Integrand size = 20, antiderivative size = 520 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^3\right )^2} \, dx=\frac {x \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}-\frac {\sqrt [3]{-1} x \left (a+b \log \left (c x^n\right )\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}+\frac {\sqrt [3]{-1} b n \log \left (-(-1)^{2/3} \sqrt [3]{d}-\sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}-\frac {b n \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{9 d^{5/3} \sqrt [3]{e}}+\frac {\sqrt [3]{-1} b n \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{9 d^{5/3} \sqrt [3]{e}}+\frac {2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {2 i \sqrt {3} \left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac {2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}+\frac {2 b n \operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {2 i \sqrt {3} b n \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac {2 b n \operatorname {PolyLog}\left (2,-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}} \]
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Time = 0.31 (sec) , antiderivative size = 520, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {205, 206, 31, 648, 631, 210, 642, 2367, 2351, 2354, 2438} \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^3\right )^2} \, dx=\frac {2 \log \left (\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {2 i \sqrt {3} \log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac {2 \log \left (\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}-\frac {\sqrt [3]{-1} x \left (a+b \log \left (c x^n\right )\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{9 d^{5/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}+\frac {2 b n \operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{9 d^{5/3} \sqrt [3]{e}}-\frac {2 i \sqrt {3} b n \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^5 d^{5/3} \sqrt [3]{e}}+\frac {2 b n \operatorname {PolyLog}\left (2,-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}+\frac {\sqrt [3]{-1} b n \log \left (-(-1)^{2/3} \sqrt [3]{d}-\sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right )^4 d^{5/3} \sqrt [3]{e}}-\frac {b n \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{9 d^{5/3} \sqrt [3]{e}}+\frac {\sqrt [3]{-1} b n \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{9 d^{5/3} \sqrt [3]{e}} \]
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Rule 31
Rule 205
Rule 206
Rule 210
Rule 631
Rule 642
Rule 648
Rule 2351
Rule 2354
Rule 2367
Rule 2438
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.83 (sec) , antiderivative size = 571, normalized size of antiderivative = 1.10 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^3\right )^2} \, dx=\frac {\frac {3 d^{2/3} x \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{d+e x^3}-\frac {2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt {3}}\right ) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{\sqrt [3]{e}}+\frac {2 \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{e}}-\frac {\left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{\sqrt [3]{e}}+\frac {3 b n \left (\frac {\left (-1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1} \sqrt [3]{e} x \log (x)+\left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (-(-1)^{2/3} \sqrt [3]{d}-\sqrt [3]{e} x\right )\right )}{(-1)^{2/3} \sqrt [3]{d} \sqrt [3]{e}+e^{2/3} x}+\sqrt [3]{-1} \left (\frac {x \log (x)}{\sqrt [3]{d}+\sqrt [3]{e} x}-\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{e}}\right )+\frac {-(-1)^{2/3} \sqrt [3]{e} x \log (x)+\left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{-\sqrt [3]{-1} \sqrt [3]{d} \sqrt [3]{e}+e^{2/3} x}+\frac {2 \sqrt [3]{-1} \left (\log (x) \log \left (1+\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )+\operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}\right )\right )}{\sqrt [3]{e}}-\frac {2 \left (\log (x) \log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )\right )}{\sqrt [3]{e}}-\frac {2 \left (-1+\sqrt [3]{-1}\right ) \left (\log (x) \log \left (1+\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )+\operatorname {PolyLog}\left (2,-\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d}}\right )\right )}{\sqrt [3]{e}}\right )}{\left (1+\sqrt [3]{-1}\right )^2}}{9 d^{5/3}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.13 (sec) , antiderivative size = 598, normalized size of antiderivative = 1.15
method | result | size |
risch | \(\frac {b x \ln \left (x^{n}\right )}{3 d \left (e \,x^{3}+d \right )}-\frac {2 b \ln \left (x +\left (\frac {d}{e}\right )^{\frac {1}{3}}\right ) n \ln \left (x \right )}{9 e \left (\frac {d}{e}\right )^{\frac {2}{3}} d}+\frac {2 b \ln \left (x +\left (\frac {d}{e}\right )^{\frac {1}{3}}\right ) \ln \left (x^{n}\right )}{9 e \left (\frac {d}{e}\right )^{\frac {2}{3}} d}+\frac {b \ln \left (x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right ) n \ln \left (x \right )}{9 e \left (\frac {d}{e}\right )^{\frac {2}{3}} d}-\frac {b \ln \left (x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right ) \ln \left (x^{n}\right )}{9 e \left (\frac {d}{e}\right )^{\frac {2}{3}} d}-\frac {2 b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}-1\right )}{3}\right ) n \ln \left (x \right )}{9 e \left (\frac {d}{e}\right )^{\frac {2}{3}} d}+\frac {2 b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}-1\right )}{3}\right ) \ln \left (x^{n}\right )}{9 e \left (\frac {d}{e}\right )^{\frac {2}{3}} d}-\frac {b n \ln \left (x +\left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{9 e \left (\frac {d}{e}\right )^{\frac {2}{3}} d}+\frac {b n \ln \left (x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{18 e \left (\frac {d}{e}\right )^{\frac {2}{3}} d}-\frac {b n \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 e \left (\frac {d}{e}\right )^{\frac {2}{3}} d}+\frac {2 b n \left (\munderset {\textit {\_R2} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{3}+d \right )}{\sum }\frac {\ln \left (x \right ) \ln \left (\frac {\textit {\_R2} -x}{\textit {\_R2}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R2} -x}{\textit {\_R2}}\right )}{\textit {\_R2}^{2}}\right )}{9 e d}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (\frac {x}{3 d \left (e \,x^{3}+d \right )}+\frac {\frac {2 \ln \left (x +\left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{9 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{9 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}}{d}\right )\) | \(598\) |
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\[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^3\right )^2} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{3} + d\right )}^{2}} \,d x } \]
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\[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^3\right )^2} \, dx=\int \frac {a + b \log {\left (c x^{n} \right )}}{\left (d + e x^{3}\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^3\right )^2} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^3\right )^2} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{3} + d\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^3\right )^2} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{{\left (e\,x^3+d\right )}^2} \,d x \]
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