Integrand size = 22, antiderivative size = 185 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \, dx=-\frac {b d^8 n \sqrt [3]{x}}{3 e^8}+\frac {b d^7 n x^{2/3}}{6 e^7}-\frac {b d^6 n x}{9 e^6}+\frac {b d^5 n x^{4/3}}{12 e^5}-\frac {b d^4 n x^{5/3}}{15 e^4}+\frac {b d^3 n x^2}{18 e^3}-\frac {b d^2 n x^{7/3}}{21 e^2}+\frac {b d n x^{8/3}}{24 e}-\frac {1}{27} b n x^3+\frac {b d^9 n \log \left (d+e \sqrt [3]{x}\right )}{3 e^9}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \]
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Time = 0.09 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2504, 2442, 45} \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \, dx=\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )+\frac {b d^9 n \log \left (d+e \sqrt [3]{x}\right )}{3 e^9}-\frac {b d^8 n \sqrt [3]{x}}{3 e^8}+\frac {b d^7 n x^{2/3}}{6 e^7}-\frac {b d^6 n x}{9 e^6}+\frac {b d^5 n x^{4/3}}{12 e^5}-\frac {b d^4 n x^{5/3}}{15 e^4}+\frac {b d^3 n x^2}{18 e^3}-\frac {b d^2 n x^{7/3}}{21 e^2}+\frac {b d n x^{8/3}}{24 e}-\frac {1}{27} b n x^3 \]
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Rule 45
Rule 2442
Rule 2504
Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int x^8 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )-\frac {1}{3} (b e n) \text {Subst}\left (\int \frac {x^9}{d+e x} \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )-\frac {1}{3} (b e n) \text {Subst}\left (\int \left (\frac {d^8}{e^9}-\frac {d^7 x}{e^8}+\frac {d^6 x^2}{e^7}-\frac {d^5 x^3}{e^6}+\frac {d^4 x^4}{e^5}-\frac {d^3 x^5}{e^4}+\frac {d^2 x^6}{e^3}-\frac {d x^7}{e^2}+\frac {x^8}{e}-\frac {d^9}{e^9 (d+e x)}\right ) \, dx,x,\sqrt [3]{x}\right ) \\ & = -\frac {b d^8 n \sqrt [3]{x}}{3 e^8}+\frac {b d^7 n x^{2/3}}{6 e^7}-\frac {b d^6 n x}{9 e^6}+\frac {b d^5 n x^{4/3}}{12 e^5}-\frac {b d^4 n x^{5/3}}{15 e^4}+\frac {b d^3 n x^2}{18 e^3}-\frac {b d^2 n x^{7/3}}{21 e^2}+\frac {b d n x^{8/3}}{24 e}-\frac {1}{27} b n x^3+\frac {b d^9 n \log \left (d+e \sqrt [3]{x}\right )}{3 e^9}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.94 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \, dx=\frac {a x^3}{3}-\frac {1}{9} b e n \left (\frac {3 d^8 \sqrt [3]{x}}{e^9}-\frac {3 d^7 x^{2/3}}{2 e^8}+\frac {d^6 x}{e^7}-\frac {3 d^5 x^{4/3}}{4 e^6}+\frac {3 d^4 x^{5/3}}{5 e^5}-\frac {d^3 x^2}{2 e^4}+\frac {3 d^2 x^{7/3}}{7 e^3}-\frac {3 d x^{8/3}}{8 e^2}+\frac {x^3}{3 e}-\frac {3 d^9 \log \left (d+e \sqrt [3]{x}\right )}{e^{10}}\right )+\frac {1}{3} b x^3 \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right ) \]
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\[\int x^{2} \left (a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{n}\right )\right )d x\]
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Time = 0.34 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.87 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \, dx=\frac {2520 \, b e^{9} x^{3} \log \left (c\right ) + 420 \, b d^{3} e^{6} n x^{2} - 840 \, b d^{6} e^{3} n x - 280 \, {\left (b e^{9} n - 9 \, a e^{9}\right )} x^{3} + 2520 \, {\left (b e^{9} n x^{3} + b d^{9} n\right )} \log \left (e x^{\frac {1}{3}} + d\right ) + 63 \, {\left (5 \, b d e^{8} n x^{2} - 8 \, b d^{4} e^{5} n x + 20 \, b d^{7} e^{2} n\right )} x^{\frac {2}{3}} - 90 \, {\left (4 \, b d^{2} e^{7} n x^{2} - 7 \, b d^{5} e^{4} n x + 28 \, b d^{8} e n\right )} x^{\frac {1}{3}}}{7560 \, e^{9}} \]
