Integrand size = 19, antiderivative size = 497 \[ \int \frac {x^2 \log (c+d x)}{a+b x^4} \, dx=\frac {\log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 \sqrt {-\sqrt {-a}} b^{3/4}}+\frac {\log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 \sqrt [4]{-a} b^{3/4}}-\frac {\log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 \sqrt {-\sqrt {-a}} b^{3/4}}-\frac {\log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 \sqrt [4]{-a} b^{3/4}}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 \sqrt {-\sqrt {-a}} b^{3/4}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 \sqrt {-\sqrt {-a}} b^{3/4}}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 \sqrt [4]{-a} b^{3/4}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 \sqrt [4]{-a} b^{3/4}} \]
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Time = 0.39 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {303, 1176, 631, 210, 1179, 642, 2463, 2456, 2441, 2440, 2438} \[ \int \frac {x^2 \log (c+d x)}{a+b x^4} \, dx=-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 \sqrt {-\sqrt {-a}} b^{3/4}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 \sqrt {-\sqrt {-a}} b^{3/4}}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 \sqrt [4]{-a} b^{3/4}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 \sqrt [4]{-a} b^{3/4}}+\frac {\log (c+d x) \log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt {-\sqrt {-a}} d+\sqrt [4]{b} c}\right )}{4 \sqrt {-\sqrt {-a}} b^{3/4}}+\frac {\log (c+d x) \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 \sqrt [4]{-a} b^{3/4}}-\frac {\log (c+d x) \log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 \sqrt {-\sqrt {-a}} b^{3/4}}-\frac {\log (c+d x) \log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 \sqrt [4]{-a} b^{3/4}} \]
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Rule 210
Rule 303
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 2438
Rule 2440
Rule 2441
Rule 2456
Rule 2463
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {\log (c+d x)}{2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x^2\right )}+\frac {\log (c+d x)}{2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x^2\right )}\right ) \, dx \\ & = -\frac {\int \frac {\log (c+d x)}{\sqrt {-a}-\sqrt {b} x^2} \, dx}{2 \sqrt {b}}+\frac {\int \frac {\log (c+d x)}{\sqrt {-a}+\sqrt {b} x^2} \, dx}{2 \sqrt {b}} \\ & = \frac {\int \left (\frac {\sqrt {-\sqrt {-a}} \log (c+d x)}{2 \sqrt {-a} \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}+\frac {\sqrt {-\sqrt {-a}} \log (c+d x)}{2 \sqrt {-a} \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}\right ) \, dx}{2 \sqrt {b}}-\frac {\int \left (\frac {\log (c+d x)}{2 \sqrt [4]{-a} \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}+\frac {\log (c+d x)}{2 \sqrt [4]{-a} \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}\right ) \, dx}{2 \sqrt {b}} \\ & = -\frac {\int \frac {\log (c+d x)}{\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x} \, dx}{4 \sqrt {-\sqrt {-a}} \sqrt {b}}-\frac {\int \frac {\log (c+d x)}{\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x} \, dx}{4 \sqrt {-\sqrt {-a}} \sqrt {b}}-\frac {\int \frac {\log (c+d x)}{\sqrt [4]{-a}-\sqrt [4]{b} x} \, dx}{4 \sqrt [4]{-a} \sqrt {b}}-\frac {\int \frac {\log (c+d x)}{\sqrt [4]{-a}+\sqrt [4]{b} x} \, dx}{4 \sqrt [4]{-a} \sqrt {b}} \\ & = \frac {\log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 \sqrt {-\sqrt {-a}} b^{3/4}}+\frac {\log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 \sqrt [4]{-a} b^{3/4}}-\frac {\log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 \sqrt {-\sqrt {-a}} b^{3/4}}-\frac {\log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 \sqrt [4]{-a} b^{3/4}}-\frac {d \int \frac {\log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{c+d x} \, dx}{4 \sqrt {-\sqrt {-a}} b^{3/4}}+\frac {d \int \frac {\log \left (\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{-\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{c+d x} \, dx}{4 \sqrt {-\sqrt {-a}} b^{3/4}}-\frac {d \int \frac {\log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{c+d x} \, dx}{4 \sqrt [4]{-a} b^{3/4}}+\frac {d \int \frac {\log \left (\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{-\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{c+d