Integrand size = 20, antiderivative size = 267 \[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx=-\frac {3 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^2}{2 e^3}+\frac {2 b^2 n^2 \left (d+e \sqrt [3]{x}\right )^3}{9 e^3}+\frac {6 b^2 d^2 n^2 \sqrt [3]{x}}{e^2}-\frac {b^2 d^3 n^2 \log ^2\left (d+e \sqrt [3]{x}\right )}{e^3}-\frac {6 b d^2 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^3}+\frac {3 b d n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^3}-\frac {2 b n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^3}+\frac {2 b d^3 n \log \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^3}+x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \]
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Time = 0.20 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2501, 2445, 2458, 45, 2372, 12, 14, 2338} \[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx=\frac {2 b d^3 n \log \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^3}-\frac {6 b d^2 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^3}+\frac {3 b d n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^3}-\frac {2 b n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^3}+x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-\frac {b^2 d^3 n^2 \log ^2\left (d+e \sqrt [3]{x}\right )}{e^3}+\frac {6 b^2 d^2 n^2 \sqrt [3]{x}}{e^2}-\frac {3 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^2}{2 e^3}+\frac {2 b^2 n^2 \left (d+e \sqrt [3]{x}\right )^3}{9 e^3} \]
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Rule 12
Rule 14
Rule 45
Rule 2338
Rule 2372
Rule 2445
Rule 2458
Rule 2501
Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right ) \\ & = x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-(2 b e n) \text {Subst}\left (\int \frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx,x,\sqrt [3]{x}\right ) \\ & = x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-(2 b n) \text {Subst}\left (\int \frac {\left (-\frac {d}{e}+\frac {x}{e}\right )^3 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e \sqrt [3]{x}\right ) \\ & = -\frac {6 b d^2 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^3}+\frac {3 b d n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^3}-\frac {2 b n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^3}+\frac {2 b d^3 n \log \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^3}+x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+\left (2 b^2 n^2\right ) \text {Subst}\left (\int \frac {18 d^2 x-9 d x^2+2 x^3-6 d^3 \log (x)}{6 e^3 x} \, dx,x,d+e \sqrt [3]{x}\right ) \\ & = -\frac {6 b d^2 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^3}+\frac {3 b d n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^3}-\frac {2 b n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^3}+\frac {2 b d^3 n \log \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^3}+x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+\frac {\left (b^2 n^2\right ) \text {Subst}\left (\int \frac {18 d^2 x-9 d x^2+2 x^3-6 d^3 \log (x)}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{3 e^3} \\ & = -\frac {6 b d^2 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^3}+\frac {3 b d n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^3}-\frac {2 b n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^3}+\frac {2 b d^3 n \log \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^3}+x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+\frac {\left (b^2 n^2\right ) \text {Subst}\left (\int \left (18 d^2-9 d x+2 x^2-\frac {6 d^3 \log (x)}{x}\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{3 e^3} \\ & = -\frac {3 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^2}{2 e^3}+\frac {2 b^2 n^2 \left (d+e \sqrt [3]{x}\right )^3}{9 e^3}+\frac {6 b^2 d^2 n^2 \sqrt [3]{x}}{e^2}-\frac {6 b d^2 