Integrand size = 20, antiderivative size = 292 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{(d+e x)^2} \, dx=-\frac {\sqrt {3} \sqrt [3]{a} \sqrt [3]{b} p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{b^{2/3} d^2+\sqrt [3]{a} \sqrt [3]{b} d e+a^{2/3} e^2}+\frac {\sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{b} d+\sqrt [3]{a} e\right ) p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b d^3-a e^3}-\frac {3 b d^2 p \log (d+e x)}{e \left (b d^3-a e^3\right )}-\frac {\sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{b} d+\sqrt [3]{a} e\right ) p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \left (b d^3-a e^3\right )}+\frac {b d^2 p \log \left (a+b x^3\right )}{e \left (b d^3-a e^3\right )}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{e (d+e x)} \]
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Time = 0.36 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2513, 6857, 1885, 1874, 31, 648, 631, 210, 642, 266} \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{(d+e x)^2} \, dx=-\frac {\sqrt {3} \sqrt [3]{a} \sqrt [3]{b} p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{2/3} e^2+\sqrt [3]{a} \sqrt [3]{b} d e+b^{2/3} d^2}-\frac {\sqrt [3]{a} \sqrt [3]{b} p \left (\sqrt [3]{a} e+\sqrt [3]{b} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \left (b d^3-a e^3\right )}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{e (d+e x)}+\frac {\sqrt [3]{a} \sqrt [3]{b} p \left (\sqrt [3]{a} e+\sqrt [3]{b} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b d^3-a e^3}+\frac {b d^2 p \log \left (a+b x^3\right )}{e \left (b d^3-a e^3\right )}-\frac {3 b d^2 p \log (d+e x)}{e \left (b d^3-a e^3\right )} \]
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Rule 31
Rule 210
Rule 266
Rule 631
Rule 642
Rule 648
Rule 1874
Rule 1885
Rule 2513
Rule 6857
Rubi steps \begin{align*} \text {integral}& = -\frac {\log \left (c \left (a+b x^3\right )^p\right )}{e (d+e x)}+\frac {(3 b p) \int \frac {x^2}{(d+e x) \left (a+b x^3\right )} \, dx}{e} \\ & = -\frac {\log \left (c \left (a+b x^3\right )^p\right )}{e (d+e x)}+\frac {(3 b p) \int \left (-\frac {d^2 e}{\left (b d^3-a e^3\right ) (d+e x)}+\frac {a d e-a e^2 x+b d^2 x^2}{\left (b d^3-a e^3\right ) \left (a+b x^3\right )}\right ) \, dx}{e} \\ & = -\frac {3 b d^2 p \log (d+e x)}{e \left (b d^3-a e^3\right )}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{e (d+e x)}+\frac {(3 b p) \int \frac {a d e-a e^2 x+b d^2 x^2}{a+b x^3} \, dx}{e \left (b d^3-a e^3\right )} \\ & = -\frac {3 b d^2 p \log (d+e x)}{e \left (b d^3-a e^3\right )}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{e (d+e x)}+\frac {(3 b p) \int \frac {a d e-a e^2 x}{a+b x^3} \, dx}{e \left (b d^3-a e^3\right )}+\frac {\left (3 b^2 d^2 p\right ) \int \frac {x^2}{a+b x^3} \, dx}{e \left (b d^3-a e^3\right )} \\ & = -\frac {3 b d^2 p \log (d+e x)}{e \left (b d^3-a e^3\right )}+\frac {b d^2 p \log \left (a+b x^3\right )}{e \left (b d^3-a e^3\right )}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{e (d+e x)}+\frac {\left (b^{2/3} p\right ) \int \frac {\sqrt [3]{a} \left (2 a \sqrt [3]{b} d e-a^{4/3} e^2\right )+\sqrt [3]{b} \left (-a \sqrt [3]{b} d e-a^{4/3} e^2\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{a^{2/3} e \left (b d^3-a e^3\right )}+\frac {\left (\sqrt [3]{a} b \left (d+\frac {\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) p\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{b d^3-a e^3} \\ & = \frac {\sqrt [3]{a} b^{2/3} \left (d+\frac {\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b d^3-a e^3}-\frac {3 b d^2 p \log (d+e x)}{e \left (b d^3-a e^3\right )}+\frac {b d^2 p \log \left (a+b x^3\right )}{e \left (b d^3-a e^3\right )}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{e (d+e x)}+\frac {\left (3 a^{2/3} b^{2/3} \left (\sqrt [3]{b} d-\sqrt [3]{a} e\right ) p\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 \left (b d^3-a e^3\right )}-\frac {\left (\sqrt [3]{a} b^{2/3} \left (d+\frac {\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) 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}{2 \left (b d^3-a e^3\right )} \\ & = \frac {\sqrt [3]{a} b^{2/3} \left (d+\frac {\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b d^3-a e^3}-\frac {3 b d^2 p \log (d+e x)}{e \left (b d^3-a e^3\right )}-\frac {\sqrt [3]{a} b^{2/3} \left (d+\frac {\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \left (b d^3-a e^3\right )}+\frac {b d^2 p \log \left (a+b x^3\right )}{e \left (b d^3-a e^3\right )}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{e (d+e x)}+\frac {\left (3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{b} d-\sqrt [3]{a} e\right ) p\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{b d^3-a e^3} \\ & = -\frac {\sqrt {3} \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{b} d-\sqrt [3]{a} e\right ) p \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{b d^3-a e^3}+\frac {\sqrt [3]{a} b^{2/3} \left (d+\frac {\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b d^3-a e^3}-\frac {3 b d^2 p \log (d+e x)}{e \left (b d^3-a e^3\right )}-\frac {\sqrt [3]{a} b^{2/3} \left (d+\frac {\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \left (b d^3-a e^3\right )}+\frac {b d^2 p \log \left (a+b x^3\right )}{e \left (b d^3-a e^3\right )}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{e (d+e x)} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.21 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.87 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{(d+e x)^2} \, dx=-\frac {2 \sqrt {3} \sqrt [3]{a} b^{2/3} d e p (d+e x) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+3 b e^2 p x^2 (d+e x) \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},-\frac {b x^3}{a}\right )-2 \sqrt [3]{a} b^{2/3} d e p (d+e x) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+6 b d^2 p (d+e x) \log (d+e x)+\sqrt [3]{a} b^{2/3} d e p (d+e x) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-2 b d^2 p (d+e x) \log \left (a+b x^3\right )+2 \left (b d^3-a e^3\right ) \log \left (c \left (a+b x^3\right )^p\right )}{2 e \left (b d^3-a e^3\right ) (d+e x)} \]
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Time = 1.73 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.95
method | result | size |
parts | \(-\frac {\ln \left (c \left (b \,x^{3}+a \right )^{p}\right )}{e \left (e x +d \right )}+\frac {3 p b \left (\frac {d^{2} \ln \left (e x +d \right )}{a \,e^{3}-b \,d^{3}}+\frac {-a d e \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+a \,e^{2} \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )-\frac {d^{2} \ln \left (b \,x^{3}+a \right )}{3}}{a \,e^{3}-b \,d^{3}}\right )}{e}\) | \(277\) |
risch | \(\text {Expression too large to display}\) | \(1068\) |
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Result contains complex when optimal does not.
Time = 1.25 (sec) , antiderivative size = 7010, normalized size of antiderivative = 24.01 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{(d+e x)^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{(d+e x)^2} \, dx=\text {Timed out} \]
