Optimal. Leaf size=520 \[ \frac {2 \log \left (\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {2 i \sqrt {3} \log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac {2 \log \left (\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}-\frac {\sqrt [3]{-1} x \left (a+b \log \left (c x^n\right )\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}+\frac {2 b n \text {Li}_2\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {2 i \sqrt {3} b n \text {Li}_2\left (\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac {2 b n \text {Li}_2\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}+\frac {\sqrt [3]{-1} b n \log \left (-(-1)^{2/3} \sqrt [3]{d}-\sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}-\frac {b n \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{9 d^{5/3} \sqrt [3]{e}}+\frac {\sqrt [3]{-1} b n \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{9 d^{5/3} \sqrt [3]{e}} \]
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Rubi [A] time = 0.46, antiderivative size = 520, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {199, 200, 31, 634, 617, 204, 628, 2330, 2314, 2317, 2391} \[ \frac {2 b n \text {PolyLog}\left (2,-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {2 i \sqrt {3} b n \text {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac {2 b n \text {PolyLog}\left (2,-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}+\frac {2 \log \left (\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {2 i \sqrt {3} \log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac {2 \log \left (\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}-\frac {\sqrt [3]{-1} x \left (a+b \log \left (c x^n\right )\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}+\frac {\sqrt [3]{-1} b n \log \left (-(-1)^{2/3} \sqrt [3]{d}-\sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}-\frac {b n \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{9 d^{5/3} \sqrt [3]{e}}+\frac {\sqrt [3]{-1} b n \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{9 d^{5/3} \sqrt [3]{e}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 199
Rule 200
Rule 204
Rule 617
Rule 628
Rule 634
Rule 2314
Rule 2317
Rule 2330
Rule 2391
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^3\right )^2} \, dx &=\int \left (\frac {a+b \log \left (c x^n\right )}{9 d^{4/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )^2}+\frac {2 \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {(-1)^{2/3} \left (a+b \log \left (c x^n\right )\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{4/3} \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )^2}-\frac {2 (-1)^{5/6} \sqrt {3} \left (a+b \log \left (c x^n\right )\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )}+\frac {a+b \log \left (c x^n\right )}{\left (-1+\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 d^{4/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )^2}+\frac {2 (-1)^{2/3} \left (a+b \log \left (c x^n\right )\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}\right ) \, dx\\ &=\frac {2 \int \frac {a+b \log \left (c x^n\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{9 d^{5/3}}+\frac {2 \int \frac {a+b \log \left (c x^n\right )}{\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x} \, dx}{9 d^{5/3}}-\frac {\left (2 (-1)^{5/6} \sqrt {3}\right ) \int \frac {a+b \log \left (c x^n\right )}{-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x} \, dx}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3}}+\frac {\int \frac {a+b \log \left (c x^n\right )}{\left (\sqrt [3]{d}+\sqrt [3]{e} x\right )^2} \, dx}{9 d^{4/3}}+\frac {\int \frac {a+b \log \left (c x^n\right )}{\left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )^2} \, dx}{9 d^{4/3}}+\frac {\int \frac {a+b \log \left (c x^n\right )}{\left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )^2} \, dx}{9 d^{4/3}}\\ &=\frac {x \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {(-1)^{2/3} x \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}+\frac {2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {2 i \sqrt {3} \left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}-\frac {2 \sqrt [3]{-1} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {(b n) \int \frac {1}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{9 d^{5/3}}+\frac {(b n) \int \frac {1}{-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x} \, dx}{9 d^{5/3}}-\frac {(b n) \int \frac {1}{\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x} \, dx}{9 d^{5/3}}-\frac {(2 b n) \int \frac {\log \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{9 d^{5/3} \sqrt [3]{e}}+\frac {\left (2 \sqrt [3]{-1} b n\right ) \int \frac {\log \left (1+\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{9 d^{5/3} \sqrt [3]{e}}+\frac {\left (2 i \sqrt {3} b n\right ) \int \frac {\log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}\\ &=\frac {x \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {(-1)^{2/3} x \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}-\frac {(-1)^{2/3} b n \log \left (-(-1)^{2/3} \sqrt [3]{d}-\sqrt [3]{e} x\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {b n \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{9 d^{5/3} \sqrt [3]{e}}+\frac {\sqrt [3]{-1} b n \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{9 d^{5/3} \sqrt [3]{e}}+\frac {2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {2 i \sqrt {3} \left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}-\frac {2 \sqrt [3]{-1} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}+\frac {2 b n \text {Li}_2\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {2 i \sqrt {3} b n \text {Li}_2\left (\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}-\frac {2 \sqrt [3]{-1} b n \text {Li}_2\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}\\ \end {align*}
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Mathematica [A] time = 1.87, size = 571, normalized size = 1.10 \[ \frac {-\frac {\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{\sqrt [3]{e}}+\frac {3 d^{2/3} x \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{d+e x^3}+\frac {2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{\sqrt [3]{e}}-\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt {3}}\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{\sqrt [3]{e}}+\frac {3 b n \left (\frac {\left (\sqrt [3]{-1}-1\right ) \left (\left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (-(-1)^{2/3} \sqrt [3]{d}-\sqrt [3]{e} x\right )+\sqrt [3]{-1} \sqrt [3]{e} x \log (x)\right )}{(-1)^{2/3} \sqrt [3]{d} \sqrt [3]{e}+e^{2/3} x}+\frac {\left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )-(-1)^{2/3} \sqrt [3]{e} x \log (x)}{e^{2/3} x-\sqrt [3]{-1} \sqrt [3]{d} \sqrt [3]{e}}+\frac {2 \sqrt [3]{-1} \left (\text {Li}_2\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )+\log (x) \log \left (\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}+1\right )\right )}{\sqrt [3]{e}}-\frac {2 \left (\text {Li}_2\left (\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )+\log (x) \log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )\right )}{\sqrt [3]{e}}-\frac {2 \left (\sqrt [3]{-1}-1\right ) \left (\text {Li}_2\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )+\log (x) \log \left (\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}+1\right )\right )}{\sqrt [3]{e}}+\sqrt [3]{-1} \left (\frac {x \log (x)}{\sqrt [3]{d}+\sqrt [3]{e} x}-\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{e}}\right )\right )}{\left (1+\sqrt [3]{-1}\right )^2}}{9 d^{5/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \log \left (c x^{n}\right ) + a}{e^{2} x^{6} + 2 \, d e x^{3} + d^{2}}, x\right ) \]
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 \[ \int \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{3} + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.37, size = 1388, normalized size = 2.67 \[ \text {result too large to display} \]
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 \[ \frac {1}{9} \, a {\left (\frac {3 \, x}{d e x^{3} + d^{2}} + \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {d}{e}\right )^{\frac {1}{3}}}\right )}{d e \left (\frac {d}{e}\right )^{\frac {2}{3}}} - \frac {\log \left (x^{2} - x \left (\frac {d}{e}\right )^{\frac {1}{3}} + \left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{d e \left (\frac {d}{e}\right )^{\frac {2}{3}}} + \frac {2 \, \log \left (x + \left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{d e \left (\frac {d}{e}\right )^{\frac {2}{3}}}\right )} + b \int \frac {\log \relax (c) + \log \left (x^{n}\right )}{e^{2} x^{6} + 2 \, d e x^{3} + d^{2}}\,{d x} \]
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 \[ \int \frac {a+b\,\ln \left (c\,x^n\right )}{{\left (e\,x^3+d\right )}^2} \,d x \]
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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