Optimal. Leaf size=737 \[ -\frac{e^2 p \text{PolyLog}\left (2,\frac{b}{a x^3}+1\right )}{3 d^3}+\frac{e^2 p \text{PolyLog}\left (2,\frac{\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{d^3}+\frac{e^2 p \text{PolyLog}\left (2,\frac{\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{d^3}+\frac{e^2 p \text{PolyLog}\left (2,\frac{\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{d^3}-\frac{3 e^2 p \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{d^3}-\frac{\sqrt [3]{a} e p \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{2 \sqrt [3]{b} d^2}-\frac{a^{2/3} p \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{4 b^{2/3} d}+\frac{a^{2/3} p \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{2 b^{2/3} d}-\frac{\sqrt{3} a^{2/3} p \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{2 b^{2/3} d}-\frac{e^2 \log \left (-\frac{b}{a x^3}\right ) \log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{3 d^3}-\frac{e^2 \log (d+e x) \log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{d^3}+\frac{e \log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{d^2 x}-\frac{\log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{2 d x^2}+\frac{e^2 p \log (d+e x) \log \left (-\frac{e \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{d^3}+\frac{e^2 p \log (d+e x) \log \left (-\frac{e \left (\sqrt [3]{a} x+(-1)^{2/3} \sqrt [3]{b}\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{d^3}+\frac{e^2 p \log (d+e x) \log \left (\frac{\sqrt [3]{-1} e \left ((-1)^{2/3} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{d^3}+\frac{\sqrt [3]{a} e p \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{b} d^2}+\frac{\sqrt{3} \sqrt [3]{a} e p \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt [3]{b} d^2}-\frac{3 e^2 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{d^3}-\frac{3 e p}{d^2 x}+\frac{3 p}{4 d x^2} \]
[Out]
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Rubi [A] time = 0.795149, antiderivative size = 737, normalized size of antiderivative = 1., number of steps used = 39, number of rules used = 19, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.826, Rules used = {2466, 2455, 263, 325, 200, 31, 634, 617, 204, 628, 292, 2454, 2394, 2315, 2462, 260, 2416, 2393, 2391} \[ -\frac{e^2 p \text{PolyLog}\left (2,\frac{b}{a x^3}+1\right )}{3 d^3}+\frac{e^2 p \text{PolyLog}\left (2,\frac{\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{d^3}+\frac{e^2 p \text{PolyLog}\left (2,\frac{\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{d^3}+\frac{e^2 p \text{PolyLog}\left (2,\frac{\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{d^3}-\frac{3 e^2 p \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{d^3}-\frac{\sqrt [3]{a} e p \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{2 \sqrt [3]{b} d^2}-\frac{a^{2/3} p \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{4 b^{2/3} d}+\frac{a^{2/3} p \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{2 b^{2/3} d}-\frac{\sqrt{3} a^{2/3} p \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{2 b^{2/3} d}-\frac{e^2 \log \left (-\frac{b}{a x^3}\right ) \log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{3 d^3}-\frac{e^2 \log (d+e x) \log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{d^3}+\frac{e \log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{d^2 x}-\frac{\log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{2 d x^2}+\frac{e^2 p \log (d+e x) \log \left (-\frac{e \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{d^3}+\frac{e^2 p \log (d+e x) \log \left (-\frac{e \left (\sqrt [3]{a} x+(-1)^{2/3} \sqrt [3]{b}\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{d^3}+\frac{e^2 p \log (d+e x) \log \left (\frac{\sqrt [3]{-1} e \left ((-1)^{2/3} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{d^3}+\frac{\sqrt [3]{a} e p \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{b} d^2}+\frac{\sqrt{3} \sqrt [3]{a} e p \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt [3]{b} d^2}-\frac{3 e^2 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{d^3}-\frac{3 e p}{d^2 x}+\frac{3 p}{4 d x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2466
Rule 2455
Rule 263
Rule 325
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rule 292
Rule 2454
Rule 2394
Rule 2315
Rule 2462
Rule 260
Rule 2416
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{\log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{x^3 (d+e x)} \, dx &=\int \left (\frac{\log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{d x^3}-\frac{e \log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{d^2 x^2}+\frac{e^2 \log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{d^3 x}-\frac{e^3 \log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{d^3 (d+e x)}\right ) \, dx\\ &=\frac{\int \frac{\log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{x^3} \, dx}{d}-\frac{e \int \frac{\log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{x^2} \, dx}{d^2}+\frac{e^2 \int \frac{\log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{x} \, dx}{d^3}-\frac{e^3 \int \frac{\log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{d+e x} \, dx}{d^3}\\ &=-\frac{\log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{2 d x^2}+\frac{e \log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{d^2 x}-\frac{e^2 \log \left (c \left (a+\frac{b}{x^3}\right )^p\right ) \log (d+e x)}{d^3}-\frac{e^2 \operatorname{Subst}\left (\int \frac{\log \left (c (a+b x)^p\right )}{x} \, dx,x,\frac{1}{x^3}\right )}{3 d^3}-\frac{(3 b p) \int \frac{1}{\left (a+\frac{b}{x^3}\right ) x^6} \, dx}{2 d}+\frac{(3 b e p) \int \frac{1}{\left (a+\frac{b}{x^3}\right ) x^5} \, dx}{d^2}-\frac{\left (3 b e^2 p\right ) \int \frac{\log (d+e x)}{\left (a+\frac{b}{x^3}\right ) x^4} \, dx}{d^3}\\ &=-\frac{\log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{2 d x^2}+\frac{e \log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{d^2 x}-\frac{e^2 \log \left (c \left (a+\frac{b}{x^3}\right )^p\right ) \log \left (-\frac{b}{a x^3}\right )}{3 d^3}-\frac{e^2 \log \left (c \left (a+\frac{b}{x^3}\right )^p\right ) \log (d+e x)}{d^3}-\frac{(3 b p) \int \frac{1}{x^3 \left (b+a x^3\right )} \, dx}{2 d}+\frac{(3 b e p) \int \frac{1}{x^2 \left (b+a x^3\right )} \, dx}{d^2}+\frac{\left (b e^2 p\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{b x}{a}\right )}{a+b x} \, dx,x,\frac{1}{x^3}\right )}{3 d^3}-\frac{\left (3 b e^2 p\right ) \int \left (\frac{\log (d+e x)}{b x}-\frac{a x^2 \log (d+e x)}{b \left (b+a x^3\right )}\right ) \, dx}{d^3}\\ &=\frac{3 p}{4 d x^2}-\frac{3 e p}{d^2 x}-\frac{\log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{2 d x^2}+\frac{e \log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{d^2 x}-\frac{e^2 \log \left (c \left (a+\frac{b}{x^3}\right )^p\right ) \log \left (-\frac{b}{a x^3}\right )}{3 d^3}-\frac{e^2 \log \left (c \left (a+\frac{b}{x^3}\right )^p\right ) \log (d+e x)}{d^3}-\frac{e^2 p \text{Li}_2\left (1+\frac{b}{a x^3}\right )}{3 d^3}+\frac{(3 a p) \int \frac{1}{b+a x^3} \, dx}{2 d}-\frac{(3 a e p) \int \frac{x}{b+a x^3} \, dx}{d^2}-\frac{\left (3 e^2 p\right ) \int \frac{\log (d+e x)}{x} \, dx}{d^3}+\frac{\left (3 a e^2 p\right ) \int \frac{x^2 \log (d+e x)}{b+a x^3} \, dx}{d^3}\\ &=\frac{3 p}{4 d x^2}-\frac{3 e p}{d^2 x}-\frac{\log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{2 d x^2}+\frac{e \log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{d^2 x}-\frac{e^2 \log \left (c \left (a+\frac{b}{x^3}\right )^p\right ) \log \left (-\frac{b}{a x^3}\right )}{3 d^3}-\frac{e^2 \log \left (c \left (a+\frac{b}{x^3}\right )^p\right ) \log (d+e x)}{d^3}-\frac{3 e^2 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{d^3}-\frac{e^2 p \text{Li}_2\left (1+\frac{b}{a x^3}\right )}{3 d^3}+\frac{(a p) \int \frac{1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{2 b^{2/3} d}+\frac{(a p) \int \frac{2 \sqrt [3]{b}-\sqrt [3]{a} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{2 b^{2/3} d}+\frac{\left (a^{2/3} e p\right ) \int \frac{1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{\sqrt [3]{b} d^2}-\frac{\left (a^{2/3} e p\right ) \int \frac{\sqrt [3]{b}+\sqrt [3]{a} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{\sqrt [3]{b} d^2}+\frac{\left (3 a e^2 p\right ) \int \left (\frac{\log (d+e x)}{3 a^{2/3} \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}+\frac{\log (d+e x)}{3 a^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{b}+\sqrt [3]{a} x\right )}+\frac{\log (d+e x)}{3 a^{2/3} \left ((-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x\right )}\right ) \, dx}{d^3}+\frac{\left (3 e^3 p\right ) \int \frac{\log \left (-\frac{e x}{d}\right )}{d+e x} \, dx}{d^3}\\ &=\frac{3 p}{4 d x^2}-\frac{3 e p}{d^2 x}-\frac{\log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{2 d x^2}+\frac{e \log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{d^2 x}-\frac{e^2 \log \left (c \left (a+\frac{b}{x^3}\right )^p\right ) \log \left (-\frac{b}{a x^3}\right )}{3 d^3}+\frac{a^{2/3} p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{2 b^{2/3} d}+\frac{\sqrt [3]{a} e p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{b} d^2}-\frac{e^2 \log \left (c \left (a+\frac{b}{x^3}\right )^p\right ) \log (d+e x)}{d^3}-\frac{3 e^2 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{d^3}-\frac{e^2 p \text{Li}_2\left (1+\frac{b}{a x^3}\right )}{3 d^3}-\frac{3 e^2 p \text{Li}_2\left (1+\frac{e x}{d}\right )}{d^3}-\frac{\left (a^{2/3} p\right ) \int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 a^{2/3} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{4 b^{2/3} d}+\frac{(3 a p) \int \frac{1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{4 \sqrt [3]{b} d}-\frac{\left (3 a^{2/3} e p\right ) \int \frac{1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{2 d^2}-\frac{\left (\sqrt [3]{a} e p\right ) \int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 a^{2/3} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{2 \sqrt [3]{b} d^2}+\frac{\left (\sqrt [3]{a} e^2 p\right ) \int \frac{\log (d+e x)}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{d^3}+\frac{\left (\sqrt [3]{a} e^2 p\right ) \int \frac{\log (d+e x)}{-\sqrt [3]{-1} \sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{d^3}+\frac{\left (\sqrt [3]{a} e^2 p\right ) \int \frac{\log (d+e x)}{(-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{d^3}\\ &=\frac{3 p}{4 d x^2}-\frac{3 e p}{d^2 x}-\frac{\log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{2 d x^2}+\frac{e \log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{d^2 x}-\frac{e^2 \log \left (c \left (a+\frac{b}{x^3}\right )^p\right ) \log \left (-\frac{b}{a x^3}\right )}{3 d^3}+\frac{a^{2/3} p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{2 b^{2/3} d}+\frac{\sqrt [3]{a} e