Integrand size = 20, antiderivative size = 391 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{(d+e x)^3} \, dx=\frac {3 b d^2 p}{2 e \left (b d^3-a e^3\right ) (d+e x)}-\frac {\sqrt {3} \sqrt [3]{a} b^{2/3} \left (2 b d^3-3 \sqrt [3]{a} b^{2/3} d^2 e+a e^3\right ) p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{2 \left (b d^3-a e^3\right )^2}+\frac {\sqrt [3]{a} b^{2/3} \left (2 b d^3+3 \sqrt [3]{a} b^{2/3} d^2 e+a e^3\right ) p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 \left (b d^3-a e^3\right )^2}-\frac {3 b d \left (b d^3+2 a e^3\right ) p \log (d+e x)}{2 e \left (b d^3-a e^3\right )^2}-\frac {\sqrt [3]{a} b^{2/3} \left (2 b d^3+3 \sqrt [3]{a} b^{2/3} d^2 e+a e^3\right ) p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 \left (b d^3-a e^3\right )^2}+\frac {b d \left (b d^3+2 a e^3\right ) p \log \left (a+b x^3\right )}{2 e \left (b d^3-a e^3\right )^2}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{2 e (d+e x)^2} \]
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Time = 0.44 (sec) , antiderivative size = 391, 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} \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{(d+e x)^3} \, dx=-\frac {\sqrt [3]{a} b^{2/3} p \left (3 \sqrt [3]{a} b^{2/3} d^2 e+a e^3+2 b d^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 \left (b d^3-a e^3\right )^2}-\frac {\sqrt {3} \sqrt [3]{a} b^{2/3} p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-3 \sqrt [3]{a} b^{2/3} d^2 e+a e^3+2 b d^3\right )}{2 \left (b d^3-a e^3\right )^2}+\frac {\sqrt [3]{a} b^{2/3} p \left (3 \sqrt [3]{a} b^{2/3} d^2 e+a e^3+2 b d^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 \left (b d^3-a e^3\right )^2}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{2 e (d+e x)^2}+\frac {b d p \left (2 a e^3+b d^3\right ) \log \left (a+b x^3\right )}{2 e \left (b d^3-a e^3\right )^2}-\frac {3 b d p \left (2 a e^3+b d^3\right ) \log (d+e x)}{2 e \left (b d^3-a e^3\right )^2}+\frac {3 b d^2 p}{2 e (d+e x) \left (b d^3-a e^3\right )} \]
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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 )}{2 e (d+e x)^2}+\frac {(3 b p) \int \frac {x^2}{(d+e x)^2 \left (a+b x^3\right )} \, dx}{2 e} \\ & = -\frac {\log \left (c \left (a+b x^3\right )^p\right )}{2 e (d+e x)^2}+\frac {(3 b p) \int \left (-\frac {d^2 e}{\left (b d^3-a e^3\right ) (d+e x)^2}-\frac {d e \left (b d^3+2 a e^3\right )}{\left (b d^3-a e^3\right )^2 (d+e x)}+\frac {a e \left (2 b d^3+a e^3\right )-3 a b d^2 e^2 x+b d \left (b d^3+2 a e^3\right ) x^2}{\left (b d^3-a e^3\right )^2 \left (a+b x^3\right )}\right ) \, dx}{2 e} \\ & = \frac {3 b d^2 p}{2 e \left (b d^3-a e^3\right ) (d+e x)}-\frac {3 b d \left (b d^3+2 a e^3\right ) p \log (d+e x)}{2 e \left (b d^3-a e^3\right )^2}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{2 e (d+e x)^2}+\frac {(3 b p) \int \frac {a e \left (2 b d^3+a e^3\right )-3 a b d^2 e^2 x+b d \left (b