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Time = 0.32 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.07 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{(d+e x)^2} \, dx=-\frac {{\left (\frac {6 \, d^{2} \log \left (e x + d\right )}{b d^{3} - a e^{3}} + \frac {2 \, \sqrt {3} {\left (a e^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} - a d e \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{{\left (b^{2} d^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b e^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (2 \, b d^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} - a e^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} - a d e\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{b^{2} d^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b e^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {2 \, {\left (b d^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} + a e^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} + a d e\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{b^{2} d^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b e^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )} b p}{2 \, e} - \frac {\log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{{\left (e x + d\right )} e} \]
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Time = 0.36 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.24 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{(d+e x)^2} \, dx=-\frac {3 \, b d^{2} p \log \left (e x + d\right )}{b d^{3} e - a e^{4}} + \frac {b d^{2} p \log \left ({\left | b x^{3} + a \right |}\right )}{b d^{3} e - a e^{4}} + \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} b p \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{b^{2} d^{2} - \left (-a b^{2}\right )^{\frac {1}{3}} b d e + \left (-a b^{2}\right )^{\frac {2}{3}} e^{2}} + \frac {{\left (a b^{3} d^{3} e^{3} p \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} b^{2} e^{6} p \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a b^{3} d^{4} e^{2} p + a^{2} b^{2} d e^{5} p\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{a b^{3} d^{6} e^{2} - 2 \, a^{2} b^{2} d^{3} e^{5} + a^{3} b e^{8}} - \frac {p \log \left (b x^{3} + a\right )}{e^{2} x + d e} + \frac {{\left (\left (-a b^{2}\right )^{\frac {1}{3}} b d p - \left (-a b^{2}\right )^{\frac {2}{3}} e p\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{2 \, {\left (b^{2} d^{3} - a b e^{3}\right )}} - \frac {\log \left (c\right )}{e^{2} x + d e} \]
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Time = 1.56 (sec) , antiderivative size = 736, normalized size of antiderivative = 2.52 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{(d+e x)^2} \, dx=\left (\sum _{k=1}^3\ln \left (-\frac {27\,a\,b^4\,d\,p^3+27\,a\,b^4\,e\,p^3\,x+{\mathrm {root}\left (b\,d^3\,e^3\,z^3-a\,e^6\,z^3-3\,b\,d^2\,e^2\,p\,z^2+3\,b\,d\,e\,p^2\,z-b\,p^3,z,k\right )}^3\,a\,b^4\,d^4\,e^3\,9+{\mathrm {root}\left (b\,d^3\,e^3\,z^3-a\,e^6\,z^3-3\,b\,d^2\,e^2\,p\,z^2+3\,b\,d\,e\,p^2\,z-b\,p^3,z,k\right )}^3\,a^2\,b^3\,d\,e^6\,45-{\mathrm {root}\left (b\,d^3\,e^3\,z^3-a\,e^6\,z^3-3\,b\,d^2\,e^2\,p\,z^2+3\,b\,d\,e\,p^2\,z-b\,p^3,z,k\right )}^2\,a^2\,b^3\,e^5\,p\,9+{\mathrm {root}\left (b\,d^3\,e^3\,z^3-a\,e^6\,z^3-3\,b\,d^2\,e^2\,p\,z^2+3\,b\,d\,e\,p^2\,z-b\,p^3,z,k\right )}^3\,a^2\,b^3\,e^7\,x\,36+{\mathrm {root}\left (b\,d^3\,e^3\,z^3-a\,e^6\,z^3-3\,b\,d^2\,e^2\,p\,z^2+3\,b\,d\,e\,p^2\,z-b\,p^3,z,k\right )}^2\,a\,b^4\,d^3\,e^2\,p\,9+{\mathrm {root}\left (b\,d^3\,e^3\,z^3-a\,e^6\,z^3-3\,b\,d^2\,e^2\,p\,z^2+3\,b\,d\,e\,p^2\,z-b\,p^3,z,k\right )}^3\,a\,b^4\,d^3\,e^4\,x\,18-\mathrm {root}\left (b\,d^3\,e^3\,z^3-a\,e^6\,z^3-3\,b\,d^2\,e^2\,p\,z^2+3\,b\,d\,e\,p^2\,z-b\,p^3,z,k\right )\,a\,b^4\,d^2\,e\,p^2\,45-\mathrm {root}\left (b\,d^3\,e^3\,z^3-a\,e^6\,z^3-3\,b\,d^2\,e^2\,p\,z^2+3\,b\,d\,e\,p^2\,z-b\,p^3,z,k\right )\,a\,b^4\,d\,e^2\,p^2\,x\,72+{\mathrm {root}\left (b\,d^3\,e^3\,z^3-a\,e^6\,z^3-3\,b\,d^2\,e^2\,p\,z^2+3\,b\,d\,e\,p^2\,z-b\,p^3,z,k\right )}^2\,a\,b^4\,d^2\,e^3\,p\,x\,27}{e^2}\right )\,\mathrm {root}\left (b\,d^3\,e^3\,z^3-a\,e^6\,z^3-3\,b\,d^2\,e^2\,p\,z^2+3\,b\,d\,e\,p^2\,z-b\,p^3,z,k\right )\right )-\frac {\ln \left (c\,{\left (b\,x^3+a\right )}^p\right )}{x\,e^2+d\,e}+\frac {3\,b\,d^2\,p\,\ln \left (d+e\,x\right )}{a\,e^4-b\,d^3\,e} \]
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