p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{b} d^2}-\frac{e^2 \log \left (c \left (a+\frac{b}{x^3}\right )^p\right ) \log (d+e x)}{d^3}-\frac{3 e^2 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{d^3}+\frac{e^2 p \log \left (-\frac{e \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right ) \log (d+e x)}{d^3}+\frac{e^2 p \log \left (-\frac{e \left ((-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right ) \log (d+e x)}{d^3}+\frac{e^2 p \log \left (\frac{\sqrt [3]{-1} e \left (\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right ) \log (d+e x)}{d^3}-\frac{a^{2/3} p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{4 b^{2/3} d}-\frac{\sqrt [3]{a} e p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{2 \sqrt [3]{b} d^2}-\frac{e^2 p \text{Li}_2\left (1+\frac{b}{a x^3}\right )}{3 d^3}-\frac{3 e^2 p \text{Li}_2\left (1+\frac{e x}{d}\right )}{d^3}+\frac{\left (3 a^{2/3} p\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{2 b^{2/3} d}-\frac{\left (3 \sqrt [3]{a} e p\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{\sqrt [3]{b} d^2}-\frac{\left (e^3 p\right ) \int \frac{\log \left (\frac{e \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{-\sqrt [3]{a} d+\sqrt [3]{b} 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]{b}+\sqrt [3]{a} x\right )}{-\sqrt [3]{a} d-\sqrt [3]{-1} \sqrt [3]{b} 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]{b}+\sqrt [3]{a} x\right )}{-\sqrt [3]{a} d+(-1)^{2/3} \sqrt [3]{b} e}\right )}{d+e x} \, dx}{d^3}\\ &=\frac{3 p}{4 d x^2}-\frac{3 e p}{d^2 x}-\frac{\sqrt{3} a^{2/3} p \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{2 b^{2/3} d}+\frac{\sqrt{3} \sqrt [3]{a} e p \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt [3]{b} d^2}-\frac{\log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{2 d x^2}+\frac{e \log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{d^2 x}-\frac{e^2 \log \left (c \left (a+\frac{b}{x^3}\right )^p\right ) \log \left (-\frac{b}{a x^3}\right )}{3 d^3}+\frac{a^{2/3} p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{2 b^{2/3} d}+\frac{\sqrt [3]{a} e p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{b} d^2}-\frac{e^2 \log \left (c \left (a+\frac{b}{x^3}\right )^p\right ) \log (d+e x)}{d^3}-\frac{3 e^2 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{d^3}+\frac{e^2 p \log \left (-\frac{e \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right ) \log (d+e x)}{d^3}+\frac{e^2 p \log \left (-\frac{e \left ((-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right ) \log (d+e x)}{d^3}+\frac{e^2 p \log \left (\frac{\sqrt [3]{-1} e \left (\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right ) \log (d+e x)}{d^3}-\frac{a^{2/3} p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{4 b^{2/3} d}-\frac{\sqrt [3]{a} e p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{2 \sqrt [3]{b} d^2}-\frac{e^2 p \text{Li}_2\left (1+\frac{b}{a x^3}\right )}{3 d^3}-\frac{3 e^2 p \text{Li}_2\left (1+\frac{e x}{d}\right )}{d^3}-\frac{\left (e^2 p\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt [3]{a} x}{-\sqrt [3]{a} d+\sqrt [3]{b} e}\right )}{x} \, dx,x,d+e x\right )}{d^3}-\frac{\left (e^2 p\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt [3]{a} x}{-\sqrt [3]{a} d-\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{x} \, dx,x,d+e x\right )}{d^3}-\frac{\left (e^2 p\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt [3]{a} x}{-\sqrt [3]{a} d+(-1)^{2/3} \sqrt [3]{b} e}\right )}{x} \, dx,x,d+e x\right )}{d^3}\\ &=\frac{3 p}{4 d x^2}-\frac{3 e p}{d^2 x}-\frac{\sqrt{3} a^{2/3} p \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{2 