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Time = 8.68 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.94 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \, dx=\frac {a x^{3}}{3} + b \left (- \frac {e n \left (- \frac {3 d^{9} \left (\begin {cases} \frac {\sqrt [3]{x}}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e \sqrt [3]{x} \right )}}{e} & \text {otherwise} \end {cases}\right )}{e^{9}} + \frac {3 d^{8} \sqrt [3]{x}}{e^{9}} - \frac {3 d^{7} x^{\frac {2}{3}}}{2 e^{8}} + \frac {d^{6} x}{e^{7}} - \frac {3 d^{5} x^{\frac {4}{3}}}{4 e^{6}} + \frac {3 d^{4} x^{\frac {5}{3}}}{5 e^{5}} - \frac {d^{3} x^{2}}{2 e^{4}} + \frac {3 d^{2} x^{\frac {7}{3}}}{7 e^{3}} - \frac {3 d x^{\frac {8}{3}}}{8 e^{2}} + \frac {x^{3}}{3 e}\right )}{9} + \frac {x^{3} \log {\left (c \left (d + e \sqrt [3]{x}\right )^{n} \right )}}{3}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.76 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \, dx=\frac {1}{3} \, b x^{3} \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) + \frac {1}{3} \, a x^{3} + \frac {1}{7560} \, b e n {\left (\frac {2520 \, d^{9} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{10}} - \frac {280 \, e^{8} x^{3} - 315 \, d e^{7} x^{\frac {8}{3}} + 360 \, d^{2} e^{6} x^{\frac {7}{3}} - 420 \, d^{3} e^{5} x^{2} + 504 \, d^{4} e^{4} x^{\frac {5}{3}} - 630 \, d^{5} e^{3} x^{\frac {4}{3}} + 840 \, d^{6} e^{2} x - 1260 \, d^{7} e x^{\frac {2}{3}} + 2520 \, d^{8} x^{\frac {1}{3}}}{e^{9}}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 390 vs. \(2 (147) = 294\).
Time = 0.30 (sec) , antiderivative size = 390, normalized size of antiderivative = 2.11 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \, dx=\frac {2520 \, b e x^{3} \log \left (c\right ) + 2520 \, a e x^{3} + {\left (\frac {2520 \, {\left (e x^{\frac {1}{3}} + d\right )}^{9} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{8}} - \frac {22680 \, {\left (e x^{\frac {1}{3}} + d\right )}^{8} d \log \left (e x^{\frac {1}{3}} + d\right )}{e^{8}} + \frac {90720 \, {\left (e x^{\frac {1}{3}} + d\right )}^{7} d^{2} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{8}} - \frac {211680 \, {\left (e x^{\frac {1}{3}} + d\right )}^{6} d^{3} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{8}} + \frac {317520 \, {\left (e x^{\frac {1}{3}} + d\right )}^{5} d^{4} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{8}} - \frac {317520 \, {\left (e x^{\frac {1}{3}} + d\right )}^{4} d^{5} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{8}} + \frac {211680 \, {\left (e x^{\frac {1}{3}} + d\right )}^{3} d^{6} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{8}} - \frac {90720 \, {\left (e x^{\frac {1}{3}} + d\right )}^{2} d^{7} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{8}} + \frac {22680 \, {\left (e x^{\frac {1}{3}} + d\right )} d^{8} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{8}} - \frac {280 \, {\left (e x^{\frac {1}{3}} + d\right )}^{9}}{e^{8}} + \frac {2835 \, {\left (e x^{\frac {1}{3}} + d\right )}^{8} d}{e^{8}} - \frac {12960 \, {\left (e x^{\frac {1}{3}} + d\right )}^{7} d^{2}}{e^{8}} + \frac {35280 \, {\left (e x^{\frac {1}{3}} + d\right )}^{6} d^{3}}{e^{8}} - \frac {63504 \, {\left (e x^{\frac {1}{3}} + d\right )}^{5} d^{4}}{e^{8}} + \frac {79380 \, {\left (e x^{\frac {1}{3}} + d\right )}^{4} d^{5}}{e^{8}} - \frac {70560 \, {\left (e x^{\frac {1}{3}} + d\right )}^{3} d^{6}}{e^{8}} + \frac {45360 \, {\left (e x^{\frac {1}{3}} + d\right )}^{2} d^{7}}{e^{8}} - \frac {22680 \, {\left (e x^{\frac {1}{3}} + d\right )} d^{8}}{e^{8}}\right )} b n}{7560 \, e} \]
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Time = 1.58 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.81 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \, dx=\frac {a\,x^3}{3}-\frac {b\,n\,x^3}{27}+\frac {b\,x^3\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{3}+\frac {b\,d\,n\,x^{8/3}}{24\,e}-\frac {b\,d^6\,n\,x}{9\,e^6}+\frac {b\,d^9\,n\,\ln \left (d+e\,x^{1/3}\right )}{3\,e^9}+\frac {b\,d^3\,n\,x^2}{18\,e^3}-\frac {b\,d^2\,n\,x^{7/3}}{21\,e^2}-\frac {b\,d^4\,n\,x^{5/3}}{15\,e^4}+\frac {b\,d^5\,n\,x^{4/3}}{12\,e^5}+\frac {b\,d^7\,n\,x^{2/3}}{6\,e^7}-\frac {b\,d^8\,n\,x^{1/3}}{3\,e^8} \]
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