Integrand size = 16, antiderivative size = 157 \[ \int x^3 \log \left (c \left (a+b x^3\right )^p\right ) \, dx=\frac {3 a p x}{4 b}-\frac {3 p x^4}{16}+\frac {\sqrt {3} a^{4/3} p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{4 b^{4/3}}-\frac {a^{4/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{4 b^{4/3}}+\frac {a^{4/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{8 b^{4/3}}+\frac {1}{4} x^4 \log \left (c \left (a+b x^3\right )^p\right ) \]
[Out]
Time = 0.08 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2505, 308, 206, 31, 648, 631, 210, 642} \[ \int x^3 \log \left (c \left (a+b x^3\right )^p\right ) \, dx=\frac {\sqrt {3} a^{4/3} p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{4 b^{4/3}}+\frac {a^{4/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{8 b^{4/3}}-\frac {a^{4/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{4 b^{4/3}}+\frac {1}{4} x^4 \log \left (c \left (a+b x^3\right )^p\right )+\frac {3 a p x}{4 b}-\frac {3 p x^4}{16} \]
[In]
[Out]
Rule 31
Rule 206
Rule 210
Rule 308
Rule 631
Rule 642
Rule 648
Rule 2505
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^4 \log \left (c \left (a+b x^3\right )^p\right )-\frac {1}{4} (3 b p) \int \frac {x^6}{a+b x^3} \, dx \\ & = \frac {1}{4} x^4 \log \left (c \left (a+b x^3\right )^p\right )-\frac {1}{4} (3 b p) \int \left (-\frac {a}{b^2}+\frac {x^3}{b}+\frac {a^2}{b^2 \left (a+b x^3\right )}\right ) \, dx \\ & = \frac {3 a p x}{4 b}-\frac {3 p x^4}{16}+\frac {1}{4} x^4 \log \left (c \left (a+b x^3\right )^p\right )-\frac {\left (3 a^2 p\right ) \int \frac {1}{a+b x^3} \, dx}{4 b} \\ & = \frac {3 a p x}{4 b}-\frac {3 p x^4}{16}+\frac {1}{4} x^4 \log \left (c \left (a+b x^3\right )^p\right )-\frac {\left (a^{4/3} p\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{4 b}-\frac {\left (a^{4/3} 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}{4 b} \\ & = \frac {3 a p x}{4 b}-\frac {3 p x^4}{16}-\frac {a^{4/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{4 b^{4/3}}+\frac {1}{4} x^4 \log \left (c \left (a+b x^3\right )^p\right )+\frac {\left (a^{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 b^{4/3}}-\frac {\left (3 a^{5/3} p\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{8 b} \\ & = \frac {3 a p x}{4 b}-\frac {3 p x^4}{16}-\frac {a^{4/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{4 b^{4/3}}+\frac {a^{4/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{8 b^{4/3}}+\frac {1}{4} x^4 \log \left (c \left (a+b x^3\right )^p\right )-\frac {\left (3 a^{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 b^{4/3}} \\ & = \frac {3 a p x}{4 b}-\frac {3 p x^4}{16}+\frac {\sqrt {3} a^{4/3} p \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{4 b^{4/3}}-\frac {a^{4/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{4 b^{4/3}}+\frac {a^{4/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{8 b^{4/3}}+\frac {1}{4} x^4 \log \left (c \left (a+b x^3\right )^p\right ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.94 \[ \int x^3 \log \left (c \left (a+b x^3\right )^p\right ) \, dx=\frac {12 a \sqrt [3]{b} p x-3 b^{4/3} p x^4+4 \sqrt {3} a^{4/3} p \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )-4 a^{4/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+2 a^{4/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+4 b^{4/3} x^4 \log \left (c \left (a+b x^3\right )^p\right )}{16 b^{4/3}} \]
[In]
[Out]
Time = 0.66 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.87
method | result | size |
parts | \(\frac {x^{4} \ln \left (c \left (b \,x^{3}+a \right )^{p}\right )}{4}-\frac {3 p b \left (-\frac {-\frac {1}{4} b \,x^{4}+a x}{b^{2}}+\frac {\left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right ) a^{2}}{b^{2}}\right )}{4}\) | \(136\) |
risch | \(\frac {x^{4} \ln \left (\left (b \,x^{3}+a \right )^{p}\right )}{4}+\frac {i \pi \,x^{4} \operatorname {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{2}}{8}-\frac {i \pi \,x^{4} \operatorname {csgn}\left (i \left (b \,x^{3}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{8}-\frac {i \pi \,x^{4} {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{3}}{8}+\frac {i \pi \,x^{4} {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{8}+\frac {\ln \left (c \right ) x^{4}}{4}-\frac {3 p \,x^{4}}{16}-\frac {a^{2} p \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{4 b^{2}}+\frac {3 a p x}{4 b}\) | \(194\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.92 \[ \int x^3 \log \left (c \left (a+b x^3\right )^p\right ) \, dx=\frac {4 \, b p x^{4} \log \left (b x^{3} + a\right ) - 3 \, b p x^{4} + 4 \, b x^{4} \log \left (c\right ) + 4 \, \sqrt {3} a p \left (-\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (-\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) - 2 \, a p \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right ) + 4 \, a p \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x - \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right ) + 12 \, a p x}{16 \, b} \]
[In]
[Out]
Timed out. \[ \int x^3 \log \left (c \left (a+b x^3\right )^p\right ) \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.92 \[ \int x^3 \log \left (c \left (a+b x^3\right )^p\right ) \, dx=\frac {1}{4} \, x^{4} \log \left ({\left (b x^{3} + a\right )}^{p} c\right ) - \frac {1}{16} \, b p {\left (\frac {4 \, \sqrt {3} a^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {2 \, a^{2} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {4 \, a^{2} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {3 \, {\left (b x^{4} - 4 \, a x\right )}}{b^{2}}\right )} \]
[In]
[Out]
none
Time = 0.36 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.02 \[ \int x^3 \log \left (c \left (a+b x^3\right )^p\right ) \, dx=\frac {1}{8} \, a^{2} b^{3} p {\left (\frac {2 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{a b^{4}} - \frac {2 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a b^{5}} - \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a b^{5}}\right )} + \frac {1}{4} \, p x^{4} \log \left (b x^{3} + a\right ) - \frac {1}{16} \, {\left (3 \, p - 4 \, \log \left (c\right )\right )} x^{4} + \frac {3 \, a p x}{4 \, b} \]
[In]
[Out]
Time = 3.84 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.82 \[ \int x^3 \log \left (c \left (a+b x^3\right )^p\right ) \, dx=\frac {x^4\,\ln \left (c\,{\left (b\,x^3+a\right )}^p\right )}{4}-\frac {3\,p\,x^4}{16}+\frac {3\,a\,p\,x}{4\,b}-\frac {a^{4/3}\,p\,\ln \left (b^{1/3}\,x+a^{1/3}\right )}{4\,b^{4/3}}+\frac {a^{4/3}\,p\,\ln \left (2\,b^{1/3}\,x-a^{1/3}-\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{4\,b^{4/3}}-\frac {a^{4/3}\,p\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{4\,b^{4/3}} \]
[In]
[Out]