b^{2/3} d}+\frac{\sqrt{3} \sqrt [3]{a} e p \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt [3]{b} d^2}-\frac{\log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{2 d x^2}+\frac{e \log \left (c \left (a+\frac{b}{x^3}\right )^p\right )}{d^2 x}-\frac{e^2 \log \left (c \left (a+\frac{b}{x^3}\right )^p\right ) \log \left (-\frac{b}{a x^3}\right )}{3 d^3}+\frac{a^{2/3} p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{2 b^{2/3} d}+\frac{\sqrt [3]{a} e p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{b} d^2}-\frac{e^2 \log \left (c \left (a+\frac{b}{x^3}\right )^p\right ) \log (d+e x)}{d^3}-\frac{3 e^2 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{d^3}+\frac{e^2 p \log \left (-\frac{e \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right ) \log (d+e x)}{d^3}+\frac{e^2 p \log \left (-\frac{e \left ((-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right ) \log (d+e x)}{d^3}+\frac{e^2 p \log \left (\frac{\sqrt [3]{-1} e \left (\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right ) \log (d+e x)}{d^3}-\frac{a^{2/3} p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{4 b^{2/3} d}-\frac{\sqrt [3]{a} e p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{2 \sqrt [3]{b} d^2}-\frac{e^2 p \text{Li}_2\left (1+\frac{b}{a x^3}\right )}{3 d^3}+\frac{e^2 p \text{Li}_2\left (\frac{\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{d^3}+\frac{e^2 p \text{Li}_2\left (\frac{\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{d^3}+\frac{e^2 p \text{Li}_2\left (\frac{\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{d^3}-\frac{3 e^2 p \text{Li}_2\left (1+\frac{e x}{d}\right )}{d^3}\\ \end{align*}
Mathematica [C] time = 0.274554, size = 520, normalized size = 0.71 \[ \frac{-10 a x^3 \left (-6 e^2 p x^2 \text{PolyLog}\left (2,\frac{\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )-6 e^2 p x^2 \text{PolyLog}\left (2,\frac{\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )-6 e^2 p x^2 \text{PolyLog}\left (2,\frac{\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )+2 e^2 p x^2 \text{PolyLog}\left (2,\frac{b}{a x^3}+1\right )+18 e^2 p x^2 \text{PolyLog}\left (2,\frac{e x}{d}+1\right )+3 d^2 \log \left (c \left (a+\frac{b}{x^3}\right )^p\right )+6 e^2 x^2 \log (d+e x) \log \left (c \left (a+\frac{b}{x^3}\right )^p\right )-6 d e x \log \left (c \left (a+\frac{b}{x^3}\right )^p\right )+2 e^2 x^2 \log \left (-\frac{b}{a x^3}\right ) \log \left (c \left (a+\frac{b}{x^3}\right )^p\right )-6 e^2 p x^2 \log (d+e x) \log \left (\frac{e \left (\sqrt [3]{-1} \sqrt [3]{b}-\sqrt [3]{a} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )-6 e^2 p x^2 \log (d+e x) \log \left (\frac{e \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{b} e-\sqrt [3]{a} d}\right )-6 e^2 p x^2 \log (d+e x) \log \left (\frac{e \left (\sqrt [3]{a} x+(-1)^{2/3} \sqrt [3]{b}\right )}{(-1)^{2/3} \sqrt [3]{b} e-\sqrt [3]{a} d}\right )+18 e^2 p x^2 \log \left (-\frac{e x}{d}\right ) \log (d+e x)\right )+18 b d^2 p \, _2F_1\left (1,\frac{5}{3};\frac{8}{3};-\frac{b}{a x^3}\right )-45 b d e p x \, _2F_1\left (1,\frac{4}{3};\frac{7}{3};-\frac{b}{a x^3}\right )}{60 a d^3 x^5} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.704, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3} \left ( ex+d \right ) }\ln \left ( c \left ( a+{\frac{b}{{x}^{3}}} \right ) ^{p} \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (c \left (\frac{a x^{3} + b}{x^{3}}\right )^{p}\right )}{e x^{4} + d x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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