Integrand size = 20, antiderivative size = 115 \[ \int (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {4 b d^2 n \sqrt {d+e x}}{5 e}-\frac {4 b d n (d+e x)^{3/2}}{15 e}-\frac {4 b n (d+e x)^{5/2}}{25 e}+\frac {4 b d^{5/2} n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{5 e}+\frac {2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e} \]
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Time = 0.04 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2356, 52, 65, 214} \[ \int (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e}+\frac {4 b d^{5/2} n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{5 e}-\frac {4 b d^2 n \sqrt {d+e x}}{5 e}-\frac {4 b d n (d+e x)^{3/2}}{15 e}-\frac {4 b n (d+e x)^{5/2}}{25 e} \]
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Rule 52
Rule 65
Rule 214
Rule 2356
Rubi steps \begin{align*} \text {integral}& = \frac {2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e}-\frac {(2 b n) \int \frac {(d+e x)^{5/2}}{x} \, dx}{5 e} \\ & = -\frac {4 b n (d+e x)^{5/2}}{25 e}+\frac {2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e}-\frac {(2 b d n) \int \frac {(d+e x)^{3/2}}{x} \, dx}{5 e} \\ & = -\frac {4 b d n (d+e x)^{3/2}}{15 e}-\frac {4 b n (d+e x)^{5/2}}{25 e}+\frac {2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e}-\frac {\left (2 b d^2 n\right ) \int \frac {\sqrt {d+e x}}{x} \, dx}{5 e} \\ & = -\frac {4 b d^2 n \sqrt {d+e x}}{5 e}-\frac {4 b d n (d+e x)^{3/2}}{15 e}-\frac {4 b n (d+e x)^{5/2}}{25 e}+\frac {2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e}-\frac {\left (2 b d^3 n\right ) \int \frac {1}{x \sqrt {d+e x}} \, dx}{5 e} \\ & = -\frac {4 b d^2 n \sqrt {d+e x}}{5 e}-\frac {4 b d n (d+e x)^{3/2}}{15 e}-\frac {4 b n (d+e x)^{5/2}}{25 e}+\frac {2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e}-\frac {\left (4 b d^3 n\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{5 e^2} \\ & = -\frac {4 b d^2 n \sqrt {d+e x}}{5 e}-\frac {4 b d n (d+e x)^{3/2}}{15 e}-\frac {4 b n (d+e x)^{5/2}}{25 e}+\frac {4 b d^{5/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{5 e}+\frac {2 (d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 e} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.76 \[ \int (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 \left (-\frac {2}{15} b n \sqrt {d+e x} \left (23 d^2+11 d e x+3 e^2 x^2\right )+2 b d^{5/2} n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+(d+e x)^{5/2} \left (a+b \log \left (c x^n\right )\right )\right )}{5 e} \]
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\[\int \left (e x +d \right )^{\frac {3}{2}} \left (a +b \ln \left (c \,x^{n}\right )\right )d x\]
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Time = 0.33 (sec) , antiderivative size = 288, normalized size of antiderivative = 2.50 \[ \int (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\left [\frac {2 \, {\left (15 \, b d^{\frac {5}{2}} n \log \left (\frac {e x + 2 \, \sqrt {e x + d} \sqrt {d} + 2 \, d}{x}\right ) - {\left (46 \, b d^{2} n - 15 \, a d^{2} + 3 \, {\left (2 \, b e^{2} n - 5 \, a e^{2}\right )} x^{2} + 2 \, {\left (11 \, b d e n - 15 \, a d e\right )} x - 15 \, {\left (b e^{2} x^{2} + 2 \, b d e x + b d^{2}\right )} \log \left (c\right ) - 15 \, {\left (b e^{2} n x^{2} + 2 \, b d e n x + b d^{2} n\right )} \log \left (x\right )\right )} \sqrt {e x + d}\right )}}{75 \, e}, -\frac {2 \, {\left (30 \, b \sqrt {-d} d^{2} n \arctan \left (\frac {\sqrt {e x + d} \sqrt {-d}}{d}\right ) + {\left (46 \, b d^{2} n - 15 \, a d^{2} + 3 \, {\left (2 \, b e^{2} n - 5 \, a e^{2}\right )} x^{2} + 2 \, {\left (11 \, b d e n - 15 \, a d e\right )} x - 15 \, {\left (b e^{2} x^{2} + 2 \, b d e x + b d^{2}\right )} \log \left (c\right ) - 15 \, {\left (b e^{2} n x^{2} + 2 \, b d e n x + b d^{2} n\right )} \log \left (x\right )\right )} \sqrt {e x + d}\right )}}{75 \, e}\right ] \]
