Integrand size = 22, antiderivative size = 134 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \, dx=\frac {b d^5 n \sqrt {x}}{3 e^5}-\frac {b d^4 n x}{6 e^4}+\frac {b d^3 n x^{3/2}}{9 e^3}-\frac {b d^2 n x^2}{12 e^2}+\frac {b d n x^{5/2}}{15 e}-\frac {1}{18} b n x^3-\frac {b d^6 n \log \left (d+e \sqrt {x}\right )}{3 e^6}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \]
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Time = 0.07 (sec) , antiderivative size = 134, 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} \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \, dx=\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )-\frac {b d^6 n \log \left (d+e \sqrt {x}\right )}{3 e^6}+\frac {b d^5 n \sqrt {x}}{3 e^5}-\frac {b d^4 n x}{6 e^4}+\frac {b d^3 n x^{3/2}}{9 e^3}-\frac {b d^2 n x^2}{12 e^2}+\frac {b d n x^{5/2}}{15 e}-\frac {1}{18} b n x^3 \]
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Rule 45
Rule 2442
Rule 2504
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int x^5 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )-\frac {1}{3} (b e n) \text {Subst}\left (\int \frac {x^6}{d+e x} \, dx,x,\sqrt {x}\right ) \\ & = \frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )-\frac {1}{3} (b e n) \text {Subst}\left (\int \left (-\frac {d^5}{e^6}+\frac {d^4 x}{e^5}-\frac {d^3 x^2}{e^4}+\frac {d^2 x^3}{e^3}-\frac {d x^4}{e^2}+\frac {x^5}{e}+\frac {d^6}{e^6 (d+e x)}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {b d^5 n \sqrt {x}}{3 e^5}-\frac {b d^4 n x}{6 e^4}+\frac {b d^3 n x^{3/2}}{9 e^3}-\frac {b d^2 n x^2}{12 e^2}+\frac {b d n x^{5/2}}{15 e}-\frac {1}{18} b n x^3-\frac {b d^6 n \log \left (d+e \sqrt {x}\right )}{3 e^6}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.96 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \, dx=\frac {a x^3}{3}-\frac {1}{6} b e n \left (-\frac {2 d^5 \sqrt {x}}{e^6}+\frac {d^4 x}{e^5}-\frac {2 d^3 x^{3/2}}{3 e^4}+\frac {d^2 x^2}{2 e^3}-\frac {2 d x^{5/2}}{5 e^2}+\frac {x^3}{3 e}+\frac {2 d^6 \log \left (d+e \sqrt {x}\right )}{e^7}\right )+\frac {1}{3} b x^3 \log \left (c \left (d+e \sqrt {x}\right )^n\right ) \]
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\[\int x^{2} \left (a +b \ln \left (c \left (d +e \sqrt {x}\right )^{n}\right )\right )d x\]
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Time = 0.31 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.91 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \, dx=\frac {60 \, b e^{6} x^{3} \log \left (c\right ) - 15 \, b d^{2} e^{4} n x^{2} - 30 \, b d^{4} e^{2} n x - 10 \, {\left (b e^{6} n - 6 \, a e^{6}\right )} x^{3} + 60 \, {\left (b e^{6} n x^{3} - b d^{6} n\right )} \log \left (e \sqrt {x} + d\right ) + 4 \, {\left (3 \, b d e^{5} n x^{2} + 5 \, b d^{3} e^{3} n x + 15 \, b d^{5} e n\right )} \sqrt {x}}{180 \, e^{6}} \]
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Time = 2.69 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.96 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \, dx=\frac {a x^{3}}{3} + b \left (- \frac {e n \left (\frac {2 d^{6} \left (\begin {cases} \frac {\sqrt {x}}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e \sqrt {x} \right )}}{e} & \text {otherwise} \end {cases}\right )}{e^{6}} - \frac {2 d^{5} \sqrt {x}}{e^{6}} + \frac {d^{4} x}{e^{5}} - \frac {2 d^{3} x^{\frac {3}{2}}}{3 e^{4}} + \frac {d^{2} x^{2}}{2 e^{3}} - \frac {2 d x^{\frac {5}{2}}}{5 e^{2}} + \frac {x^{3}}{3 e}\right )}{6} + \frac {x^{3} \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}}{3}\right ) \]
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Time = 0.22 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.79 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \, dx=\frac {1}{3} \, b x^{3} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + \frac {1}{3} \, a x^{3} - \frac {1}{180} \, b e n {\left (\frac {60 \, d^{6} \log \left (e \sqrt {x} + d\right )}{e^{7}} + \frac {10 \, e^{5} x^{3} - 12 \, d e^{4} x^{\frac {5}{2}} + 15 \, d^{2} e^{3} x^{2} - 20 \, d^{3} e^{2} x^{\frac {3}{2}} + 30 \, d^{4} e x - 60 \, d^{5} \sqrt {x}}{e^{6}}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (108) = 216\).
Time = 0.32 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.97 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \, dx=\frac {60 \, b e x^{3} \log \left (c\right ) + 60 \, a e x^{3} + {\left (\frac {60 \, {\left (e \sqrt {x} + d\right )}^{6} \log \left (e \sqrt {x} + d\right )}{e^{5}} - \frac {360 \, {\left (e \sqrt {x} + d\right )}^{5} d \log \left (e \sqrt {x} + d\right )}{e^{5}} + \frac {900 \, {\left (e \sqrt {x} + d\right )}^{4} d^{2} \log \left (e \sqrt {x} + d\right )}{e^{5}} - \frac {1200 \, {\left (e \sqrt {x} + d\right )}^{3} d^{3} \log \left (e \sqrt {x} + d\right )}{e^{5}} + \frac {900 \, {\left (e \sqrt {x} + d\right )}^{2} d^{4} \log \left (e \sqrt {x} + d\right )}{e^{5}} - \frac {360 \, {\left (e \sqrt {x} + d\right )} d^{5} \log \left (e \sqrt {x} + d\right )}{e^{5}} - \frac {10 \, {\left (e \sqrt {x} + d\right )}^{6}}{e^{5}} + \frac {72 \, {\left (e \sqrt {x} + d\right )}^{5} d}{e^{5}} - \frac {225 \, {\left (e \sqrt {x} + d\right )}^{4} d^{2}}{e^{5}} + \frac {400 \, {\left (e \sqrt {x} + d\right )}^{3} d^{3}}{e^{5}} - \frac {450 \, {\left (e \sqrt {x} + d\right )}^{2} d^{4}}{e^{5}} + \frac {360 \, {\left (e \sqrt {x} + d\right )} d^{5}}{e^{5}}\right )} b n}{180 \, e} \]
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Time = 1.72 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.83 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \, dx=\frac {a\,x^3}{3}-\frac {b\,n\,x^3}{18}+\frac {b\,x^3\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{3}+\frac {b\,d\,n\,x^{5/2}}{15\,e}-\frac {b\,d^4\,n\,x}{6\,e^4}-\frac {b\,d^6\,n\,\ln \left (d+e\,\sqrt {x}\right )}{3\,e^6}-\frac {b\,d^2\,n\,x^2}{12\,e^2}+\frac {b\,d^3\,n\,x^{3/2}}{9\,e^3}+\frac {b\,d^5\,n\,\sqrt {x}}{3\,e^5} \]
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