d^3+2 a e^3\right ) x^2}{a+b x^3} \, dx}{2 e \left (b d^3-a e^3\right )^2} \\ & = \frac {3 b d^2 p}{2 e \left (b d^3-a e^3\right ) (d+e x)}-\frac {3 b d \left (b d^3+2 a e^3\right ) p \log (d+e x)}{2 e \left (b d^3-a e^3\right )^2}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{2 e (d+e x)^2}+\frac {(3 b p) \int \frac {a e \left (2 b d^3+a e^3\right )-3 a b d^2 e^2 x}{a+b x^3} \, dx}{2 e \left (b d^3-a e^3\right )^2}+\frac {\left (3 b^2 d \left (b d^3+2 a e^3\right ) p\right ) \int \frac {x^2}{a+b x^3} \, dx}{2 e \left (b d^3-a e^3\right )^2} \\ & = \frac {3 b d^2 p}{2 e \left (b d^3-a e^3\right ) (d+e x)}-\frac {3 b d \left (b d^3+2 a e^3\right ) p \log (d+e x)}{2 e \left (b d^3-a e^3\right )^2}+\frac {b d \left (b d^3+2 a e^3\right ) p \log \left (a+b x^3\right )}{2 e \left (b d^3-a e^3\right )^2}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{2 e (d+e x)^2}+\frac {\left (b^{2/3} p\right ) \int \frac {\sqrt [3]{a} \left (-3 a^{4/3} b d^2 e^2+2 a \sqrt [3]{b} e \left (2 b d^3+a e^3\right )\right )+\sqrt [3]{b} \left (-3 a^{4/3} b d^2 e^2-a \sqrt [3]{b} e \left (2 b d^3+a e^3\right )\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 a^{2/3} e \left (b d^3-a e^3\right )^2}+\frac {\left (\sqrt [3]{a} b \left (2 b d^3+3 \sqrt [3]{a} b^{2/3} d^2 e+a e^3\right ) p\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{2 \left (b d^3-a e^3\right )^2} \\ & = \frac {3 b d^2 p}{2 e \left (b d^3-a e^3\right ) (d+e x)}+\frac {\sqrt [3]{a} b^{2/3} \left (2 b d^3+3 \sqrt [3]{a} b^{2/3} d^2 e+a e^3\right ) p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 \left (b d^3-a e^3\right )^2}-\frac {3 b d \left (b d^3+2 a e^3\right ) p \log (d+e x)}{2 e \left (b d^3-a e^3\right )^2}+\frac {b d \left (b d^3+2 a e^3\right ) p \log \left (a+b x^3\right )}{2 e \left (b d^3-a e^3\right )^2}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{2 e (d+e x)^2}+\frac {\left (3 a^{2/3} b \left (2 b d^3-3 \sqrt [3]{a} b^{2/3} d^2 e+a e^3\right ) p\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{4 \left (b d^3-a e^3\right )^2}-\frac {\left (\sqrt [3]{a} b^{2/3} \left (2 b d^3+3 \sqrt [3]{a} b^{2/3} d^2 e+a e^3\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}{4 \left (b d^3-a e^3\right )^2} \\ & = \frac {3 b d^2 p}{2 e \left (b d^3-a e^3\right ) (d+e x)}+\frac {\sqrt [3]{a} b^{2/3} \left (2 b d^3+3 \sqrt [3]{a} b^{2/3} d^2 e+a e^3\right ) p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 \left (b d^3-a e^3\right )^2}-\frac {3 b d \left (b d^3+2 a e^3\right ) p \log (d+e x)}{2 e \left (b d^3-a e^3\right )^2}-\frac {\sqrt [3]{a} b^{2/3} \left (2 b d^3+3 \sqrt [3]{a} b^{2/3} d^2 e+a e^3\right ) p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 \left (b d^3-a e^3\right )^2}+\frac {b d \left (b d^3+2 a e^3\right ) p \log \left (a+b x^3\right )}{2 e \left (b d^3-a e^3\right )^2}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{2 e (d+e x)^2}+\frac {\left (3 \sqrt [3]{a} b^{2/3} \left (2 