Integrand size = 23, antiderivative size = 323 \[ \int \frac {\sqrt {d+e x} \log (a+b x)}{a+b x} \, dx=-\frac {4 \sqrt {d+e x}}{b}+\frac {4 \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2}}+\frac {2 \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{b^{3/2}}+\frac {2 \sqrt {d+e x} \log (a+b x)}{b}-\frac {2 \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{b^{3/2}}-\frac {4 \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{b^{3/2}}-\frac {2 \sqrt {b d-a e} \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{b^{3/2}} \]
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Time = 0.61 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {2458, 2388, 65, 214, 2390, 12, 1601, 6873, 6131, 6055, 2449, 2352, 2356, 52} \[ \int \frac {\sqrt {d+e x} \log (a+b x)}{a+b x} \, dx=\frac {2 \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{b^{3/2}}+\frac {4 \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2}}-\frac {2 \sqrt {b d-a e} \log (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2}}-\frac {4 \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{b^{3/2}}-\frac {2 \sqrt {b d-a e} \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{b^{3/2}}+\frac {2 \sqrt {d+e x} \log (a+b x)}{b}-\frac {4 \sqrt {d+e x}}{b} \]
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Rule 12
Rule 52
Rule 65
Rule 214
Rule 1601
Rule 2352
Rule 2356
Rule 2388
Rule 2390
Rule 2449
Rule 2458
Rule 6055
Rule 6131
Rule 6873
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\sqrt {\frac {b d-a e}{b}+\frac {e x}{b}} \log (x)}{x} \, dx,x,a+b x\right )}{b} \\ & = \frac {e \text {Subst}\left (\int \frac {\log (x)}{\sqrt {\frac {b d-a e}{b}+\frac {e x}{b}}} \, dx,x,a+b x\right )}{b^2}+\frac {(b d-a e) \text {Subst}\left (\int \frac {\log (x)}{x \sqrt {\frac {b d-a e}{b}+\frac {e x}{b}}} \, dx,x,a+b x\right )}{b^2} \\ & = \frac {2 \sqrt {d+e x} \log (a+b x)}{b}-\frac {2 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{b^{3/2}}-\frac {2 \text {Subst}\left (\int \frac {\sqrt {\frac {b d-a e}{b}+\frac {e x}{b}}}{x} \, dx,x,a+b x\right )}{b}-\frac {(b d-a e) \text {Subst}\left (\int -\frac {2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d-\frac {a e}{b}+\frac {e x}{b}}}{\sqrt {b d-a e}}\right )}{\sqrt {b d-a e} x} \, dx,x,a+b x\right )}{b^2} \\ & = -\frac {4 \sqrt {d+e x}}{b}+\frac {2 \sqrt {d+e x} \log (a+b x)}{b}-\frac {2 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{b^{3/2}}+\frac {\left (2 \sqrt {b d-a e}\right ) \text {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d-\frac {a e}{b}+\frac {e x}{b}}}{\sqrt {b d-a e}}\right )}{x} \, dx,x,a+b x\right )}{b^{3/2}}-\frac {(2 (b d-a e)) \text {Subst}\left (\int \frac {1}{x \sqrt {\frac {b d-a e}{b}+\frac {e x}{b}}} \, dx,x,a+b x\right )}{b^2} \\ & = -\frac {4 \sqrt {d+e x}}{b}+\frac {2 \sqrt {d+e x} \log (a+b x)}{b}-\frac {2 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{b^{3/2}}+\frac {\left (4 \sqrt {b d-a e}\right ) \text {Subst}\left (\int \frac {x \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b d-a e}}\right )}{a e+b \left (-d+x^2\right )} \, dx,x,\sqrt {d+e x}\right )}{\sqrt {b}}-\frac {(4 (b d-a e)) \text {Subst}\left (\int \frac {1}{-\frac {b d-a e}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{b e} \\ & = -\frac {4 \sqrt {d+e x}}{b}+\frac {4 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2}}+\frac {2 \sqrt {d+e x} \log (a+b x)}{b}-\frac {2 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{b^{3/2}}+\frac {\left (4 \sqrt {b d-a e}\right ) \text {Subst}\left (\int \frac {x \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b d-a e}}\right )}{-b d+a e+b x^2} \, dx,x,\sqrt {d+e x}\right )}{\sqrt {b}} \\ & = -\frac {4 \sqrt {d+e x}}{b}+\frac {4 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2}}+\frac {2 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{b^{3/2}}+\frac {2 \sqrt {d+e x} \log (a+b x)}{b}-\frac {2 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{b^{3/2}}-\frac {4 \text {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b d-a e}}\right )}{1-\frac {\sqrt {b} x}{\sqrt {b d-a e}}} \, dx,x,\sqrt {d+e x}\right )}{b} \\ & = -\frac {4 \sqrt {d+e x}}{b}+\frac {4 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2}}+\frac {2 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{b^{3/2}}+\frac {2 \sqrt {d+e x} \log (a+b x)}{b}-\frac {2 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{b^{3/2}}-\frac {4 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{b^{3/2}}+\frac {4 \text {Subst}\left (\int \frac {\log \left (\frac {2}{1-\frac {\sqrt {b} x}{\sqrt {b d-a e}}}\right )}{1-\frac {b x^2}{b d-a e}} \, dx,x,\sqrt {d+e x}\right )}{b} \\ & = -\frac {4 \sqrt {d+e x}}{b}+\frac {4 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2}}+\frac {2 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{b^{3/2}}+\frac {2 \sqrt {d+e x} \log (a+b x)}{b}-\frac {2 