Optimal. Leaf size=488 \[ -\frac {\sqrt [3]{b} p \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{2 \sqrt [3]{a} e}-\frac {d \log (d+e x) \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^2}+\frac {x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e}+\frac {d p \text {Li}_2\left (\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{e^2}+\frac {d p \text {Li}_2\left (\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{e^2}+\frac {d p \text {Li}_2\left (\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{e^2}+\frac {d p \log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{e^2}+\frac {d p \log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{a} x+(-1)^{2/3} \sqrt [3]{b}\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{e^2}+\frac {d p \log (d+e x) \log \left (\frac {\sqrt [3]{-1} e \left ((-1)^{2/3} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{e^2}+\frac {\sqrt [3]{b} p \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{a} e}-\frac {\sqrt {3} \sqrt [3]{b} p \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt [3]{a} e}-\frac {3 d p \text {Li}_2\left (\frac {e x}{d}+1\right )}{e^2}-\frac {3 d p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^2} \]
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Rubi [A] time = 0.60, antiderivative size = 488, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 16, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.762, Rules used = {2466, 2448, 263, 200, 31, 634, 617, 204, 628, 2462, 260, 2416, 2394, 2315, 2393, 2391} \[ \frac {d p \text {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{e^2}+\frac {d p \text {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{e^2}+\frac {d p \text {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{e^2}-\frac {3 d p \text {PolyLog}\left (2,\frac {e x}{d}+1\right )}{e^2}-\frac {\sqrt [3]{b} p \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{2 \sqrt [3]{a} e}-\frac {d \log (d+e x) \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^2}+\frac {x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e}+\frac {d p \log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{e^2}+\frac {d p \log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{a} x+(-1)^{2/3} \sqrt [3]{b}\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{e^2}+\frac {d p \log (d+e x) \log \left (\frac {\sqrt [3]{-1} e \left ((-1)^{2/3} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{e^2}+\frac {\sqrt [3]{b} p \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{a} e}-\frac {\sqrt {3} \sqrt [3]{b} p \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt [3]{a} e}-\frac {3 d p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 260
Rule 263
Rule 617
Rule 628
Rule 634
Rule 2315
Rule 2391
Rule 2393
Rule 2394
Rule 2416
Rule 2448
Rule 2462
Rule 2466
Rubi steps
\begin {align*} \int \frac {x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d+e x} \, dx &=\int \left (\frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e}-\frac {d \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e (d+e x)}\right ) \, dx\\ &=\frac {\int \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \, dx}{e}-\frac {d \int \frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d+e x} \, dx}{e}\\ &=\frac {x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e}-\frac {d \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{e^2}-\frac {(3 b d p) \int \frac {\log (d+e x)}{\left (a+\frac {b}{x^3}\right ) x^4} \, dx}{e^2}+\frac {(3 b p) \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^3} \, dx}{e}\\ &=\frac {x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e}-\frac {d \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{e^2}-\frac {(3 b d p) \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}{e^2}+\frac {(3 b p) \int \frac {1}{b+a x^3} \, dx}{e}\\ &=\frac {x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e}-\frac {d \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{e^2}-\frac {(3 d p) \int \frac {\log (d+e x)}{x} \, dx}{e^2}+\frac {(3 a d p) \int \frac {x^2 \log (d+e x)}{b+a x^3} \, dx}{e^2}+\frac {\left (\sqrt [3]{b} p\right ) \int \frac {1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{e}+\frac {\left (\sqrt [3]{b} p\right ) \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}{e}\\ &=\frac {x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e}+\frac {\sqrt [3]{b} p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} e}-\frac {d \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{e^2}-\frac {3 d p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^2}+\frac {(3 a d p) \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}{e^2}-\frac {\left (\sqrt [3]{b} 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]{a} e}+\frac {\left (3 b^{2/3} p\right ) \int \frac {1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{2 e}+\frac {(3 d p) \int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx}{e}\\ &=\frac {x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e}+\frac {\sqrt [3]{b} p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} e}-\frac {d \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{e^2}-\frac {3 d p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^2}-\frac {\sqrt [3]{b} p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{2 \sqrt [3]{a} e}-\frac {3 d p \text {Li}_2\left (1+\frac {e x}{d}\right )}{e^2}+\frac {\left (\sqrt [3]{a} d p\right ) \int \frac {\log (d+e x)}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{e^2}+\frac {\left (\sqrt [3]{a} d p\right ) \int \frac {\log (d+e x)}{-\sqrt [3]{-1} \sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{e^2}+\frac {\left (\sqrt [3]{a} d p\right ) \int \frac {\log (d+e x)}{(-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{e^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]{a} x}{\sqrt [3]{b}}\right )}{\sqrt [3]{a} e}\\ &=-\frac {\sqrt {3} \sqrt [3]{b} p \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt [3]{a} e}+\frac {x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e}+\frac {\sqrt [3]{b} p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} e}-\frac {d \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{e^2}-\frac {3 d p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^2}+\frac {d 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)}{e^2}+\frac {d 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)}{e^2}+\frac {d 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)}{e^2}-\frac {\sqrt [3]{b} p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{2 \sqrt [3]{a} e}-\frac {3 d p \text {Li}_2\left (1+\frac {e x}{d}\right )}{e^2}-\frac {(d p) \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}{e}-\frac {(d p) \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}{e}-\frac {(d p) \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}{e}\\ &=-\frac {\sqrt {3} \sqrt [3]{b} p \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt [3]{a} e}+\frac {x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e}+\frac {\sqrt [3]{b} p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} e}-\frac {d \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{e^2}-\frac {3 d p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^2}+\frac {d 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)}{e^2}+\frac {d 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)}{e^2}+\frac {d 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)}{e^2}-\frac {\sqrt [3]{b} p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{2 \sqrt [3]{a} e}-\frac {3 d p \text {Li}_2\left (1+\frac {e x}{d}\right )}{e^2}-\frac {(d p) \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 )}{e^2}-\frac {(d p) \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 )}{e^2}-\frac {(d p) \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 )}{e^2}\\ &=-\frac {\sqrt {3} \sqrt [3]{b} p \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt [3]{a} e}+\frac {x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e}+\frac {\sqrt [3]{b} p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} e}-\frac {d \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{e^2}-\frac {3 d p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^2}+\frac {d 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)}{e^2}+\frac {d 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)}{e^2}+\frac {d 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)}{e^2}-\frac {\sqrt [3]{b} p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{2 \sqrt [3]{a} e}+\frac {d p \text {Li}_2\left (\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{e^2}+\frac {d p \text {Li}_2\left (\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{e^2}+\frac {d p \text {Li}_2\left (\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{e^2}-\frac {3 d p \text {Li}_2\left (1+\frac {e x}{d}\right )}{e^2}\\ \end {align*}
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Mathematica [C] time = 0.12, size = 403, normalized size = 0.83 \[ -\frac {d \log (d+e x) \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^2}+\frac {x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e}+\frac {d p \text {Li}_2\left (\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{e^2}+\frac {d p \text {Li}_2\left (\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{e^2}+\frac {d p \text {Li}_2\left (\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{e^2}+\frac {d p \log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{e^2}+\frac {d p \log (d+e x) \log \left (-\frac {(-1)^{2/3} e \left (\sqrt [3]{b}-\sqrt [3]{-1} \sqrt [3]{a} x\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{e^2}+\frac {d p \log (d+e x) \log \left (\frac {\sqrt [3]{-1} e \left ((-1)^{2/3} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{e^2}-\frac {3 b p \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {b}{a x^3}\right )}{2 a e x^2}-\frac {3 d p \text {Li}_2\left (\frac {d+e x}{d}\right )}{e^2}-\frac {3 d p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x \log \left (c \left (\frac {a x^{3} + b}{x^{3}}\right )^{p}\right )}{e x + d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \log \left ({\left (a + \frac {b}{x^{3}}\right )}^{p} c\right )}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.42, size = 0, normalized size = 0.00 \[ \int \frac {x \ln \left (c \left (a +\frac {b}{x^{3}}\right )^{p}\right )}{e x +d}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \log \left ({\left (a + \frac {b}{x^{3}}\right )}^{p} c\right )}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x\,\ln \left (c\,{\left (a+\frac {b}{x^3}\right )}^p\right )}{d+e\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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