Optimal. Leaf size=510 \[ -\frac{e p \text{PolyLog}\left (2,\frac{b x^3}{a}+1\right )}{3 d^2}-\frac{e p \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{d^2}-\frac{e p \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (d+e x)}{\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d}\right )}{d^2}-\frac{e p \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{d^2}+\frac{\sqrt [3]{b} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{a} d}-\frac{e \log \left (-\frac{b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 d^2}+\frac{e \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{d^2}-\frac{\log \left (c \left (a+b x^3\right )^p\right )}{d x}-\frac{e p \log (d+e x) \log \left (-\frac{e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{d^2}-\frac{e p \log (d+e x) \log \left (-\frac{e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{d^2}-\frac{e p \log (d+e x) \log \left (\frac{\sqrt [3]{-1} e \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d}\right )}{d^2}-\frac{\sqrt [3]{b} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a} d}-\frac{\sqrt{3} \sqrt [3]{b} p \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt [3]{a} d} \]
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Rubi [A] time = 0.677497, antiderivative size = 510, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 16, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.696, Rules used = {2466, 2455, 292, 31, 634, 617, 204, 628, 2454, 2394, 2315, 2462, 260, 2416, 2393, 2391} \[ -\frac{e p \text{PolyLog}\left (2,\frac{b x^3}{a}+1\right )}{3 d^2}-\frac{e p \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{d^2}-\frac{e p \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (d+e x)}{\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d}\right )}{d^2}-\frac{e p \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{d^2}+\frac{\sqrt [3]{b} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{a} d}-\frac{e \log \left (-\frac{b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 d^2}+\frac{e \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{d^2}-\frac{\log \left (c \left (a+b x^3\right )^p\right )}{d x}-\frac{e p \log (d+e x) \log \left (-\frac{e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{d^2}-\frac{e p \log (d+e x) \log \left (-\frac{e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{d^2}-\frac{e p \log (d+e x) \log \left (\frac{\sqrt [3]{-1} e \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d}\right )}{d^2}-\frac{\sqrt [3]{b} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a} d}-\frac{\sqrt{3} \sqrt [3]{b} p \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt [3]{a} d} \]
Antiderivative was successfully verified.
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Rule 2466
Rule 2455
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
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+b x^3\right )^p\right )}{x^2 (d+e x)} \, dx &=\int \left (\frac{\log \left (c \left (a+b x^3\right )^p\right )}{d x^2}-\frac{e \log \left (c \left (a+b x^3\right )^p\right )}{d^2 x}+\frac{e^2 \log \left (c \left (a+b x^3\right )^p\right )}{d^2 (d+e x)}\right ) \, dx\\ &=\frac{\int \frac{\log \left (c \left (a+b x^3\right )^p\right )}{x^2} \, dx}{d}-\frac{e \int \frac{\log \left (c \left (a+b x^3\right )^p\right )}{x} \, dx}{d^2}+\frac{e^2 \int \frac{\log \left (c \left (a+b x^3\right )^p\right )}{d+e x} \, dx}{d^2}\\ &=-\frac{\log \left (c \left (a+b x^3\right )^p\right )}{d x}+\frac{e \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{d^2}-\frac{e \operatorname{Subst}\left (\int \frac{\log \left (c (a+b x)^p\right )}{x} \, dx,x,x^3\right )}{3 d^2}+\frac{(3 b p) \int \frac{x}{a+b x^3} \, dx}{d}-\frac{(3 b e p) \int \frac{x^2 \log (d+e x)}{a+b x^3} \, dx}{d^2}\\ &=-\frac{\log \left (c \left (a+b x^3\right )^p\right )}{d x}-\frac{e \log \left (-\frac{b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 d^2}+\frac{e \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{d^2}-\frac{\left (b^{2/3} p\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{\sqrt [3]{a} d}+\frac{\left (b^{2/3} 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}+\frac{(b e p) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{b x}{a}\right )}{a+b x} \, dx,x,x^3\right )}{3 d^2}-\frac{(3 b e p) \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^2}\\ &=-\frac{\sqrt [3]{b} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a} d}-\frac{\log \left (c \left (a+b x^3\right )^p\right )}{d x}-\frac{e \log \left (-\frac{b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 d^2}+\frac{e \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{d^2}-\frac{e p \text{Li}_2\left (1+\frac{b x^3}{a}\right )}{3 d^2}+\frac{\left (\sqrt [3]{b} 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}+\frac{\left (3 b^{2/3} p\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 