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Time = 88.80 (sec) , antiderivative size = 377, normalized size of antiderivative = 3.28 \[ \int (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=a d \left (\begin {cases} \frac {2 \left (d + e x\right )^{\frac {3}{2}}}{3 e} & \text {for}\: e \neq 0 \\\sqrt {d} x & \text {otherwise} \end {cases}\right ) + a e \left (\begin {cases} - \frac {2 d \left (d + e x\right )^{\frac {3}{2}}}{3 e^{2}} + \frac {2 \left (d + e x\right )^{\frac {5}{2}}}{5 e^{2}} & \text {for}\: e \neq 0 \\\frac {\sqrt {d} x^{2}}{2} & \text {otherwise} \end {cases}\right ) - b d n \left (\begin {cases} \frac {16 d^{\frac {3}{2}} \sqrt {1 + \frac {e x}{d}}}{9 e} + \frac {2 d^{\frac {3}{2}} \log {\left (\frac {e x}{d} \right )}}{3 e} - \frac {4 d^{\frac {3}{2}} \log {\left (\sqrt {1 + \frac {e x}{d}} + 1 \right )}}{3 e} + \frac {4 \sqrt {d} x \sqrt {1 + \frac {e x}{d}}}{9} & \text {for}\: e > -\infty \wedge e < \infty \wedge e \neq 0 \\\sqrt {d} x & \text {otherwise} \end {cases}\right ) + b d \left (\begin {cases} \frac {2 \left (d + e x\right )^{\frac {3}{2}}}{3 e} & \text {for}\: e \neq 0 \\\sqrt {d} x & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} - b e n \left (\begin {cases} - \frac {124 d^{\frac {5}{2}} \sqrt {1 + \frac {e x}{d}}}{225 e^{2}} - \frac {4 d^{\frac {5}{2}} \log {\left (\frac {e x}{d} \right )}}{15 e^{2}} + \frac {8 d^{\frac {5}{2}} \log {\left (\sqrt {1 + \frac {e x}{d}} + 1 \right )}}{15 e^{2}} + \frac {32 d^{\frac {3}{2}} x \sqrt {1 + \frac {e x}{d}}}{225 e} + \frac {4 \sqrt {d} x^{2} \sqrt {1 + \frac {e x}{d}}}{25} & \text {for}\: e > -\infty \wedge e < \infty \wedge e \neq 0 \\\frac {\sqrt {d} x^{2}}{4} & \text {otherwise} \end {cases}\right ) + b e \left (\begin {cases} - \frac {2 d \left (d + e x\right )^{\frac {3}{2}}}{3 e^{2}} + \frac {2 \left (d + e x\right )^{\frac {5}{2}}}{5 e^{2}} & \text {for}\: e \neq 0 \\\frac {\sqrt {d} x^{2}}{2} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} \]
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Time = 0.28 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.91 \[ \int (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 \, {\left (e x + d\right )}^{\frac {5}{2}} b \log \left (c x^{n}\right )}{5 \, e} + \frac {2 \, {\left (e x + d\right )}^{\frac {5}{2}} a}{5 \, e} - \frac {2 \, {\left (15 \, d^{\frac {5}{2}} \log \left (\frac {\sqrt {e x + d} - \sqrt {d}}{\sqrt {e x + d} + \sqrt {d}}\right ) + 6 \, {\left (e x + d\right )}^{\frac {5}{2}} + 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 30 \, \sqrt {e x + d} d^{2}\right )} b n}{75 \, e} \]
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\[ \int (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { {\left (e x + d\right )}^{\frac {3}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )} \,d x } \]
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Timed out. \[ \int (d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx=\int \left (a+b\,\ln \left (c\,x^n\right )\right )\,{\left (d+e\,x\right )}^{3/2} \,d x \]
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