Integrand size = 17, antiderivative size = 359 \[ \int \frac {x \log (c+d x)}{a+b x^3} \, dx=-\frac {\log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 \sqrt [3]{a} b^{2/3}}+\frac {\sqrt [3]{-1} \log \left (\frac {d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right ) \log (c+d x)}{3 \sqrt [3]{a} b^{2/3}}-\frac {(-1)^{2/3} \log \left (-\frac {d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 \sqrt [3]{a} b^{2/3}}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac {(-1)^{2/3} \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{b} (c+d x)}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 \sqrt [3]{a} b^{2/3}}+\frac {\sqrt [3]{-1} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{b} (c+d x)}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{3 \sqrt [3]{a} b^{2/3}} \]
[Out]
Time = 0.21 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.588, Rules used = {298, 31, 648, 631, 210, 642, 2463, 2441, 2440, 2438} \[ \int \frac {x \log (c+d x)}{a+b x^3} \, dx=-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac {(-1)^{2/3} \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{b} (c+d x)}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 \sqrt [3]{a} b^{2/3}}+\frac {\sqrt [3]{-1} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{b} (c+d x)}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac {\log (c+d x) \log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 \sqrt [3]{a} b^{2/3}}+\frac {\sqrt [3]{-1} \log (c+d x) \log \left (\frac {d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} c}\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac {(-1)^{2/3} \log (c+d x) \log \left (-\frac {d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 \sqrt [3]{a} b^{2/3}} \]
[In]
[Out]
Rule 31
Rule 210
Rule 298
Rule 631
Rule 642
Rule 648
Rule 2438
Rule 2440
Rule 2441
Rule 2463
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {\log (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {(-1)^{2/3} \log (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}+\frac {\sqrt [3]{-1} \log (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx \\ & = -\frac {\int \frac {\log (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 \sqrt [3]{a} \sqrt [3]{b}}+\frac {\sqrt [3]{-1} \int \frac {\log (c+d x)}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {(-1)^{2/3} \int \frac {\log (c+d x)}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{3 \sqrt [3]{a} \sqrt [3]{b}} \\ & = -\frac {\log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 \sqrt [3]{a} b^{2/3}}+\frac {\sqrt [3]{-1} \log \left (\frac {d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right ) \log (c+d x)}{3 \sqrt [3]{a} b^{2/3}}-\frac {(-1)^{2/3} \log \left (-\frac {d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 \sqrt [3]{a} b^{2/3}}+\frac {d \int \frac {\log \left (\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{c+d x} \, dx}{3 \sqrt [3]{a} b^{2/3}}-\frac {\left (\sqrt [3]{-1} d\right ) \int \frac {\log \left (\frac {d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{c+d x} \, dx}{3 \sqrt [3]{a} b^{2/3}}+\frac {\left ((-1)^{2/3} d\right ) \int \frac {\log \left (\frac {d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{-(-1)^{2/3} \sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{c+d x} \, dx}{3 \sqrt [3]{a} b^{2/3}} \\ & = -\frac {\log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 \sqrt [3]{a} b^{2/3}}+\frac {\sqrt [3]{-1} \log \left (\frac {d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right ) \log (c+d x)}{3 \sqrt [3]{a} b^{2/3}}-\frac {(-1)^{2/3} \log \left (-\frac {d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 \sqrt [3]{a} b^{2/3}}+\frac {\text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{b} x}{-\sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{x} \, dx,x,c+d x\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac {\sqrt [3]{-1} \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{b} x}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{x} \, dx,x,c+d x\right )}{3 \sqrt [3]{a} b^{2/3}}+\frac {(-1)^{2/3} \text {Subst}\left (\int \frac {\log \left (1+\frac {(-1)^{2/3} \sqrt [3]{b} x}{-(-1)^{2/3} \sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{x} \, dx,x,c+d x\right )}{3 \sqrt [3]{a} b^{2/3}} \\ & = -\frac {\log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 \sqrt [3]{a} b^{2/3}}+\frac {\sqrt [3]{-1} \log \left (\frac {d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right ) \log (c+d x)}{3 \sqrt [3]{a} b^{2/3}}-\frac {(-1)^{2/3} \log \left (-\frac {d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 \sqrt [3]{a} b^{2/3}}-\frac {\text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac {(-1)^{2/3} \text {Li}_2\left (\frac {(-1)^{2/3} \sqrt [3]{b} (c+d x)}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 \sqrt [3]{a} b^{2/3}}+\frac {\sqrt [3]{-1} \text {Li}_2\left (\frac {\sqrt [3]{-1} \sqrt [3]{b} (c+d x)}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{3 \sqrt [3]{a} b^{2/3}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.83 \[ \int \frac {x \log (c+d x)}{a+b x^3} \, dx=\frac {-\log \left (\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} c+\sqrt [3]{a} d}\right ) \log (c+d x)+\sqrt [3]{-1} \log \left (\frac {d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right ) \log (c+d x)-(-1)^{2/3} \log \left (\frac {d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{-(-1)^{2/3} \sqrt [3]{b} c+\sqrt [3]{a} d}\right ) \log (c+d x)-\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )-(-1)^{2/3} \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{b} (c+d x)}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )+\sqrt [3]{-1} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{b} (c+d x)}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{3 \sqrt [3]{a} b^{2/3}} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.56 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.24
method | result | size |
derivativedivides | \(-\frac {d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{-\textit {\_R1} +c}\right )}{3 b}\) | \(86\) |
default | \(-\frac {d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{-\textit {\_R1} +c}\right )}{3 b}\) | \(86\) |
risch | \(-\frac {d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{-\textit {\_R1} +c}\right )}{3 b}\) | \(86\) |
[In]
[Out]
\[ \int \frac {x \log (c+d x)}{a+b x^3} \, dx=\int { \frac {x \log \left (d x + c\right )}{b x^{3} + a} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x \log (c+d x)}{a+b x^3} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {x \log (c+d x)}{a+b x^3} \, dx=\int { \frac {x \log \left (d x + c\right )}{b x^{3} + a} \,d x } \]
[In]
[Out]
\[ \int \frac {x \log (c+d x)}{a+b x^3} \, dx=\int { \frac {x \log \left (d x + c\right )}{b x^{3} + a} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x \log (c+d x)}{a+b x^3} \, dx=\int \frac {x\,\ln \left (c+d\,x\right )}{b\,x^3+a} \,d x \]
[In]
[Out]