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{b^{3/2}}-\frac {4 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{b^{3/2}}-\frac {\left (4 \sqrt {b d-a e}\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{b^{3/2}} \\ & = -\frac {4 \sqrt {d+e x}}{b}+\frac {4 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2}}+\frac {2 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{b^{3/2}}+\frac {2 \sqrt {d+e x} \log (a+b x)}{b}-\frac {2 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{b^{3/2}}-\frac {4 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{b^{3/2}}-\frac {2 \sqrt {b d-a e} \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{b^{3/2}} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 534, normalized size of antiderivative = 1.65 \[ \int \frac {\sqrt {d+e x} \log (a+b x)}{a+b x} \, dx=\frac {-8 \sqrt {b} \sqrt {d+e x}+8 \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )+4 \sqrt {b} \sqrt {d+e x} \log (a+b x)+2 \sqrt {b d-a e} \log (a+b x) \log \left (\sqrt {b d-a e}-\sqrt {b} \sqrt {d+e x}\right )-\sqrt {b d-a e} \log ^2\left (\sqrt {b d-a e}-\sqrt {b} \sqrt {d+e x}\right )-2 \sqrt {b d-a e} \log (a+b x) \log \left (\sqrt {b d-a e}+\sqrt {b} \sqrt {d+e x}\right )+\sqrt {b d-a e} \log ^2\left (\sqrt {b d-a e}+\sqrt {b} \sqrt {d+e x}\right )+2 \sqrt {b d-a e} \log \left (\sqrt {b d-a e}+\sqrt {b} \sqrt {d+e x}\right ) \log \left (\frac {1}{2}-\frac {\sqrt {b} \sqrt {d+e x}}{2 \sqrt {b d-a e}}\right )-2 \sqrt {b d-a e} \log \left (\sqrt {b d-a e}-\sqrt {b} \sqrt {d+e x}\right ) \log \left (\frac {1}{2} \left (1+\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )\right )-2 \sqrt {b d-a e} \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {\sqrt {b} \sqrt {d+e x}}{2 \sqrt {b d-a e}}\right )+2 \sqrt {b d-a e} \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )\right )}{2 b^{3/2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.90 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.79
method | result | size |
derivativedivides | \(\frac {2 \sqrt {e x +d}\, \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{e}\right )}{b}-\frac {4 \sqrt {e x +d}}{b}-\frac {4 \left (-a e +b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{b \sqrt {\left (a e -b d \right ) b}}+2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (b \,\textit {\_Z}^{2}+a e -b d \right )}{\sum }\frac {\left (\ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{e}\right )-2 b \left (\frac {\ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{4 \underline {\hspace {1.25 ex}}\alpha b}+\frac {\underline {\hspace {1.25 ex}}\alpha \ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\sqrt {e x +d}+\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{2 a e -2 b d}+\frac {\underline {\hspace {1.25 ex}}\alpha \operatorname {dilog}\left (\frac {\sqrt {e x +d}+\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{2 a e -2 b d}\right )\right ) \left (-a e +b d \right )}{2 b^{2} \underline {\hspace {1.25 ex}}\alpha }\right )\) | \(256\) |
default | \(\frac {2 \sqrt {e x +d}\, \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{e}\right )}{b}-\frac {4 \sqrt {e x +d}}{b}-\frac {4 \left (-a e +b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{b \sqrt {\left (a e -b d \right ) b}}+2 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (b \,\textit {\_Z}^{2}+a e -b d \right )}{\sum }\frac {\left (\ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{e}\right )-2 b \left (\frac {\ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{4 \underline {\hspace {1.25 ex}}\alpha b}+\frac {\underline {\hspace {1.25 ex}}\alpha \ln \left (\sqrt {e x +d}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\sqrt {e x +d}+\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{2 a e -2 b d}+\frac {\underline {\hspace {1.25 ex}}\alpha \operatorname {dilog}\left (\frac {\sqrt {e x +d}+\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{2 a e -2 b d}\right )\right ) \left (-a e +b d \right )}{2 b^{2} \underline {\hspace {1.25 ex}}\alpha }\right )\) | \(256\) |
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\[ \int \frac {\sqrt {d+e x} \log (a+b x)}{a+b x} \, dx=\int { \frac {\sqrt {e x + d} \log \left (b x + a\right )}{b x + a} \,d x } \]
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\[ \int \frac {\sqrt {d+e x} \log (a+b x)}{a+b x} \, dx=\int \frac {\sqrt {d + e x} \log {\left (a + b x \right )}}{a + b x}\, dx \]
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Exception generated. \[ \int \frac {\sqrt {d+e x} \log (a+b x)}{a+b x} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {\sqrt {d+e x} \log (a+b x)}{a+b x} \, dx=\int { \frac {\sqrt {e x + d} \log \left (b x + a\right )}{b x + a} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {d+e x} \log (a+b x)}{a+b x} \, dx=\int \frac {\ln \left (a+b\,x\right )\,\sqrt {d+e\,x}}{a+b\,x} \,d x \]
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