Integrand size = 16, antiderivative size = 151 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^5} \, dx=-\frac {3 b p}{4 a x}+\frac {\sqrt {3} b^{4/3} p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{4 a^{4/3}}+\frac {b^{4/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{4 a^{4/3}}-\frac {b^{4/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{8 a^{4/3}}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{4 x^4} \]
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Time = 0.06 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2505, 331, 298, 31, 648, 631, 210, 642} \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^5} \, dx=\frac {\sqrt {3} b^{4/3} p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{4 a^{4/3}}-\frac {b^{4/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{8 a^{4/3}}+\frac {b^{4/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{4 a^{4/3}}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{4 x^4}-\frac {3 b p}{4 a x} \]
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Rule 31
Rule 210
Rule 298
Rule 331
Rule 631
Rule 642
Rule 648
Rule 2505
Rubi steps \begin{align*} \text {integral}& = -\frac {\log \left (c \left (a+b x^3\right )^p\right )}{4 x^4}+\frac {1}{4} (3 b p) \int \frac {1}{x^2 \left (a+b x^3\right )} \, dx \\ & = -\frac {3 b p}{4 a x}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{4 x^4}-\frac {\left (3 b^2 p\right ) \int \frac {x}{a+b x^3} \, dx}{4 a} \\ & = -\frac {3 b p}{4 a x}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{4 x^4}+\frac {\left (b^{5/3} p\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{4 a^{4/3}}-\frac {\left (b^{5/3} p\right ) \int \frac {\sqrt [3]{a}+\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{4 a^{4/3}} \\ & = -\frac {3 b p}{4 a x}+\frac {b^{4/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{4 a^{4/3}}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{4 x^4}-\frac {\left (b^{4/3} p\right ) \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{8 a^{4/3}}-\frac {\left (3 b^{5/3} p\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{8 a} \\ & = -\frac {3 b p}{4 a x}+\frac {b^{4/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{4 a^{4/3}}-\frac {b^{4/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{8 a^{4/3}}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{4 x^4}-\frac {\left (3 b^{4/3} p\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{4 a^{4/3}} \\ & = -\frac {3 b p}{4 a x}+\frac {\sqrt {3} b^{4/3} p \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{4 a^{4/3}}+\frac {b^{4/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{4 a^{4/3}}-\frac {b^{4/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{8 a^{4/3}}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{4 x^4} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.00 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.32 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^5} \, dx=-\frac {3 b p \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},1,\frac {2}{3},-\frac {b x^3}{a}\right )}{4 a x}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{4 x^4} \]
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Time = 0.93 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.85
method | result | size |
parts | \(-\frac {\ln \left (c \left (b \,x^{3}+a \right )^{p}\right )}{4 x^{4}}+\frac {3 p b \left (-\frac {\left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) b}{a}-\frac {1}{a x}\right )}{4}\) | \(128\) |
risch | \(-\frac {\ln \left (\left (b \,x^{3}+a \right )^{p}\right )}{4 x^{4}}-\frac {i \pi a \,\operatorname {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{2}-i \pi a \,\operatorname {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-i \pi a {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{3}+i \pi a {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )-2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{4} \textit {\_Z}^{3}-b^{4} p^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 a^{4} \textit {\_R}^{3}+3 b^{4} p^{3}\right ) x -a^{3} b p \,\textit {\_R}^{2}\right )\right ) a \,x^{4}+6 b p \,x^{3}+2 \ln \left (c \right ) a}{8 a \,x^{4}}\) | \(215\) |
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Time = 0.36 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.91 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^5} \, dx=-\frac {2 \, \sqrt {3} b p x^{4} \left (\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (\frac {b}{a}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + b p x^{4} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x^{2} - a x \left (\frac {b}{a}\right )^{\frac {2}{3}} + a \left (\frac {b}{a}\right )^{\frac {1}{3}}\right ) - 2 \, b p x^{4} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x + a \left (\frac {b}{a}\right )^{\frac {2}{3}}\right ) + 6 \, b p x^{3} + 2 \, a p \log \left (b x^{3} + a\right ) + 2 \, a \log \left (c\right )}{8 \, a x^{4}} \]
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Timed out. \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^5} \, dx=\text {Timed out} \]
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Time = 0.28 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.84 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^5} \, dx=-\frac {1}{8} \, b p {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {2 \, \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{a \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {6}{a x}\right )} - \frac {\log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{4 \, x^{4}} \]
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Time = 0.31 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.01 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^5} \, dx=\frac {1}{8} \, b^{2} p {\left (\frac {2 \, \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{a^{2}} + \frac {2 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a^{2} b^{2}} - \frac {\left (-a b^{2}\right )^{\frac {2}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a^{2} b^{2}}\right )} - \frac {p \log \left (b x^{3} + a\right )}{4 \, x^{4}} - \frac {3 \, b p x^{3} + a \log \left (c\right )}{4 \, a x^{4}} \]
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Time = 3.35 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.83 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^5} \, dx=\frac {b^{4/3}\,p\,\ln \left (b^{1/3}\,x+a^{1/3}\right )}{4\,a^{4/3}}-\frac {\ln \left (c\,{\left (b\,x^3+a\right )}^p\right )}{4\,x^4}-\frac {3\,b\,p}{4\,a\,x}+\frac {b^{4/3}\,p\,\ln \left (4\,b^{1/3}\,x-2\,a^{1/3}-\sqrt {3}\,a^{1/3}\,2{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{4\,a^{4/3}}-\frac {b^{4/3}\,p\,\ln \left (4\,b^{1/3}\,x-2\,a^{1/3}+\sqrt {3}\,a^{1/3}\,2{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{4\,a^{4/3}} \]
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