Integrand size = 23, antiderivative size = 674 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^3 (d+e x)} \, dx=-\frac {\sqrt {3} b^{2/3} p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{2 a^{2/3} d}+\frac {\sqrt {3} \sqrt [3]{b} e p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{a} d^2}+\frac {b^{2/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 a^{2/3} d}+\frac {\sqrt [3]{b} e p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a} d^2}+\frac {e^2 p \log \left (-\frac {e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (-\frac {e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right ) \log (d+e x)}{d^3}-\frac {b^{2/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 a^{2/3} d}-\frac {\sqrt [3]{b} e p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{a} d^2}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{2 d x^2}+\frac {e \log \left (c \left (a+b x^3\right )^p\right )}{d^2 x}+\frac {e^2 \log \left (-\frac {b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 d^3}-\frac {e^2 \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,1+\frac {b x^3}{a}\right )}{3 d^3} \]
[Out]
Time = 0.42 (sec) , antiderivative size = 674, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.739, Rules used = {2516, 2505, 206, 31, 648, 631, 210, 642, 298, 2504, 2441, 2352, 2512, 266, 2463, 2440, 2438} \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^3 (d+e x)} \, dx=-\frac {\sqrt {3} b^{2/3} p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{2 a^{2/3} d}-\frac {\sqrt [3]{b} e p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{a} d^2}-\frac {b^{2/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 a^{2/3} d}+\frac {b^{2/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 a^{2/3} d}+\frac {\sqrt {3} \sqrt [3]{b} e p \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{a} d^2}+\frac {e^2 \log \left (-\frac {b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 d^3}-\frac {e^2 \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{d^3}+\frac {e \log \left (c \left (a+b x^3\right )^p\right )}{d^2 x}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{2 d x^2}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {b x^3}{a}+1\right )}{3 d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{d^3}+\frac {e^2 p \log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{d^3}+\frac {e^2 p \log (d+e x) \log \left (-\frac {e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{d^3}+\frac {e^2 p \log (d+e x) \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d}\right )}{d^3}+\frac {\sqrt [3]{b} e p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a} d^2} \]
[In]
[Out]
Rule 31
Rule 206
Rule 210
Rule 266
Rule 298
Rule 631
Rule 642
Rule 648
Rule 2352
Rule 2438
Rule 2440
Rule 2441
Rule 2463
Rule 2504
Rule 2505
Rule 2512
Rule 2516
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\log \left (c \left (a+b x^3\right )^p\right )}{d x^3}-\frac {e \log \left (c \left (a+b x^3\right )^p\right )}{d^2 x^2}+\frac {e^2 \log \left (c \left (a+b x^3\right )^p\right )}{d^3 x}-\frac {e^3 \log \left (c \left (a+b x^3\right )^p\right )}{d^3 (d+e x)}\right ) \, dx \\ & = \frac {\int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^3} \, dx}{d}-\frac {e \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^2} \, dx}{d^2}+\frac {e^2 \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x} \, dx}{d^3}-\frac {e^3 \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{d+e x} \, dx}{d^3} \\ & = -\frac {\log \left (c \left (a+b x^3\right )^p\right )}{2 d x^2}+\frac {e \log \left (c \left (a+b x^3\right )^p\right )}{d^2 x}-\frac {e^2 \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{d^3}+\frac {e^2 \text {Subst}\left (\int \frac {\log \left (c (a+b x)^p\right )}{x} \, dx,x,x^3\right )}{3 d^3}+\frac {(3 b p) \int \frac {1}{a+b x^3} \, dx}{2 d}-\frac {(3 b e p) \int \frac {x}{a+b x^3} \, dx}{d^2}+\frac {\left (3 b e^2 p\right ) \int \frac {x^2 \log (d+e x)}{a+b x^3} \, dx}{d^3} \\ & = -\frac {\log \left (c \left (a+b x^3\right )^p\right )}{2 d x^2}+\frac {e \log \left (c \left (a+b x^3\right )^p\right )}{d^2 x}+\frac {e^2 \log \left (-\frac {b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 d^3}-\frac {e^2 \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{d^3}+\frac {(b p) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{2 a^{2/3} d}+\frac {(b p) \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}{2 a^{2/3} d}+\frac {\left (b^{2/3} e p\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{\sqrt [3]{a} d^2}-\frac {\left (b^{2/3} e 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}{\sqrt [3]{a} d^2}-\frac {\left (b e^2 p\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {b x}{a}\right )}{a+b x} \, dx,x,x^3\right )}{3 d^3}+\frac {\left (3 b e^2 p\right ) \int \left (\frac {\log (d+e x)}{3 b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {\log (d+e x)}{3 b^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {\log (d+e x)}{3 b^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}\right ) \, dx}{d^3} \\ & = \frac {b^{2/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 a^{2/3} d}+\frac {\sqrt [3]{b} e p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a} d^2}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{2 d x^2}+\frac {e \log \left (c \left (a+b x^3\right )^p\right )}{d^2 x}+\frac {e^2 \log \left (-\frac {b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 d^3}-\frac {e^2 \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{d^3}+\frac {e^2 p \text {Li}_2\left (1+\frac {b x^3}{a}\right )}{3 d^3}-\frac {\left (b^{2/3} 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}{4 a^{2/3} d}+\frac {(3 b p) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{4 \sqrt [3]{a} d}-\frac {\left (\sqrt [3]{b} e 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 \sqrt [3]{a} d^2}-\frac {\left (3 b^{2/3} e p\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 d^2}+\frac {\left (\sqrt [3]{b} e^2 p\right ) \int \frac {\log (d+e x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{d^3}+\frac {\left (\sqrt [3]{b} e^2 p\right ) \int \frac {\log (d+e x)}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{d^3}+\frac {\left (\sqrt [3]{b} e^2 p\right ) \int \frac {\log (d+e x)}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{d^3} \\ & = \frac {b^{2/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 a^{2/3} d}+\frac {\sqrt [3]{b} e p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a} d^2}+\frac {e^2 p \log \left (-\frac {e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (-\frac {e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right ) \log (d+e x)}{d^3}-\frac {b^{2/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 a^{2/3} d}-\frac {\sqrt [3]{b} e p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{a} d^2}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{2 d x^2}+\frac {e \log \left (c \left (a+b x^3\right )^p\right )}{d^2 x}+\frac {e^2 \log \left (-\frac {b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 d^3}-\frac {e^2 \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{d^3}+\frac {e^2 p \text {Li}_2\left (1+\frac {b x^3}{a}\right )}{3 d^3}+\frac {\left (3 b^{2/3} p\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{2 a^{2/3} d}-\frac {\left (3 \sqrt [3]{b} e p\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{a} d^2}-\frac {\left (e^3 p\right ) \int \frac {\log \left (\frac {e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} d+\sqrt [3]{a} e}\right )}{d+e x} \, dx}{d^3}-\frac {\left (e^3 p\right ) \int \frac {\log \left (\frac {e \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} d-\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{d+e x} \, dx}{d^3}-\frac {\left (e^3 p\right ) \int \frac {\log \left (\frac {e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} d+(-1)^{2/3} \sqrt [3]{a} e}\right )}{d+e x} \, dx}{d^3} \\ & = -\frac {\sqrt {3} b^{2/3} p \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{2 a^{2/3} d}+\frac {\sqrt {3} \sqrt [3]{b} e p \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{a} d^2}+\frac {b^{2/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 a^{2/3} d}+\frac {\sqrt [3]{b} e p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a} d^2}+\frac {e^2 p \log \left (-\frac {e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (-\frac {e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right ) \log (d+e x)}{d^3}-\frac {b^{2/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 a^{2/3} d}-\frac {\sqrt [3]{b} e p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{a} d^2}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{2 d x^2}+\frac {e \log \left (c \left (a+b x^3\right )^p\right )}{d^2 x}+\frac {e^2 \log \left (-\frac {b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 d^3}-\frac {e^2 \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{d^3}+\frac {e^2 p \text {Li}_2\left (1+\frac {b x^3}{a}\right )}{3 d^3}-\frac {\left (e^2 p\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{b} x}{-\sqrt [3]{b} d+\sqrt [3]{a} e}\right )}{x} \, dx,x,d+e x\right )}{d^3}-\frac {\left (e^2 p\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{b} x}{-\sqrt [3]{b} d-\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{x} \, dx,x,d+e x\right )}{d^3}-\frac {\left (e^2 p\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{b} x}{-\sqrt [3]{b} d+(-1)^{2/3} \sqrt [3]{a} e}\right )}{x} \, dx,x,d+e x\right )}{d^3} \\ & = -\frac {\sqrt {3} b^{2/3} p \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{2 a^{2/3} d}+\frac {\sqrt {3} \sqrt [3]{b} e p \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{a} d^2}+\frac {b^{2/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 a^{2/3} d}+\frac {\sqrt [3]{b} e p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a} d^2}+\frac {e^2 p \log \left (-\frac {e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (-\frac {e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right ) \log (d+e x)}{d^3}-\frac {b^{2/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 a^{2/3} d}-\frac {\sqrt [3]{b} e p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{a} d^2}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{2 d x^2}+\frac {e \log \left (c \left (a+b x^3\right )^p\right )}{d^2 x}+\frac {e^2 \log \left (-\frac {b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 d^3}-\frac {e^2 \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{d^3}+\frac {e^2 p \text {Li}_2\left (\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{d^3}+\frac {e^2 p \text {Li}_2\left (\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{d^3}+\frac {e^2 p \text {Li}_2\left (\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{d^3}+\frac {e^2 p \text {Li}_2\left (1+\frac {b x^3}{a}\right )}{3 d^3} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.25 (sec) , antiderivative size = 542, normalized size of antiderivative = 0.80 \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^3 (d+e x)} \, dx=\frac {-\frac {6 \sqrt {3} b^{2/3} d^2 p \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{2/3}}-\frac {18 b d e p x^2 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},-\frac {b x^3}{a}\right )}{a}+\frac {6 b^{2/3} d^2 p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{2/3}}+12 e^2 p \log \left (\frac {e \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right ) \log (d+e x)+12 e^2 p \log \left (\frac {e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} d+\sqrt [3]{a} e}\right ) \log (d+e x)+12 e^2 p \log \left (\frac {e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} d+(-1)^{2/3} \sqrt [3]{a} e}\right ) \log (d+e x)-\frac {3 b^{2/3} d^2 p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{2/3}}-\frac {6 d^2 \log \left (c \left (a+b x^3\right )^p\right )}{x^2}+\frac {12 d e \log \left (c \left (a+b x^3\right )^p\right )}{x}+4 e^2 \log \left (-\frac {b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )-12 e^2 \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )+12 e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )+12 e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )+12 e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )+4 e^2 p \operatorname {PolyLog}\left (2,1+\frac {b x^3}{a}\right )}{12 d^3} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.92 (sec) , antiderivative size = 421, normalized size of antiderivative = 0.62
method | result | size |
parts | \(-\frac {e^{2} \ln \left (e x +d \right ) \ln \left (c \left (b \,x^{3}+a \right )^{p}\right )}{d^{3}}-\frac {\ln \left (c \left (b \,x^{3}+a \right )^{p}\right )}{2 d \,x^{2}}+\frac {\ln \left (c \left (b \,x^{3}+a \right )^{p}\right ) e^{2} \ln \left (x \right )}{d^{3}}+\frac {e \ln \left (c \left (b \,x^{3}+a \right )^{p}\right )}{d^{2} x}-\frac {3 p b \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 d 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 d 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 d b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {2 e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 d^{2} b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {e \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 d^{2} b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {2 e \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d^{2} b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {2 e^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b +a \right )}{\sum }\left (\ln \left (x \right ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )\right )\right )}{3 d^{3} b}-\frac {2 e^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b -3 b d \,\textit {\_Z}^{2}+3 b \,d^{2} \textit {\_Z} +a \,e^{3}-b \,d^{3}\right )}{\sum }\left (\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )\right )\right )}{3 d^{3} b}\right )}{2}\) | \(421\) |
risch | \(-\frac {\ln \left (\left (b \,x^{3}+a \right )^{p}\right ) e^{2} \ln \left (e x +d \right )}{d^{3}}-\frac {\ln \left (\left (b \,x^{3}+a \right )^{p}\right )}{2 d \,x^{2}}+\frac {\ln \left (\left (b \,x^{3}+a \right )^{p}\right ) e^{2} \ln \left (x \right )}{d^{3}}+\frac {\ln \left (\left (b \,x^{3}+a \right )^{p}\right ) e}{d^{2} x}+\frac {p \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{2 d \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {p \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{4 d \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {p \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{2 d \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {p e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{d^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {p e \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{2 d^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {p e \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{d^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {p \,e^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b +a \right )}{\sum }\left (\ln \left (x \right ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )\right )\right )}{d^{3}}+\frac {p \,e^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b -3 b d \,\textit {\_Z}^{2}+3 b \,d^{2} \textit {\_Z} +a \,e^{3}-b \,d^{3}\right )}{\sum }\left (\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )\right )\right )}{d^{3}}+\left (\frac {i \pi \,\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}}{2}-\frac {i \pi \,\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 )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (b \,x^{3}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (-\frac {e^{2} \ln \left (e x +d \right )}{d^{3}}-\frac {1}{2 d \,x^{2}}+\frac {e^{2} \ln \left (x \right )}{d^{3}}+\frac {e}{d^{2} x}\right )\) | \(548\) |
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\[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^3 (d+e x)} \, dx=\int { \frac {\log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{{\left (e x + d\right )} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^3 (d+e x)} \, dx=\text {Timed out} \]
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\[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^3 (d+e x)} \, dx=\int { \frac {\log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{{\left (e x + d\right )} x^{3}} \,d x } \]
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\[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^3 (d+e x)} \, dx=\int { \frac {\log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{{\left (e x + d\right )} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^3 (d+e x)} \, dx=\int \frac {\ln \left (c\,{\left (b\,x^3+a\right )}^p\right )}{x^3\,\left (d+e\,x\right )} \,d x \]
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