Integrand size = 17, antiderivative size = 139 \[ \int \frac {1}{(a+b x) \sqrt [3]{c+d x}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{b^{2/3} \sqrt [3]{b c-a d}}-\frac {\log (a+b x)}{2 b^{2/3} \sqrt [3]{b c-a d}}+\frac {3 \log \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{2 b^{2/3} \sqrt [3]{b c-a d}} \]
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Time = 0.08 (sec) , antiderivative size = 139, 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 = {57, 631, 210, 31} \[ \int \frac {1}{(a+b x) \sqrt [3]{c+d x}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}}+1}{\sqrt {3}}\right )}{b^{2/3} \sqrt [3]{b c-a d}}-\frac {\log (a+b x)}{2 b^{2/3} \sqrt [3]{b c-a d}}+\frac {3 \log \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{2 b^{2/3} \sqrt [3]{b c-a d}} \]
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Rule 31
Rule 57
Rule 210
Rule 631
Rubi steps \begin{align*} \text {integral}& = -\frac {\log (a+b x)}{2 b^{2/3} \sqrt [3]{b c-a d}}+\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}-\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 b^{2/3} \sqrt [3]{b c-a d}} \\ & = -\frac {\log (a+b x)}{2 b^{2/3} \sqrt [3]{b c-a d}}+\frac {3 \log \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{2 b^{2/3} \sqrt [3]{b c-a d}}-\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 )}{b^{2/3} \sqrt [3]{b c-a d}} \\ & = \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 )}{b^{2/3} \sqrt [3]{b c-a d}}-\frac {\log (a+b x)}{2 b^{2/3} \sqrt [3]{b c-a d}}+\frac {3 \log \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{2 b^{2/3} \sqrt [3]{b c-a d}} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.11 \[ \int \frac {1}{(a+b x) \sqrt [3]{c+d x}} \, dx=\frac {-2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{-b c+a d}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{-b c+a d}+\sqrt [3]{b} \sqrt [3]{c+d x}\right )+\log \left ((-b c+a d)^{2/3}-\sqrt [3]{b} \sqrt [3]{-b c+a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}\right )}{2 b^{2/3} \sqrt [3]{-b c+a d}} \]
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Time = 0.67 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.03
method | result | size |
pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (2 \left (d x +c \right )^{\frac {1}{3}}-\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}\right )-2 \ln \left (\left (d x +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}\right )+\ln \left (\left (d x +c \right )^{\frac {2}{3}}-\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}} \left (d x +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}\right )}{2 b \left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}\) | \(143\) |
derivativedivides | \(-\frac {\ln \left (\left (d x +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}\right )}{b \left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\left (d x +c \right )^{\frac {2}{3}}-\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}} \left (d x +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}\right )}{2 b \left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (d x +c \right )^{\frac {1}{3}}}{\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{b \left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}\) | \(161\) |
default | \(-\frac {\ln \left (\left (d x +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}\right )}{b \left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\left (d x +c \right )^{\frac {2}{3}}-\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}} \left (d x +c \right )^{\frac {1}{3}}+\left (\frac {a d -b c}{b}\right )^{\frac {2}{3}}\right )}{2 b \left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (d x +c \right )^{\frac {1}{3}}}{\left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{b \left (\frac {a d -b c}{b}\right )^{\frac {1}{3}}}\) | \(161\) |
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Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (108) = 216\).