b d^3-3 \sqrt [3]{a} b^{2/3} d^2 e+a e^3\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 )}{2 \left (b d^3-a e^3\right )^2} \\ & = \frac {3 b d^2 p}{2 e \left (b d^3-a e^3\right ) (d+e x)}-\frac {\sqrt {3} \sqrt [3]{a} b^{2/3} \left (2 b d^3-3 \sqrt [3]{a} b^{2/3} d^2 e+a e^3\right ) p \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{2 \left (b d^3-a e^3\right )^2}+\frac {\sqrt [3]{a} b^{2/3} \left (2 b d^3+3 \sqrt [3]{a} b^{2/3} d^2 e+a e^3\right ) p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 \left (b d^3-a e^3\right )^2}-\frac {3 b d \left (b d^3+2 a e^3\right ) p \log (d+e x)}{2 e \left (b d^3-a e^3\right )^2}-\frac {\sqrt [3]{a} b^{2/3} \left (2 b d^3+3 \sqrt [3]{a} b^{2/3} d^2 e+a e^3\right ) p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 \left (b d^3-a e^3\right )^2}+\frac {b d \left (b d^3+2 a e^3\right ) p \log \left (a+b x^3\right )}{2 e \left (b d^3-a e^3\right )^2}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{2 e (d+e x)^2} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.39 (sec) , antiderivative size = 325, normalized size of antiderivative = 0.83 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{(d+e x)^3} \, dx=\frac {\frac {b^{2/3} p (d+e x) \left (6 \sqrt [3]{b} d^2 \left (b d^3-a e^3\right )-2 \sqrt {3} \sqrt [3]{a} e \left (2 b d^3+a e^3\right ) (d+e x) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )-9 b^{4/3} d^2 e^2 x^2 (d+e x) \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},-\frac {b x^3}{a}\right )+2 \sqrt [3]{a} e \left (2 b d^3+a e^3\right ) (d+e x) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-6 \sqrt [3]{b} d \left (b d^3+2 a e^3\right ) (d+e x) \log (d+e x)-\sqrt [3]{a} e \left (2 b d^3+a e^3\right ) (d+e x) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 \sqrt [3]{b} d \left (b d^3+2 a e^3\right ) (d+e x) \log \left (a+b x^3\right )\right )}{\left (b d^3-a e^3\right )^2}-2 \log \left (c \left (a+b x^3\right )^p\right )}{4 e (d+e x)^2} \]
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Time = 2.62 (sec) , antiderivative size = 347, normalized size of antiderivative = 0.89
method | result | size |
parts | \(-\frac {\ln \left (c \left (b \,x^{3}+a \right )^{p}\right )}{2 e \left (e x +d \right )^{2}}+\frac {3 p b \left (-\frac {d^{2}}{\left (a \,e^{3}-b \,d^{3}\right ) \left (e x +d \right )}-\frac {d \left (2 a \,e^{3}+b \,d^{3}\right ) \ln \left (e x +d \right )}{\left (a \,e^{3}-b \,d^{3}\right )^{2}}+\frac {\left (a^{2} e^{4}+2 a b \,d^{3} e \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )-3 a b \,d^{2} e^{2} \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+\frac {\left (2 a b d \,e^{3}+b^{2} d^{4}\right ) \ln \left (b \,x^{3}+a \right )}{3 b}}{\left (a \,e^{3}-b \,d^{3}\right )^{2}}\right )}{2 e}\) | \(347\) |
risch | \(\text {Expression too large to display}\) | \(4085\) |
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Result contains complex when optimal does not.