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^3}+\frac {3 b d n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^3}-\frac {2 b n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^3}+\frac {2 b d^3 n \log \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^3}+x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-\frac {\left (2 b^2 d^3 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{e^3} \\ & = -\frac {3 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^2}{2 e^3}+\frac {2 b^2 n^2 \left (d+e \sqrt [3]{x}\right )^3}{9 e^3}+\frac {6 b^2 d^2 n^2 \sqrt [3]{x}}{e^2}-\frac {b^2 d^3 n^2 \log ^2\left (d+e \sqrt [3]{x}\right )}{e^3}-\frac {6 b d^2 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^3}+\frac {3 b d n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^3}-\frac {2 b n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^3}+\frac {2 b d^3 n \log \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{e^3}+x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.74 \[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx=\frac {b^2 e n^2 \left (66 d^2-15 d e \sqrt [3]{x}+4 e^2 x^{2/3}\right ) \sqrt [3]{x}+6 a b n \left (7 d^3-6 d^2 e \sqrt [3]{x}+3 d e^2 x^{2/3}-2 e^3 x\right )+18 a^2 \left (d^3+e^3 x\right )+6 b \left (6 a \left (d^3+e^3 x\right )-b n \left (11 d^3+6 d^2 e \sqrt [3]{x}-3 d e^2 x^{2/3}+2 e^3 x\right )\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )+18 b^2 \left (d^3+e^3 x\right ) \log ^2\left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{18 e^3} \]
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\[\int {\left (a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{n}\right )\right )}^{2}d x\]
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Time = 0.34 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.07 \[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx=\frac {18 \, b^{2} e^{3} x \log \left (c\right )^{2} + 18 \, {\left (b^{2} e^{3} n^{2} x + b^{2} d^{3} n^{2}\right )} \log \left (e x^{\frac {1}{3}} + d\right )^{2} - 12 \, {\left (b^{2} e^{3} n - 3 \, a b e^{3}\right )} x \log \left (c\right ) + 2 \, {\left (2 \, b^{2} e^{3} n^{2} - 6 \, a b e^{3} n + 9 \, a^{2} e^{3}\right )} x + 6 \, {\left (3 \, b^{2} d e^{2} n^{2} x^{\frac {2}{3}} - 6 \, b^{2} d^{2} e n^{2} x^{\frac {1}{3}} - 11 \, b^{2} d^{3} n^{2} + 6 \, a b d^{3} n - 2 \, {\left (b^{2} e^{3} n^{2} - 3 \, a b e^{3} n\right )} x + 6 \, {\left (b^{2} e^{3} n x + b^{2} d^{3} n\right )} \log \left (c\right )\right )} \log \left (e x^{\frac {1}{3}} + d\right ) - 3 \, {\left (5 \, b^{2} d e^{2} n^{2} - 6 \, b^{2} d e^{2} n \log \left (c\right ) - 6 \, a b d e^{2} n\right )} x^{\frac {2}{3}} + 6 \, {\left (11 \, b^{2} d^{2} e n^{2} - 6 \, b^{2} d^{2} e n \log \left (c\right ) - 6 \, a b d^{2} e n\right )} x^{\frac {1}{3}}}{18 \, e^{3}} \]
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\[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx=\int \left (a + b \log {\left (c \left (d + e \sqrt [3]{x}\right )^{n} \right )}\right )^{2}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.81 \[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx=\frac {1}{3} \, {\left (e n {\left (\frac {6 \, d^{3} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{4}} - \frac {2 \, e^{2} x - 3 \, d e x^{\frac {2}{3}} + 6 \, d^{2} x^{\frac {1}{3}}}{e^{3}}\right )} + 6 \, x \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right )\right )} a b + \frac {1}{18} \, {\left (6 \, e n {\left (\frac {6 \, d^{3} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{4}} - \frac {2 \, e^{2} x - 3 \, d e x^{\frac {2}{3}} + 6 \, d^{2} x^{\frac {1}{3}}}{e^{3}}\right )} \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) + 18 \, x \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right )^{2} - \frac {{\left (18 \, d^{3} \log \left (e x^{\frac {1}{3}} + d\right )^{2} - 4 \, e^{3} x + 66 \, d^{3} \log \left (e x^{\frac {1}{3}} + d\right ) + 15 \, d e^{2} x^{\frac {2}{3}} - 66 \, d^{2} e x^{\frac {1}{3}}\right )} n^{2}}{e^{3}}\right )} b^{2} + a^{2} x \]