Time = 0.24 (sec) , antiderivative size = 570, normalized size of antiderivative = 4.10 \[ \int \frac {1}{(a+b x) \sqrt [3]{c+d x}} \, dx=\left [\frac {\sqrt {3} {\left (b^{2} c - a b d\right )} \sqrt {-\frac {{\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}}}{b c - a d}} \log \left (\frac {2 \, b^{2} d x + 3 \, b^{2} c - a b d - \sqrt {3} {\left ({\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}} {\left (b c - a d\right )} + {\left (b^{2} c - a b d\right )} {\left (d x + c\right )}^{\frac {1}{3}} - 2 \, {\left (b^{3} c - a b^{2} d\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {2}{3}}\right )} \sqrt {-\frac {{\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}}}{b c - a d}} - 3 \, {\left (b^{3} c - a b^{2} d\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}{b x + a}\right ) - {\left (b^{3} c - a b^{2} d\right )}^{\frac {2}{3}} \log \left ({\left (d x + c\right )}^{\frac {2}{3}} b^{2} + {\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b + {\left (b^{3} c - a b^{2} d\right )}^{\frac {2}{3}}\right ) + 2 \, {\left (b^{3} c - a b^{2} d\right )}^{\frac {2}{3}} \log \left ({\left (d x + c\right )}^{\frac {1}{3}} b - {\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}}\right )}{2 \, {\left (b^{3} c - a b^{2} d\right )}}, \frac {2 \, \sqrt {3} {\left (b^{2} c - a b d\right )} \sqrt {\frac {{\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}}}{b c - a d}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (d x + c\right )}^{\frac {1}{3}} b + {\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}}\right )} \sqrt {\frac {{\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}}}{b c - a d}}}{3 \, b}\right ) - {\left (b^{3} c - a b^{2} d\right )}^{\frac {2}{3}} \log \left ({\left (d x + c\right )}^{\frac {2}{3}} b^{2} + {\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b + {\left (b^{3} c - a b^{2} d\right )}^{\frac {2}{3}}\right ) + 2 \, {\left (b^{3} c - a b^{2} d\right )}^{\frac {2}{3}} \log \left ({\left (d x + c\right )}^{\frac {1}{3}} b - {\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}}\right )}{2 \, {\left (b^{3} c - a b^{2} d\right )}}\right ] \]
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\[ \int \frac {1}{(a+b x) \sqrt [3]{c+d x}} \, dx=\int \frac {1}{\left (a + b x\right ) \sqrt [3]{c + d x}}\, dx \]
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Exception generated. \[ \int \frac {1}{(a+b x) \sqrt [3]{c+d x}} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.34 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.41 \[ \int \frac {1}{(a+b x) \sqrt [3]{c+d x}} \, dx=\frac {3 \, {\left (b^{3} c - a b^{2} d\right )}^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (d x + c\right )}^{\frac {1}{3}} + \left (\frac {b c - a d}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b c - a d}{b}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} b^{3} c - \sqrt {3} a b^{2} d} - \frac {\log \left ({\left (d x + c\right )}^{\frac {2}{3}} + {\left (d x + c\right )}^{\frac {1}{3}} \left (\frac {b c - a d}{b}\right )^{\frac {1}{3}} + \left (\frac {b c - a d}{b}\right )^{\frac {2}{3}}\right )}{2 \, {\left (b^{3} c - a b^{2} d\right )}^{\frac {1}{3}}} + \frac {\left (\frac {b c - a d}{b}\right )^{\frac {2}{3}} \log \left ({\left | {\left (d x + c\right )}^{\frac {1}{3}} - \left (\frac {b c - a d}{b}\right )^{\frac {1}{3}} \right |}\right )}{b c - a d} \]
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Time = 0.18 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.47 \[ \int \frac {1}{(a+b x) \sqrt [3]{c+d x}} \, dx=\frac {\ln \left (9\,b\,{\left (c+d\,x\right )}^{1/3}-\frac {9\,b^3\,c-9\,a\,b^2\,d}{b^{4/3}\,{\left (b\,c-a\,d\right )}^{2/3}}\right )}{b^{2/3}\,{\left (b\,c-a\,d\right )}^{1/3}}+\frac {\ln \left (9\,b\,{\left (c+d\,x\right )}^{1/3}-\frac {{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (9\,b^3\,c-9\,a\,b^2\,d\right )}{4\,b^{4/3}\,{\left (b\,c-a\,d\right )}^{2/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,b^{2/3}\,{\left (b\,c-a\,d\right )}^{1/3}}-\frac {\ln \left (9\,b\,{\left (c+d\,x\right )}^{1/3}-\frac {{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (9\,b^3\,c-9\,a\,b^2\,d\right )}{4\,b^{4/3}\,{\left (b\,c-a\,d\right )}^{2/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,b^{2/3}\,{\left (b\,c-a\,d\right )}^{1/3}} \]
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