Integrand size = 16, antiderivative size = 151 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^6} \, dx=-\frac {3 b p}{10 a x^2}+\frac {\sqrt {3} b^{5/3} p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{5 a^{5/3}}-\frac {b^{5/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{5 a^{5/3}}+\frac {b^{5/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{10 a^{5/3}}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{5 x^5} \]
[Out]
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, 206, 31, 648, 631, 210, 642} \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^6} \, dx=\frac {\sqrt {3} b^{5/3} p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{5 a^{5/3}}+\frac {b^{5/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{10 a^{5/3}}-\frac {b^{5/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{5 a^{5/3}}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{5 x^5}-\frac {3 b p}{10 a x^2} \]
[In]
[Out]
Rule 31
Rule 206
Rule 210
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 )}{5 x^5}+\frac {1}{5} (3 b p) \int \frac {1}{x^3 \left (a+b x^3\right )} \, dx \\ & = -\frac {3 b p}{10 a x^2}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{5 x^5}-\frac {\left (3 b^2 p\right ) \int \frac {1}{a+b x^3} \, dx}{5 a} \\ & = -\frac {3 b p}{10 a x^2}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{5 x^5}-\frac {\left (b^2 p\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{5 a^{5/3}}-\frac {\left (b^2 p\right ) \int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{5 a^{5/3}} \\ & = -\frac {3 b p}{10 a x^2}-\frac {b^{5/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{5 a^{5/3}}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{5 x^5}+\frac {\left (b^{5/3} 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}{10 a^{5/3}}-\frac {\left (3 b^2 p\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{10 a^{4/3}} \\ & = -\frac {3 b p}{10 a x^2}-\frac {b^{5/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{5 a^{5/3}}+\frac {b^{5/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{10 a^{5/3}}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{5 x^5}-\frac {\left (3 b^{5/3} p\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{5 a^{5/3}} \\ & = -\frac {3 b p}{10 a x^2}+\frac {\sqrt {3} b^{5/3} p \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{5 a^{5/3}}-\frac {b^{5/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{5 a^{5/3}}+\frac {b^{5/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{10 a^{5/3}}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{5 x^5} \\ \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 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^6} \, dx=-\frac {3 b p \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},1,\frac {1}{3},-\frac {b x^3}{a}\right )}{10 a x^2}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{5 x^5} \]
[In]
[Out]
Time = 1.31 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.85
method | result | size |
parts | \(-\frac {\ln \left (c \left (b \,x^{3}+a \right )^{p}\right )}{5 x^{5}}+\frac {3 p b \left (-\frac {1}{2 a \,x^{2}}-\frac {\left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right ) b}{a}\right )}{5}\) | \(128\) |
risch | \(-\frac {\ln \left (\left (b \,x^{3}+a \right )^{p}\right )}{5 x^{5}}-\frac {-2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{5} \textit {\_Z}^{3}+b^{5} p^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{5}-3 b^{5} p^{3}\right ) x -a^{2} b^{3} p^{2} \textit {\_R} \right )\right ) a \,x^{5}+i \pi a \,\operatorname {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{2}-i \pi a \,\operatorname {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-i \pi a {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{3}+i \pi a {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+3 b p \,x^{3}+2 \ln \left (c \right ) a}{10 a \,x^{5}}\) | \(216\) |
[In]
[Out]
none
Time = 0.34 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.14 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^6} \, dx=\frac {2 \, \sqrt {3} b p x^{5} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) - b p x^{5} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b^{2} x^{2} + a b x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} + a^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right ) + 2 \, b p x^{5} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x - a \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) - 3 \, b p x^{3} - 2 \, a p \log \left (b x^{3} + a\right ) - 2 \, a \log \left (c\right )}{10 \, a x^{5}} \]
[In]
[Out]
Timed out. \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^6} \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.85 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^6} \, dx=-\frac {1}{10} \, b p {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {\log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {2 \, \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{a \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {3}{a x^{2}}\right )} - \frac {\log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{5 \, x^{5}} \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.99 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^6} \, dx=\frac {b^{2} p \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{5 \, a^{2}} - \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} b p \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{5 \, a^{2}} - \frac {\left (-a b^{2}\right )^{\frac {1}{3}} b p \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{10 \, a^{2}} - \frac {p \log \left (b x^{3} + a\right )}{5 \, x^{5}} - \frac {3 \, b p x^{3} + 2 \, a \log \left (c\right )}{10 \, a x^{5}} \]
[In]
[Out]
Time = 3.66 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.03 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^6} \, dx=\frac {{\left (-b\right )}^{5/3}\,p\,\ln \left (a^{1/3}\,{\left (-b\right )}^{11/3}-b^4\,x\right )}{5\,a^{5/3}}-\frac {\ln \left (c\,{\left (b\,x^3+a\right )}^p\right )}{5\,x^5}-\frac {3\,b\,p}{10\,a\,x^2}+\frac {{\left (-b\right )}^{5/3}\,p\,\ln \left (225\,a^2\,b^4\,p\,x-225\,a^{7/3}\,{\left (-b\right )}^{11/3}\,p\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{5\,a^{5/3}}-\frac {{\left (-b\right )}^{5/3}\,p\,\ln \left (225\,a^2\,b^4\,p\,x+225\,a^{7/3}\,{\left (-b\right )}^{11/3}\,p\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{5\,a^{5/3}} \]
[In]
[Out]