x} \, dx}{4 \sqrt [4]{-a} b^{3/4}} \\ & = \frac {\log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 \sqrt {-\sqrt {-a}} b^{3/4}}+\frac {\log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 \sqrt [4]{-a} b^{3/4}}-\frac {\log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 \sqrt {-\sqrt {-a}} b^{3/4}}-\frac {\log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 \sqrt [4]{-a} b^{3/4}}+\frac {\text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [4]{b} x}{-\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{x} \, dx,x,c+d x\right )}{4 \sqrt {-\sqrt {-a}} b^{3/4}}-\frac {\text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt [4]{b} x}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{x} \, dx,x,c+d x\right )}{4 \sqrt {-\sqrt {-a}} b^{3/4}}+\frac {\text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [4]{b} x}{-\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{x} \, dx,x,c+d x\right )}{4 \sqrt [4]{-a} b^{3/4}}-\frac {\text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt [4]{b} x}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{x} \, dx,x,c+d x\right )}{4 \sqrt [4]{-a} b^{3/4}} \\ & = \frac {\log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 \sqrt {-\sqrt {-a}} b^{3/4}}+\frac {\log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 \sqrt [4]{-a} b^{3/4}}-\frac {\log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 \sqrt {-\sqrt {-a}} b^{3/4}}-\frac {\log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 \sqrt [4]{-a} b^{3/4}}-\frac {\text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 \sqrt {-\sqrt {-a}} b^{3/4}}+\frac {\text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 \sqrt {-\sqrt {-a}} b^{3/4}}-\frac {\text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 \sqrt [4]{-a} b^{3/4}}+\frac {\text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 \sqrt [4]{-a} b^{3/4}} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 464, normalized size of antiderivative = 0.93 \[ \int \frac {x^2 \log (c+d x)}{a+b x^4} \, dx=\frac {\sqrt [4]{-a} \log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)+\sqrt {-\sqrt {-a}} \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)-\sqrt [4]{-a} \log \left (\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{-\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)-\sqrt {-\sqrt {-a}} \log \left (\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{-\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)-\sqrt [4]{-a} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )+\sqrt [4]{-a} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )-\sqrt {-\sqrt {-a}} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )+\sqrt {-\sqrt {-a}} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 \sqrt {-\sqrt {-a}} \sqrt [4]{-a} b^{3/4}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.57 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.19
method | result | size |
derivativedivides | \(-\frac {d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{-\textit {\_R1} +c}\right )}{4 b}\) | \(94\) |
default | \(-\frac {d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{-\textit {\_R1} +c}\right )}{4 b}\) | \(94\) |
risch | \(-\frac {d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{-\textit {\_R1} +c}\right )}{4 b}\) | \(94\) |
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\[ \int \frac {x^2 \log (c+d x)}{a+b x^4} \, dx=\int { \frac {x^{2} \log \left (d x + c\right )}{b x^{4} + a} \,d x } \]
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Timed out. \[ \int \frac {x^2 \log (c+d x)}{a+b x^4} \, dx=\text {Timed out} \]
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\[ \int \frac {x^2 \log (c+d x)}{a+b x^4} \, dx=\int { \frac {x^{2} \log \left (d x + c\right )}{b x^{4} + a} \,d x } \]
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\[ \int \frac {x^2 \log (c+d x)}{a+b x^4} \, dx=\int { \frac {x^{2} \log \left (d x + c\right )}{b x^{4} + a} \,d x } \]
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Timed out. \[ \int \frac {x^2 \log (c+d x)}{a+b x^4} \, dx=\int \frac {x^2\,\ln \left (c+d\,x\right )}{b\,x^4+a} \,d x \]
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