Integrand size = 22, antiderivative size = 860 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^3\right )^2} \, dx=\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{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 )^2}{\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 )^2}{9 d^{5/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}-\frac {2 b n \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 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}+\frac {2 \sqrt [3]{-1} b n \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 )^4 d^{5/3} \sqrt [3]{e}}-\frac {2 i \sqrt {3} \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 )}{\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 ) \log \left (1+\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}+\frac {2 \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 )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}-\frac {2 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}+\frac {4 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}+\frac {2 \sqrt [3]{-1} b^2 n^2 \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}-\frac {4 i \sqrt {3} b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {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 \sqrt [3]{-1} b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}+\frac {4 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {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 {4 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}+\frac {4 i \sqrt {3} b^2 n^2 \operatorname {PolyLog}\left (3,\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 b^2 n^2 \operatorname {PolyLog}\left (3,-\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}} \]
[Out]
Time = 0.57 (sec) , antiderivative size = 860, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2367, 2355, 2354, 2438, 2421, 6724} \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^3\right )^2} \, dx=-\frac {2 b^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^2}{9 d^{5/3} \sqrt [3]{e}}+\frac {2 \sqrt [3]{-1} b^2 \operatorname {PolyLog}\left (2,\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 \operatorname {PolyLog}\left (2,-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^2}{9 d^{5/3} \sqrt [3]{e}}-\frac {4 b^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n^2}{9 d^{5/3} \sqrt [3]{e}}+\frac {4 i \sqrt {3} b^2 \operatorname {PolyLog}\left (3,\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 {4 b^2 \operatorname {PolyLog}\left (3,-\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 {2 b \left (a+b \log \left (c x^n\right )\right ) \log \left (\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}+1\right ) n}{9 d^{5/3} \sqrt [3]{e}}+\frac {2 \sqrt [3]{-1} b \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 ) n}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}+\frac {2 \sqrt [3]{-1} b \left (a+b \log \left (c x^n\right )\right ) \log \left (\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}+1\right ) n}{9 d^{5/3} \sqrt [3]{e}}+\frac {4 b \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right ) n}{9 d^{5/3} \sqrt [3]{e}}-\frac {4 i \sqrt {3} b \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,\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 {4 b \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\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 )^2}{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 )^2}{\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 )^2}{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 )^2 \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 )^2 \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 )^2 \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}} \]
[In]
[Out]
Rule 2354
Rule 2355
Rule 2367
Rule 2421
Rule 2438
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^2}{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 )^2}{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 )^2}{\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 )^2}{\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 )^2}{\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 )^2}{\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 )^2}{\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 )^2}{\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 )^2}{-\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 )^2}{\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 )^2}{\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 )^2}{\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 )^2}{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 )^2}{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 )^2}{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 )^2 \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 )^2 \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 )^2 \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) \int \frac {a+b \log \left (c x^n\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{9 d^{5/3}}+\frac {(2 b n) \int \frac {a+b \log \left (c x^n\right )}{-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{e} x} \, dx}{9 d^{5/3}}-\frac {(2 b n) \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 {(4 b n) \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}{9 d^{5/3} \sqrt [3]{e}}+\frac {\left (4 \sqrt [3]{-1} b n\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}{9 d^{5/3} \sqrt [3]{e}}+\frac {\left (4 i \sqrt {3} b n\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}{\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 )^2}{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 )^2}{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 )^2}{9 d^{5/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}-\frac {2 b n \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 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {2 (-1)^{2/3} b n \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 )}{9 d^{5/3} \sqrt [3]{e}}-\frac {2 i \sqrt {3} \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 )}{\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 ) \log \left (1+\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {2 \sqrt [3]{-1} \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 )}{9 d^{5/3} \sqrt [3]{e}}+\frac {4 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {4 i \sqrt {3} b n \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 )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}-\frac {4 \sqrt [3]{-1} b n \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 )}{9 d^{5/3} \sqrt [3]{e}}+\frac {\left (2 b^2 n^2\right ) \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 (4 b^2 n^2\right ) \int \frac {\text {Li}_2\left (-\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^2 n^2\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 (4 \sqrt [3]{-1} b^2 n^2\right ) \int \frac {\text {Li}_2\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{9 d^{5/3} \sqrt [3]{e}}+\frac {\left (2 (-1)^{2/3} b^2 n^2\right ) \int \frac {\log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{x} \, dx}{9 d^{5/3} \sqrt [3]{e}}+\frac {\left (4 i \sqrt {3} b^2 n^2\right ) \int \frac {\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 )^2}{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 )^2}{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 )^2}{9 d^{5/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}-\frac {2 b n \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 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {2 (-1)^{2/3} b n \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 )}{9 d^{5/3} \sqrt [3]{e}}-\frac {2 i \sqrt {3} \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 )}{\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 ) \log \left (1+\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {2 \sqrt [3]{-1} \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 )}{9 d^{5/3} \sqrt [3]{e}}-\frac {2 b^2 n^2 \text {Li}_2\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}+\frac {4 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {2 (-1)^{2/3} b^2 n^2 \text {Li}_2\left (\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {4 i \sqrt {3} b n \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 )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac {2 \sqrt [3]{-1} b^2 n^2 \text {Li}_2\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {4 \sqrt [3]{-1} b n \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 )}{9 d^{5/3} \sqrt [3]{e}}-\frac {4 b^2 n^2 \text {Li}_3\left (-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}+\frac {4 i \sqrt {3} b^2 n^2 \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 \text {Li}_3\left (-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}} \\ \end{align*}
Time = 2.72 (sec) , antiderivative size = 1180, normalized size of antiderivative = 1.37 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^3\right )^2} \, dx=\frac {\frac {3 d^{2/3} x \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^2}{d+e x^3}+\frac {2 \sqrt {3} \arctan \left (\frac {-\sqrt [3]{d}+2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right ) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^2}{\sqrt [3]{e}}+\frac {2 \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{e}}-\frac {\left (a-b n \log (x)+b \log \left (c x^n\right )\right )^2 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{\sqrt [3]{e}}+\frac {6 b n \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \left (\frac {\left (-1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1} \sqrt [3]{e} x \log (x)+\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 )\right )}{(-1)^{2/3} \sqrt [3]{d} \sqrt [3]{e}+e^{2/3} x}+\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 )+\frac {-(-1)^{2/3} \sqrt [3]{e} x \log (x)+\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 )}{-\sqrt [3]{-1} \sqrt [3]{d} \sqrt [3]{e}+e^{2/3} x}+\frac {2 \sqrt [3]{-1} \left (\log (x) \log \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )+\operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )\right )}{\sqrt [3]{e}}-\frac {2 \left (\log (x) \log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )\right )}{\sqrt [3]{e}}-\frac {2 \left (-1+\sqrt [3]{-1}\right ) \left (\log (x) \log \left (1+\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )+\operatorname {PolyLog}\left (2,-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )\right )}{\sqrt [3]{e}}\right )}{\left (1+\sqrt [3]{-1}\right )^2}+\frac {3 b^2 n^2 \left (\frac {\sqrt [3]{-1} \left (\log (x) \left (\frac {\sqrt [3]{e} x \log (x)}{\sqrt [3]{d}+\sqrt [3]{e} x}-2 \log \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )\right )-2 \operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )\right )}{\sqrt [3]{e}}-\frac {\left (-1+\sqrt [3]{-1}\right ) \sqrt [3]{d} \left (\log (x) \left (\left (-\frac {\sqrt [3]{-1}}{\sqrt [3]{d}}-\frac {1}{(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x}\right ) \log (x)+\frac {2 \sqrt [3]{-1} \log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\sqrt [3]{d}}\right )+\frac {2 \sqrt [3]{-1} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\sqrt [3]{d}}\right )}{\sqrt [3]{e}}+\frac {\log (x) \left ((-1)^{2/3} \sqrt [3]{e} x \log (x)-2 \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (1+\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )\right )-2 \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \operatorname {PolyLog}\left (2,-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\sqrt [3]{-1} \sqrt [3]{d} \sqrt [3]{e}-e^{2/3} x}+\frac {2 \sqrt [3]{-1} \left (\log ^2(x) \log \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )+2 \log (x) \operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )-2 \operatorname {PolyLog}\left (3,-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )\right )}{\sqrt [3]{e}}-\frac {2 \left (\log ^2(x) \log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )+2 \log (x) \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )-2 \operatorname {PolyLog}\left (3,\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )\right )}{\sqrt [3]{e}}-\frac {2 \left (-1+\sqrt [3]{-1}\right ) \left (\log ^2(x) \log \left (1+\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )+2 \log (x) \operatorname {PolyLog}\left (2,-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )-2 \operatorname {PolyLog}\left (3,-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )\right )}{\sqrt [3]{e}}\right )}{\left (1+\sqrt [3]{-1}\right )^2}}{9 d^{5/3}} \]
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\[\int \frac {{\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{\left (e \,x^{3}+d \right )^{2}}d x\]
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\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^3\right )^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x^{3} + d\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^3\right )^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^3\right )^2} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^3\right )^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x^{3} + d\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^3\right )^2} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{{\left (e\,x^3+d\right )}^2} \,d x \]
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