d}-\frac{\left (\sqrt [3]{b} e p\right ) \int \frac{\log (d+e x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{d^2}-\frac{\left (\sqrt [3]{b} e p\right ) \int \frac{\log (d+e x)}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{d^2}-\frac{\left (\sqrt [3]{b} e p\right ) \int \frac{\log (d+e x)}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{d^2}\\ &=-\frac{\sqrt [3]{b} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a} d}-\frac{e 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^2}-\frac{e 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^2}-\frac{e 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^2}+\frac{\sqrt [3]{b} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{a} d}-\frac{\log \left (c \left (a+b x^3\right )^p\right )}{d x}-\frac{e \log \left (-\frac{b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 d^2}+\frac{e \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{d^2}-\frac{e p \text{Li}_2\left (1+\frac{b x^3}{a}\right )}{3 d^2}+\frac{\left (3 \sqrt [3]{b} p\right ) \operatorname{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}+\frac{\left (e^2 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^2}+\frac{\left (e^2 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^2}+\frac{\left (e^2 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^2}\\ &=-\frac{\sqrt{3} \sqrt [3]{b} p \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt [3]{a} d}-\frac{\sqrt [3]{b} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a} d}-\frac{e 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^2}-\frac{e 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^2}-\frac{e 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^2}+\frac{\sqrt [3]{b} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{a} d}-\frac{\log \left (c \left (a+b x^3\right )^p\right )}{d x}-\frac{e \log \left (-\frac{b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 d^2}+\frac{e \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{d^2}-\frac{e p \text{Li}_2\left (1+\frac{b x^3}{a}\right )}{3 d^2}+\frac{(e p) \operatorname{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^2}+\frac{(e p) \operatorname{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^2}+\frac{(e p) \operatorname{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^2}\\ &=-\frac{\sqrt{3} \sqrt [3]{b} p \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt [3]{a} d}-\frac{\sqrt [3]{b} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a} d}-\frac{e 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^2}-\frac{e 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^2}-\frac{e 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^2}+\frac{\sqrt [3]{b} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{a} d}-\frac{\log \left (c \left (a+b x^3\right )^p\right )}{d x}-\frac{e \log \left (-\frac{b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 d^2}+\frac{e \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{d^2}-\frac{e p \text{Li}_2\left (\frac{\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{d^2}-\frac{e 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^2}-\frac{e 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^2}-\frac{e p \text{Li}_2\left (1+\frac{b x^3}{a}\right )}{3 d^2}\\ \end{align*}
Mathematica [C] time = 0.0606668, size = 424, normalized size = 0.83 \[ -\frac{e p \text{PolyLog}\left (2,\frac{a+b x^3}{a}\right )}{3 d^2}-\frac{e p \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{d^2}-\frac{e p \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (d+e x)}{\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d}\right )}{d^2}-\frac{e p \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{d^2}-\frac{e \log \left (-\frac{b x^3}{a}\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 d^2}+\frac{e \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{d^2}-\frac{\log \left (c \left (a+b x^3\right )^p\right )}{d x}-\frac{e p \log (d+e x) \log \left (-\frac{e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{d^2}-\frac{e p \log (d+e x) \log \left (-\frac{(-1)^{2/3} e \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{d^2}-\frac{e p \log (d+e x) \log \left (\frac{\sqrt [3]{-1} e \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d}\right )}{d^2}+\frac{3 b p x^2 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};-\frac{b x^3}{a}\right )}{2 a d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.592, size = 732, normalized size = 1.4 \begin{align*} \text{result too large to display} \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 ({\left (b x^{3} + a\right )}^{p} c\right )}{e x^{3} + d x^{2}}, 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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{{\left (e x + d\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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