Integrand size = 19, antiderivative size = 414 \[ \int \frac {\log (c+d x)}{x^4 \left (a+b x^3\right )} \, dx=-\frac {d}{6 a c x^2}+\frac {d^2}{3 a c^2 x}+\frac {d^3 \log (x)}{3 a c^3}-\frac {d^3 \log (c+d x)}{3 a c^3}-\frac {\log (c+d x)}{3 a x^3}-\frac {b \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{a^2}+\frac {b \log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}+\frac {b \log \left (-\frac {d \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}+\frac {b \log \left (\frac {\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^2}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{3 a^2}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 a^2}-\frac {b \operatorname {PolyLog}\left (2,1+\frac {d x}{c}\right )}{a^2} \]
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Time = 0.32 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {272, 46, 2463, 2442, 2441, 2352, 266, 2440, 2438} \[ \int \frac {\log (c+d x)}{x^4 \left (a+b x^3\right )} \, dx=\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^2}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{3 a^2}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 a^2}-\frac {b \operatorname {PolyLog}\left (2,\frac {d x}{c}+1\right )}{a^2}-\frac {b \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{a^2}+\frac {b \log (c+d x) \log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^2}+\frac {b \log (c+d x) \log \left (-\frac {d \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 a^2}+\frac {b \log (c+d x) \log \left (\frac {\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{a} d+\sqrt [3]{b} c}\right )}{3 a^2}+\frac {d^3 \log (x)}{3 a c^3}-\frac {d^3 \log (c+d x)}{3 a c^3}+\frac {d^2}{3 a c^2 x}-\frac {\log (c+d x)}{3 a x^3}-\frac {d}{6 a c x^2} \]
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Rule 46
Rule 266
Rule 272
Rule 2352
Rule 2438
Rule 2440
Rule 2441
Rule 2442
Rule 2463
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\log (c+d x)}{a x^4}-\frac {b \log (c+d x)}{a^2 x}+\frac {b^2 x^2 \log (c+d x)}{a^2 \left (a+b x^3\right )}\right ) \, dx \\ & = \frac {\int \frac {\log (c+d x)}{x^4} \, dx}{a}-\frac {b \int \frac {\log (c+d x)}{x} \, dx}{a^2}+\frac {b^2 \int \frac {x^2 \log (c+d x)}{a+b x^3} \, dx}{a^2} \\ & = -\frac {\log (c+d x)}{3 a x^3}-\frac {b \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{a^2}+\frac {b^2 \int \left (\frac {\log (c+d x)}{3 b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {\log (c+d x)}{3 b^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {\log (c+d x)}{3 b^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}\right ) \, dx}{a^2}+\frac {d \int \frac {1}{x^3 (c+d x)} \, dx}{3 a}+\frac {(b d) \int \frac {\log \left (-\frac {d x}{c}\right )}{c+d x} \, dx}{a^2} \\ & = -\frac {\log (c+d x)}{3 a x^3}-\frac {b \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{a^2}-\frac {b \text {Li}_2\left (1+\frac {d x}{c}\right )}{a^2}+\frac {b^{4/3} \int \frac {\log (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^2}+\frac {b^{4/3} \int \frac {\log (c+d x)}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^2}+\frac {b^{4/3} \int \frac {\log (c+d x)}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^2}+\frac {d \int \left (\frac {1}{c x^3}-\frac {d}{c^2 x^2}+\frac {d^2}{c^3 x}-\frac {d^3}{c^3 (c+d x)}\right ) \, dx}{3 a} \\ & = -\frac {d}{6 a c x^2}+\frac {d^2}{3 a c^2 x}+\frac {d^3 \log (x)}{3 a c^3}-\frac {d^3 \log (c+d x)}{3 a c^3}-\frac {\log (c+d x)}{3 a x^3}-\frac {b \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{a^2}+\frac {b \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 a^2}+\frac {b \log \left (-\frac {d \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}+\frac {b \log \left (\frac {\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}-\frac {b \text {Li}_2\left (1+\frac {d x}{c}\right )}{a^2}-\frac {(b 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 a^2}-\frac {(b d) \int \frac {\log \left (\frac {d \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} c-\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{c+d x} \, dx}{3 a^2}-\frac {(b d) \int \frac {\log \left (\frac {d \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} c+(-1)^{2/3} \sqrt [3]{a} d}\right )}{c+d x} \, dx}{3 a^2} \\ & = -\frac {d}{6 a c x^2}+\frac {d^2}{3 a c^2 x}+\frac {d^3 \log (x)}{3 a c^3}-\frac {d^3 \log (c+d x)}{3 a c^3}-\frac {\log (c+d x)}{3 a x^3}-\frac {b \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{a^2}+\frac {b \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 a^2}+\frac {b \log \left (-\frac {d \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}+\frac {b \log \left (\frac {\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}-\frac {b \text {Li}_2\left (1+\frac {d x}{c}\right )}{a^2}-\frac {b \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 a^2}-\frac {b \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{b} x}{-\sqrt [3]{b} c-\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{x} \, dx,x,c+d x\right )}{3 a^2}-\frac {b \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{b} x}{-\sqrt [3]{b} c+(-1)^{2/3} \sqrt [3]{a} d}\right )}{x} \, dx,x,c+d x\right )}{3 a^2} \\ & = -\frac {d}{6 a c x^2}+\frac {d^2}{3 a c^2 x}+\frac {d^3 \log (x)}{3 a c^3}-\frac {d^3 \log (c+d x)}{3 a c^3}-\frac {\log (c+d x)}{3 a x^3}-\frac {b \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{a^2}+\frac {b \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 a^2}+\frac {b \log \left (-\frac {d \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}+\frac {b \log \left (\frac {\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}+\frac {b \text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^2}+\frac {b \text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{3 a^2}+\frac {b \text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 a^2}-\frac {b \text {Li}_2\left (1+\frac {d x}{c}\right )}{a^2} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 405, normalized size of antiderivative = 0.98 \[ \int \frac {\log (c+d x)}{x^4 \left (a+b x^3\right )} \, dx=-\frac {\log (c+d x)}{3 a x^3}-\frac {b \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{a^2}+\frac {b \log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}+\frac {b \log \left (-\frac {(-1)^{2/3} d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}+\frac {b \log \left (\frac {\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}-\frac {d \left (\frac {1}{c x^2}-\frac {2 d}{c^2 x}-\frac {2 d^2 \log (x)}{c^3}+\frac {2 d^2 \log (c+d x)}{c^3}\right )}{6 a}-\frac {b \operatorname {PolyLog}\left (2,\frac {c+d x}{c}\right )}{a^2}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^2}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{3 a^2}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 a^2} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.73 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.45
method | result | size |
risch | \(-\frac {d}{6 a c \,x^{2}}+\frac {d^{2}}{3 a \,c^{2} x}+\frac {d^{3} \ln \left (-d x \right )}{3 a \,c^{3}}-\frac {d^{3} \ln \left (d x +c \right )}{3 a \,c^{3}}-\frac {\ln \left (d x +c \right )}{3 a \,x^{3}}+\frac {b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )\right )}{3 a^{2}}-\frac {b \ln \left (-\frac {x d}{c}\right ) \ln \left (d x +c \right )}{a^{2}}-\frac {b \operatorname {dilog}\left (-\frac {x d}{c}\right )}{a^{2}}\) | \(186\) |
derivativedivides | \(d^{3} \left (\frac {-\frac {1}{6 c \,d^{2} x^{2}}+\frac {1}{3 c^{2} d x}+\frac {\ln \left (-d x \right )}{3 c^{3}}-\frac {\ln \left (d x +c \right ) \left (d x +c \right ) \left (3 c^{2}-3 c \left (d x +c \right )+\left (d x +c \right )^{2}\right )}{3 c^{3} d^{3} x^{3}}}{a}+\frac {b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )\right )}{3 a^{2} d^{3}}-\frac {b \left (\operatorname {dilog}\left (-\frac {x d}{c}\right )+\ln \left (d x +c \right ) \ln \left (-\frac {x d}{c}\right )\right )}{a^{2} d^{3}}\right )\) | \(199\) |
default | \(d^{3} \left (\frac {-\frac {1}{6 c \,d^{2} x^{2}}+\frac {1}{3 c^{2} d x}+\frac {\ln \left (-d x \right )}{3 c^{3}}-\frac {\ln \left (d x +c \right ) \left (d x +c \right ) \left (3 c^{2}-3 c \left (d x +c \right )+\left (d x +c \right )^{2}\right )}{3 c^{3} d^{3} x^{3}}}{a}+\frac {b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )\right )}{3 a^{2} d^{3}}-\frac {b \left (\operatorname {dilog}\left (-\frac {x d}{c}\right )+\ln \left (d x +c \right ) \ln \left (-\frac {x d}{c}\right )\right )}{a^{2} d^{3}}\right )\) | \(199\) |
parts | \(-\frac {\ln \left (d x +c \right )}{3 a \,x^{3}}-\frac {\ln \left (d x +c \right ) b \ln \left (x \right )}{a^{2}}+\frac {\ln \left (d x +c \right ) b \ln \left (b \,x^{3}+a \right )}{3 a^{2}}-\frac {d \left (\frac {b \left (\frac {\ln \left (d x +c \right ) \ln \left (b \,x^{3}+a \right )}{d}-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )}{d}\right )}{a^{2}}-\frac {-\frac {d^{2} \ln \left (d x +c \right )}{c^{3}}-\frac {1}{2 c \,x^{2}}+\frac {d^{2} \ln \left (x \right )}{c^{3}}+\frac {d}{c^{2} x}}{a}-\frac {3 b \operatorname {dilog}\left (\frac {d x +c}{c}\right )}{a^{2} d}-\frac {3 b \ln \left (x \right ) \ln \left (\frac {d x +c}{c}\right )}{a^{2} d}\right )}{3}\) | \(239\) |
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\[ \int \frac {\log (c+d x)}{x^4 \left (a+b x^3\right )} \, dx=\int { \frac {\log \left (d x + c\right )}{{\left (b x^{3} + a\right )} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {\log (c+d x)}{x^4 \left (a+b x^3\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {\log (c+d x)}{x^4 \left (a+b x^3\right )} \, dx=\int { \frac {\log \left (d x + c\right )}{{\left (b x^{3} + a\right )} x^{4}} \,d x } \]
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\[ \int \frac {\log (c+d x)}{x^4 \left (a+b x^3\right )} \, dx=\int { \frac {\log \left (d x + c\right )}{{\left (b x^{3} + a\right )} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {\log (c+d x)}{x^4 \left (a+b x^3\right )} \, dx=\int \frac {\ln \left (c+d\,x\right )}{x^4\,\left (b\,x^3+a\right )} \,d x \]
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