Time = 9.42 (sec) , antiderivative size = 13236, normalized size of antiderivative = 33.85 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{(d+e x)^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{(d+e x)^3} \, dx=\text {Timed out} \]
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Time = 0.32 (sec) , antiderivative size = 517, normalized size of antiderivative = 1.32 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{(d+e x)^3} \, dx=-\frac {{\left (\frac {2 \, \sqrt {3} {\left (3 \, a b d^{2} e^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a b d^{3} e \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} e^{4} \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^{3} d^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a b^{2} d^{3} e^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}} + a^{2} b e^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {6 \, d^{2}}{b d^{4} - a d e^{3} + {\left (b d^{3} e - a e^{4}\right )} x} + \frac {6 \, {\left (b d^{4} + 2 \, a d e^{3}\right )} \log \left (e x + d\right )}{b^{2} d^{6} - 2 \, a b d^{3} e^{3} + a^{2} e^{6}} - \frac {{\left (2 \, b^{2} d^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}} + 4 \, a b d e^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 3 \, a b d^{2} e^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a b d^{3} e - a^{2} e^{4}\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{b^{3} d^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a b^{2} d^{3} e^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}} + a^{2} b e^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {2 \, {\left (b^{2} d^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, a b d e^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}} + 3 \, a b d^{2} e^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a b d^{3} e + a^{2} e^{4}\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{b^{3} d^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a b^{2} d^{3} e^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}} + a^{2} b e^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )} b p}{4 \, e} - \frac {\log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{2 \, {\left (e x + d\right )}^{2} e} \]
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Time = 0.46 (sec) , antiderivative size = 660, normalized size of antiderivative = 1.69 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{(d+e x)^3} \, dx=\frac {{\left (3 \, a b^{5} d^{8} e^{3} p \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 6 \, a^{2} b^{4} d^{5} e^{6} p \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 3 \, a^{3} b^{3} d^{2} e^{9} p \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a b^{5} d^{9} e^{2} p + 3 \, a^{2} b^{4} d^{6} e^{5} p - a^{4} b^{2} e^{11} p\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{2 \, {\left (a b^{5} d^{12} e^{2} - 4 \, a^{2} b^{4} d^{9} e^{5} + 6 \, a^{3} b^{3} d^{6} e^{8} - 4 \, a^{4} b^{2} d^{3} e^{11} + a^{5} b e^{14}\right )}} + \frac {3 \, {\left (2 \, \left (-a b^{2}\right )^{\frac {1}{3}} b d p - \left (-a b^{2}\right )^{\frac {2}{3}} e p\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 )}{2 \, {\left (\sqrt {3} b^{2} d^{4} + 2 \, \sqrt {3} a b d e^{3} - 2 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} b d^{3} e - \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} a e^{4} + 3 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} d^{2} e^{2}\right )}} - \frac {p \log \left (b x^{3} + a\right )}{2 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} - \frac {3 \, {\left (b^{2} d^{4} p + 2 \, a b d e^{3} p\right )} \log \left (e x + d\right )}{2 \, {\left (b^{2} d^{6} e - 2 \, a b d^{3} e^{4} + a^{2} e^{7}\right )}} + \frac {{\left (2 \, \left (-a b^{2}\right )^{\frac {1}{3}} b d^{3} p + \left (-a b^{2}\right )^{\frac {1}{3}} a e^{3} p - 3 \, \left (-a b^{2}\right )^{\frac {2}{3}} d^{2} 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 )}{4 \, {\left (b^{2} d^{6} - 2 \, a b d^{3} e^{3} + a^{2} e^{6}\right )}} + \frac {{\left (b^{2} d^{4} p + 2 \, a b d e^{3} p\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{2 \, {\left (b^{2} d^{6} e - 2 \, a b d^{3} e^{4} + a^{2} e^{7}\right )}} + \frac {3 \, b d^{2} e p x + 3 \, b d^{3} p - b d^{3} \log \left (c\right ) + a e^{3} \log \left (c\right )}{2 \, {\left (b d^{3} e^{3} x^{2} - a e^{6} x^{2} + 2 \, b d^{4} e^{2} x - 2 \, a d e^{5} x + b d^{5} e - a d^{2} e^{4}\right )}} \]
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Time = 1.92 (sec) , antiderivative size = 2227, normalized size of antiderivative = 5.70 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{(d+e x)^3} \, dx=\text {Too large to display} \]
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