Optimal. Leaf size=1198 \[ \frac {2 b^3 \text {Li}_3\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^3}{3 d^{5/3} \sqrt [3]{e}}-\frac {6 \sqrt [3]{-1} b^3 \text {Li}_3\left (\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^3}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}-\frac {2 \sqrt [3]{-1} b^3 \text {Li}_3\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^3}{3 d^{5/3} \sqrt [3]{e}}+\frac {4 b^3 \text {Li}_4\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^3}{3 d^{5/3} \sqrt [3]{e}}-\frac {12 i \sqrt {3} b^3 \text {Li}_4\left (\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^3}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac {12 b^3 \text {Li}_4\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^3}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}-\frac {2 b^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^2}{3 d^{5/3} \sqrt [3]{e}}+\frac {6 \sqrt [3]{-1} b^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^2}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}+\frac {2 \sqrt [3]{-1} b^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^2}{3 d^{5/3} \sqrt [3]{e}}-\frac {4 b^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^2}{3 d^{5/3} \sqrt [3]{e}}+\frac {12 i \sqrt {3} b^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^2}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}-\frac {12 b^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^2}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}-\frac {b \left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}+1\right ) n}{3 d^{5/3} \sqrt [3]{e}}+\frac {3 \sqrt [3]{-1} b \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}+\frac {\sqrt [3]{-1} b \left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}+1\right ) n}{3 d^{5/3} \sqrt [3]{e}}+\frac {2 b \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n}{3 d^{5/3} \sqrt [3]{e}}-\frac {6 i \sqrt {3} b \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac {6 b \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n}{\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 )^3}{9 d^{5/3} \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}-\frac {\sqrt [3]{-1} x \left (a+b \log \left (c x^n\right )\right )^3}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \left (\sqrt [3]{e} x+(-1)^{2/3} \sqrt [3]{d}\right )}+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{9 d^{5/3} \left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\right )}+\frac {2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}+1\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {2 i \sqrt {3} \left (a+b \log \left (c x^n\right )\right )^3 \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 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}+1\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}} \]
[Out]
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Rubi [A] time = 1.43, antiderivative size = 1198, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2330, 2318, 2317, 2374, 6589, 2383} \[ \text {result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2317
Rule 2318
Rule 2330
Rule 2374
Rule 2383
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{\left (d+e x^3\right )^2} \, dx &=\int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^3}{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 )^3}{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 )^3}{\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 )^3}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )}+\frac {\left (a+b \log \left (c x^n\right )\right )^3}{\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 )^3}{\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 {\left (a+b \log \left (c x^n\right )\right )^3}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{9 d^{5/3}}+\frac {2 \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{\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 {\left (a+b \log \left (c x^n\right )\right )^3}{-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x} \, dx}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3}}+\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{\left (\sqrt [3]{d}+\sqrt [3]{e} x\right )^2} \, dx}{9 d^{4/3}}+\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{\left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x\right )^2} \, dx}{9 d^{4/3}}+\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{\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 )^3}{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 )^3}{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 )^3}{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 )^3 \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 )^3 \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 )^3 \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 {\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{3 d^{5/3}}+\frac {(b n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x} \, dx}{3 d^{5/3}}-\frac {(b n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x} \, dx}{3 d^{5/3}}-\frac {(2 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{3 d^{5/3} \sqrt [3]{e}}+\frac {\left (2 \sqrt [3]{-1} b n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{3 d^{5/3} \sqrt [3]{e}}+\frac {\left (6 i \sqrt {3} b n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \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 )^3}{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 )^3}{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 )^3}{9 d^{5/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}-\frac {b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}+\frac {2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {(-1)^{2/3} b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {2 i \sqrt {3} \left (a+b \log \left (c x^n\right )\right )^3 \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 {\sqrt [3]{-1} b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {2 \sqrt [3]{-1} \left (a+b \log \left (c x^n\right )\right )^3 \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 \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {6 i \sqrt {3} b n \left (a+b \log \left (c x^n\right )\right )^2 \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 \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}+\frac {\left (2 b^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{3 d^{5/3} \sqrt [3]{e}}-\frac {\left (4 b^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{3 d^{5/3} \sqrt [3]{e}}-\frac {\left (2 \sqrt [3]{-1} b^2 n^2\right ) \int \frac {\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 )}{x} \, dx}{3 d^{5/3} \sqrt [3]{e}}+\frac {\left (4 \sqrt [3]{-1} b^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{3 d^{5/3} \sqrt [3]{e}}+\frac {\left (2 (-1)^{2/3} b^2 n^2\right ) \int \frac {\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 )}{x} \, dx}{3 d^{5/3} \sqrt [3]{e}}+\frac {\left (12 i \sqrt {3} b^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\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 )^3}{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 )^3}{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 )^3}{9 d^{5/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}-\frac {b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}+\frac {2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {(-1)^{2/3} b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {2 i \sqrt {3} \left (a+b \log \left (c x^n\right )\right )^3 \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 {\sqrt [3]{-1} b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {2 \sqrt [3]{-1} \left (a+b \log \left (c x^n\right )\right )^3 \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^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {2 (-1)^{2/3} b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {6 i \sqrt {3} b n \left (a+b \log \left (c x^n\right )\right )^2 \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^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {2 \sqrt [3]{-1} b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {4 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}+\frac {12 i \sqrt {3} b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\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 {4 \sqrt [3]{-1} b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}+\frac {\left (2 b^3 n^3\right ) \int \frac {\text {Li}_2\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{3 d^{5/3} \sqrt [3]{e}}+\frac {\left (4 b^3 n^3\right ) \int \frac {\text {Li}_3\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{3 d^{5/3} \sqrt [3]{e}}-\frac {\left (2 \sqrt [3]{-1} b^3 n^3\right ) \int \frac {\text {Li}_2\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{3 d^{5/3} \sqrt [3]{e}}-\frac {\left (4 \sqrt [3]{-1} b^3 n^3\right ) \int \frac {\text {Li}_3\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{3 d^{5/3} \sqrt [3]{e}}+\frac {\left (2 (-1)^{2/3} b^3 n^3\right ) \int \frac {\text {Li}_2\left (\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{3 d^{5/3} \sqrt [3]{e}}-\frac {\left (12 i \sqrt {3} b^3 n^3\right ) \int \frac {\text {Li}_3\left (\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 )^3}{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 )^3}{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 )^3}{9 d^{5/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}-\frac {b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}+\frac {2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {(-1)^{2/3} b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {2 i \sqrt {3} \left (a+b \log \left (c x^n\right )\right )^3 \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 {\sqrt [3]{-1} b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {2 \sqrt [3]{-1} \left (a+b \log \left (c x^n\right )\right )^3 \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^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {2 (-1)^{2/3} b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {6 i \sqrt {3} b n \left (a+b \log \left (c x^n\right )\right )^2 \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^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {2 \sqrt [3]{-1} b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}+\frac {2 b^3 n^3 \text {Li}_3\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {4 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}+\frac {2 (-1)^{2/3} b^3 n^3 \text {Li}_3\left (\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}+\frac {12 i \sqrt {3} b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\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^3 n^3 \text {Li}_3\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}+\frac {4 \sqrt [3]{-1} b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}+\frac {4 b^3 n^3 \text {Li}_4\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}-\frac {12 i \sqrt {3} b^3 n^3 \text {Li}_4\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 {4 \sqrt [3]{-1} b^3 n^3 \text {Li}_4\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} \sqrt [3]{e}}\\ \end {align*}
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Mathematica [A] time = 7.80, size = 2215, normalized size = 1.85 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.39, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b \log \left (c x^{n}\right ) + a^{3}}{e^{2} x^{6} + 2 \, d e x^{3} + d^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3}}{{\left (e x^{3} + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 31.12, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{3}}{\left (e \,x^{3}+d \right )^{2}}\, dx \]
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 \[ \frac {1}{9} \, a^{3} {\left (\frac {3 \, x}{d e x^{3} + d^{2}} + \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {d}{e}\right )^{\frac {1}{3}}}\right )}{d e \left (\frac {d}{e}\right )^{\frac {2}{3}}} - \frac {\log \left (x^{2} - x \left (\frac {d}{e}\right )^{\frac {1}{3}} + \left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{d e \left (\frac {d}{e}\right )^{\frac {2}{3}}} + \frac {2 \, \log \left (x + \left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{d e \left (\frac {d}{e}\right )^{\frac {2}{3}}}\right )} + \int \frac {b^{3} \log \relax (c)^{3} + b^{3} \log \left (x^{n}\right )^{3} + 3 \, a b^{2} \log \relax (c)^{2} + 3 \, a^{2} b \log \relax (c) + 3 \, {\left (b^{3} \log \relax (c) + a b^{2}\right )} \log \left (x^{n}\right )^{2} + 3 \, {\left (b^{3} \log \relax (c)^{2} + 2 \, a b^{2} \log \relax (c) + a^{2} b\right )} \log \left (x^{n}\right )}{e^{2} x^{6} + 2 \, d e x^{3} + d^{2}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{{\left (e\,x^3+d\right )}^2} \,d x \]
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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