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Time = 0.31 (sec) , antiderivative size = 465, normalized size of antiderivative = 1.74 \[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx=\frac {18 \, b^{2} e x \log \left (c\right )^{2} + {\left (\frac {18 \, {\left (e x^{\frac {1}{3}} + d\right )}^{3} \log \left (e x^{\frac {1}{3}} + d\right )^{2}}{e^{2}} - \frac {54 \, {\left (e x^{\frac {1}{3}} + d\right )}^{2} d \log \left (e x^{\frac {1}{3}} + d\right )^{2}}{e^{2}} + \frac {54 \, {\left (e x^{\frac {1}{3}} + d\right )} d^{2} \log \left (e x^{\frac {1}{3}} + d\right )^{2}}{e^{2}} - \frac {12 \, {\left (e x^{\frac {1}{3}} + d\right )}^{3} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{2}} + \frac {54 \, {\left (e x^{\frac {1}{3}} + d\right )}^{2} d \log \left (e x^{\frac {1}{3}} + d\right )}{e^{2}} - \frac {108 \, {\left (e x^{\frac {1}{3}} + d\right )} d^{2} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{2}} + \frac {4 \, {\left (e x^{\frac {1}{3}} + d\right )}^{3}}{e^{2}} - \frac {27 \, {\left (e x^{\frac {1}{3}} + d\right )}^{2} d}{e^{2}} + \frac {108 \, {\left (e x^{\frac {1}{3}} + d\right )} d^{2}}{e^{2}}\right )} b^{2} n^{2} + 6 \, {\left (\frac {6 \, {\left (e x^{\frac {1}{3}} + d\right )}^{3} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{2}} - \frac {18 \, {\left (e x^{\frac {1}{3}} + d\right )}^{2} d \log \left (e x^{\frac {1}{3}} + d\right )}{e^{2}} + \frac {18 \, {\left (e x^{\frac {1}{3}} + d\right )} d^{2} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{2}} - \frac {2 \, {\left (e x^{\frac {1}{3}} + d\right )}^{3}}{e^{2}} + \frac {9 \, {\left (e x^{\frac {1}{3}} + d\right )}^{2} d}{e^{2}} - \frac {18 \, {\left (e x^{\frac {1}{3}} + d\right )} d^{2}}{e^{2}}\right )} b^{2} n \log \left (c\right ) + 36 \, a b e x \log \left (c\right ) + 6 \, {\left (\frac {6 \, {\left (e x^{\frac {1}{3}} + d\right )}^{3} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{2}} - \frac {18 \, {\left (e x^{\frac {1}{3}} + d\right )}^{2} d \log \left (e x^{\frac {1}{3}} + d\right )}{e^{2}} + \frac {18 \, {\left (e x^{\frac {1}{3}} + d\right )} d^{2} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{2}} - \frac {2 \, {\left (e x^{\frac {1}{3}} + d\right )}^{3}}{e^{2}} + \frac {9 \, {\left (e x^{\frac {1}{3}} + d\right )}^{2} d}{e^{2}} - \frac {18 \, {\left (e x^{\frac {1}{3}} + d\right )} d^{2}}{e^{2}}\right )} a b n + 18 \, a^{2} e x}{18 \, e} \]
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Time = 1.80 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.09 \[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \, dx=\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )\,\left (\frac {2\,b\,x\,\left (3\,a-b\,n\right )}{3}-x^{2/3}\,\left (\frac {b\,d\,\left (3\,a-b\,n\right )}{e}-\frac {3\,a\,b\,d}{e}\right )+\frac {d\,x^{1/3}\,\left (\frac {2\,b\,d\,\left (3\,a-b\,n\right )}{e}-\frac {6\,a\,b\,d}{e}\right )}{e}\right )-x^{2/3}\,\left (\frac {d\,\left (3\,a^2-2\,a\,b\,n+\frac {2\,b^2\,n^2}{3}\right )}{2\,e}-\frac {d\,\left (3\,a^2-b^2\,n^2\right )}{2\,e}\right )+x^{1/3}\,\left (\frac {d\,\left (\frac {d\,\left (3\,a^2-2\,a\,b\,n+\frac {2\,b^2\,n^2}{3}\right )}{e}-\frac {d\,\left (3\,a^2-b^2\,n^2\right )}{e}\right )}{e}+\frac {2\,b^2\,d^2\,n^2}{e^2}\right )+x\,\left (a^2-\frac {2\,a\,b\,n}{3}+\frac {2\,b^2\,n^2}{9}\right )+{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2\,\left (b^2\,x+\frac {b^2\,d^3}{e^3}\right )-\frac {\ln \left (d+e\,x^{1/3}\right )\,\left (11\,b^2\,d^3\,n^2-6\,a\,b\,d^3\,n\right )}{3\,e^3} \]
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