt [3]{b} \sqrt [3]{c+d x}\right )+\log \left ((-b c+a d)^{2/3}-\sqrt [3]{b} \sqrt [3]{-b c+a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}\right )}{2 \sqrt [3]{b} (-b c+a d)^{2/3}} \]
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Time = 0.31 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.04
method | result | size |
pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (2 \left (d x +c \right )^{\frac {1}{3}}-\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}\right )+2 \ln \left (\left (d x +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}\right )-\ln \left (\left (d x +c \right )^{\frac {2}{3}}-\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}} \left (d x +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}\right )}{2 b \left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}}\) | \(145\) |
derivativedivides | \(\frac {\ln \left (\left (d x +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}\right )}{b \left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (d x +c \right )^{\frac {2}{3}}-\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}} \left (d x +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}\right )}{2 b \left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (d x +c \right )^{\frac {1}{3}}}{\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{b \left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}}\) | \(160\) |
default | \(\frac {\ln \left (\left (d x +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}\right )}{b \left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (d x +c \right )^{\frac {2}{3}}-\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}} \left (d x +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}\right )}{2 b \left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (d x +c \right )^{\frac {1}{3}}}{\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{b \left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}}\) | \(160\) |
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Leaf count of result is larger than twice the leaf count of optimal. 405 vs. \(2 (109) = 218\).
Time = 0.26 (sec) , antiderivative size = 900, normalized size of antiderivative = 6.43 \[ \int \frac {1}{(a+b x) (c+d x)^{2/3}} \, dx=\left [-\frac {\sqrt {3} {\left (b^{2} c - a b d\right )} \sqrt {-\frac {{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {1}{3}}}{b}} \log \left (-\frac {3 \, b^{2} c^{2} - 4 \, a b c d + a^{2} d^{2} + 2 \, {\left (b^{2} c d - a b d^{2}\right )} x + \sqrt {3} {\left (2 \, {\left (b^{2} c - a b d\right )} {\left (d x + c\right )}^{\frac {2}{3}} - {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {1}{3}} {\left (b c - a d\right )} - {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}\right )} \sqrt {-\frac {{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {1}{3}}}{b}} - 3 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {1}{3}} {\left (b c - a d\right )} {\left (d x + c\right )}^{\frac {1}{3}}}{b x + a}\right ) + {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {2}{3}} \log \left (-{\left (b^{2} c - a b d\right )} {\left (d x + c\right )}^{\frac {2}{3}} - {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {1}{3}} {\left (b c - a d\right )} - {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}\right ) - 2 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {2}{3}} \log \left (-{\left (b^{2} c - a b d\right )} {\left (d x + c\right )}^{\frac {1}{3}} + {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {2}{3}}\right )}{2 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}}, -\frac {2 \, \sqrt {3} {\left (b^{2} c - a b d\right )} \sqrt {\frac {{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {3} {\left ({\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {1}{3}} {\left (b c - a d\right )} + 2 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}\right )} \sqrt {\frac {{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {1}{3}}}{b}}}{3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}}\right ) + {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {2}{3}} \log \left (-{\left (b^{2} c - a b d\right )} {\left (d x + c\right )}^{\frac {2}{3}} - {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {1}{3}} {\left (b c - a d\right )} - {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}\right ) - 2 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {2}{3}} \log \left (-{\left (b^{2} c - a b d\right )} {\left (d x + c\right )}^{\frac {1}{3}} + {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}^{\frac {2}{3}}\right )}{2 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}}\right ] \]
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\[ \int \frac {1}{(a+b x) (c+d x)^{2/3}} \, dx=\int \frac {1}{\left (a + b x\right ) \left (c + d x\right )^{\frac {2}{3}}}\, dx \]
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Exception generated. \[ \int \frac {1}{(a+b x) (c+d x)^{2/3}} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.33 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.48 \[ \int \frac {1}{(a+b x) (c+d x)^{2/3}} \, dx=-\frac {3 \, {\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (d x + c\right )}^{\frac {1}{3}} + \left (\frac {b c - a d}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b c - a d}{b}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} b^{2} c - \sqrt {3} a b d} - \frac {{\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}} \log \left ({\left (d x + c\right )}^{\frac {2}{3}} + {\left (d x + c\right )}^{\frac {1}{3}} \left (\frac {b c - a d}{b}\right )^{\frac {1}{3}} + \left (\frac {b c - a d}{b}\right )^{\frac {2}{3}}\right )}{2 \, {\left (b^{2} c - a b d\right )}} + \frac {\left (\frac {b c - a d}{b}\right )^{\frac {1}{3}} \log \left ({\left | {\left (d x + c\right )}^{\frac {1}{3}} - \left (\frac {b c - a d}{b}\right )^{\frac {1}{3}} \right |}\right )}{b c - a d} \]
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Time = 0.35 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.47 \[ \int \frac {1}{(a+b x) (c+d x)^{2/3}} \, dx=\frac {\ln \left (9\,b^2\,{\left (c+d\,x\right )}^{1/3}-\frac {9\,b^3\,c-9\,a\,b^2\,d}{b^{1/3}\,{\left (a\,d-b\,c\right )}^{2/3}}\right )}{b^{1/3}\,{\left (a\,d-b\,c\right )}^{2/3}}+\frac {\ln \left (9\,b^2\,{\left (c+d\,x\right )}^{1/3}-\frac {\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (9\,b^3\,c-9\,a\,b^2\,d\right )}{2\,b^{1/3}\,{\left (a\,d-b\,c\right )}^{2/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,b^{1/3}\,{\left (a\,d-b\,c\right )}^{2/3}}-\frac {\ln \left (9\,b^2\,{\left (c+d\,x\right )}^{1/3}+\frac {\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (9\,b^3\,c-9\,a\,b^2\,d\right )}{2\,b^{1/3}\,{\left (a\,d-b\,c\right )}^{2/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,b^{1/3}\,{\left (a\,d-b\,c\right )}^{2/3}} \]
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