Integrand size = 20, antiderivative size = 102 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \, dx=\frac {b d^3 n \sqrt {x}}{2 e^3}-\frac {b d^2 n x}{4 e^2}+\frac {b d n x^{3/2}}{6 e}-\frac {1}{8} b n x^2-\frac {b d^4 n \log \left (d+e \sqrt {x}\right )}{2 e^4}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \]
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Time = 0.05 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2504, 2442, 45} \[ \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \, dx=\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )-\frac {b d^4 n \log \left (d+e \sqrt {x}\right )}{2 e^4}+\frac {b d^3 n \sqrt {x}}{2 e^3}-\frac {b d^2 n x}{4 e^2}+\frac {b d n x^{3/2}}{6 e}-\frac {1}{8} b n x^2 \]
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Rule 45
Rule 2442
Rule 2504
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int x^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )-\frac {1}{2} (b e n) \text {Subst}\left (\int \frac {x^4}{d+e x} \, dx,x,\sqrt {x}\right ) \\ & = \frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )-\frac {1}{2} (b e n) \text {Subst}\left (\int \left (-\frac {d^3}{e^4}+\frac {d^2 x}{e^3}-\frac {d x^2}{e^2}+\frac {x^3}{e}+\frac {d^4}{e^4 (d+e x)}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {b d^3 n \sqrt {x}}{2 e^3}-\frac {b d^2 n x}{4 e^2}+\frac {b d n x^{3/2}}{6 e}-\frac {1}{8} b n x^2-\frac {b d^4 n \log \left (d+e \sqrt {x}\right )}{2 e^4}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.99 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \, dx=\frac {a x^2}{2}-\frac {1}{4} b e n \left (-\frac {2 d^3 \sqrt {x}}{e^4}+\frac {d^2 x}{e^3}-\frac {2 d x^{3/2}}{3 e^2}+\frac {x^2}{2 e}+\frac {2 d^4 \log \left (d+e \sqrt {x}\right )}{e^5}\right )+\frac {1}{2} b x^2 \log \left (c \left (d+e \sqrt {x}\right )^n\right ) \]
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\[\int x \left (a +b \ln \left (c \left (d +e \sqrt {x}\right )^{n}\right )\right )d x\]
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none
Time = 0.30 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.93 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \, dx=\frac {12 \, b e^{4} x^{2} \log \left (c\right ) - 6 \, b d^{2} e^{2} n x - 3 \, {\left (b e^{4} n - 4 \, a e^{4}\right )} x^{2} + 12 \, {\left (b e^{4} n x^{2} - b d^{4} n\right )} \log \left (e \sqrt {x} + d\right ) + 4 \, {\left (b d e^{3} n x + 3 \, b d^{3} e n\right )} \sqrt {x}}{24 \, e^{4}} \]
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Time = 1.23 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.98 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \, dx=\frac {a x^{2}}{2} + b \left (- \frac {e n \left (\frac {2 d^{4} \left (\begin {cases} \frac {\sqrt {x}}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e \sqrt {x} \right )}}{e} & \text {otherwise} \end {cases}\right )}{e^{4}} - \frac {2 d^{3} \sqrt {x}}{e^{4}} + \frac {d^{2} x}{e^{3}} - \frac {2 d x^{\frac {3}{2}}}{3 e^{2}} + \frac {x^{2}}{2 e}\right )}{4} + \frac {x^{2} \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}}{2}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.82 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \, dx=-\frac {1}{24} \, b e n {\left (\frac {12 \, d^{4} \log \left (e \sqrt {x} + d\right )}{e^{5}} + \frac {3 \, e^{3} x^{2} - 4 \, d e^{2} x^{\frac {3}{2}} + 6 \, d^{2} e x - 12 \, d^{3} \sqrt {x}}{e^{4}}\right )} + \frac {1}{2} \, b x^{2} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + \frac {1}{2} \, a x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (82) = 164\).
Time = 0.33 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.76 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \, dx=\frac {12 \, b e x^{2} \log \left (c\right ) + 12 \, a e x^{2} + {\left (\frac {12 \, {\left (e \sqrt {x} + d\right )}^{4} \log \left (e \sqrt {x} + d\right )}{e^{3}} - \frac {48 \, {\left (e \sqrt {x} + d\right )}^{3} d \log \left (e \sqrt {x} + d\right )}{e^{3}} + \frac {72 \, {\left (e \sqrt {x} + d\right )}^{2} d^{2} \log \left (e \sqrt {x} + d\right )}{e^{3}} - \frac {48 \, {\left (e \sqrt {x} + d\right )} d^{3} \log \left (e \sqrt {x} + d\right )}{e^{3}} - \frac {3 \, {\left (e \sqrt {x} + d\right )}^{4}}{e^{3}} + \frac {16 \, {\left (e \sqrt {x} + d\right )}^{3} d}{e^{3}} - \frac {36 \, {\left (e \sqrt {x} + d\right )}^{2} d^{2}}{e^{3}} + \frac {48 \, {\left (e \sqrt {x} + d\right )} d^{3}}{e^{3}}\right )} b n}{24 \, e} \]
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Time = 1.74 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.83 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \, dx=\frac {a\,x^2}{2}-\frac {b\,n\,x^2}{8}+\frac {b\,x^2\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{2}-\frac {b\,d^2\,n\,x}{4\,e^2}+\frac {b\,d\,n\,x^{3/2}}{6\,e}-\frac {b\,d^4\,n\,\ln \left (d+e\,\sqrt {x}\right )}{2\,e^4}+\frac {b\,d^3\,n\,\sqrt {x}}{2\,e^3} \]
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