Integrand size = 22, antiderivative size = 171 \[ \int x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx=\frac {b e^7 n \sqrt {x}}{4 d^7}-\frac {b e^6 n x}{8 d^6}+\frac {b e^5 n x^{3/2}}{12 d^5}-\frac {b e^4 n x^2}{16 d^4}+\frac {b e^3 n x^{5/2}}{20 d^3}-\frac {b e^2 n x^3}{24 d^2}+\frac {b e n x^{7/2}}{28 d}-\frac {b e^8 n \log \left (d+\frac {e}{\sqrt {x}}\right )}{4 d^8}+\frac {1}{4} x^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )-\frac {b e^8 n \log (x)}{8 d^8} \]
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Time = 0.08 (sec) , antiderivative size = 171, 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, 46} \[ \int x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx=\frac {1}{4} x^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )-\frac {b e^8 n \log \left (d+\frac {e}{\sqrt {x}}\right )}{4 d^8}-\frac {b e^8 n \log (x)}{8 d^8}+\frac {b e^7 n \sqrt {x}}{4 d^7}-\frac {b e^6 n x}{8 d^6}+\frac {b e^5 n x^{3/2}}{12 d^5}-\frac {b e^4 n x^2}{16 d^4}+\frac {b e^3 n x^{5/2}}{20 d^3}-\frac {b e^2 n x^3}{24 d^2}+\frac {b e n x^{7/2}}{28 d} \]
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Rule 46
Rule 2442
Rule 2504
Rubi steps \begin{align*} \text {integral}& = -\left (2 \text {Subst}\left (\int \frac {a+b \log \left (c (d+e x)^n\right )}{x^9} \, dx,x,\frac {1}{\sqrt {x}}\right )\right ) \\ & = \frac {1}{4} x^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )-\frac {1}{4} (b e n) \text {Subst}\left (\int \frac {1}{x^8 (d+e x)} \, dx,x,\frac {1}{\sqrt {x}}\right ) \\ & = \frac {1}{4} x^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )-\frac {1}{4} (b e n) \text {Subst}\left (\int \left (\frac {1}{d x^8}-\frac {e}{d^2 x^7}+\frac {e^2}{d^3 x^6}-\frac {e^3}{d^4 x^5}+\frac {e^4}{d^5 x^4}-\frac {e^5}{d^6 x^3}+\frac {e^6}{d^7 x^2}-\frac {e^7}{d^8 x}+\frac {e^8}{d^8 (d+e x)}\right ) \, dx,x,\frac {1}{\sqrt {x}}\right ) \\ & = \frac {b e^7 n \sqrt {x}}{4 d^7}-\frac {b e^6 n x}{8 d^6}+\frac {b e^5 n x^{3/2}}{12 d^5}-\frac {b e^4 n x^2}{16 d^4}+\frac {b e^3 n x^{5/2}}{20 d^3}-\frac {b e^2 n x^3}{24 d^2}+\frac {b e n x^{7/2}}{28 d}-\frac {b e^8 n \log \left (d+\frac {e}{\sqrt {x}}\right )}{4 d^8}+\frac {1}{4} x^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )-\frac {b e^8 n \log (x)}{8 d^8} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.91 \[ \int x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx=\frac {a x^4}{4}+\frac {1}{4} b x^4 \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )+\frac {1}{8} b e n \left (\frac {2 e^6 \sqrt {x}}{d^7}-\frac {e^5 x}{d^6}+\frac {2 e^4 x^{3/2}}{3 d^5}-\frac {e^3 x^2}{2 d^4}+\frac {2 e^2 x^{5/2}}{5 d^3}-\frac {e x^3}{3 d^2}+\frac {2 x^{7/2}}{7 d}-\frac {2 e^7 \log \left (d+\frac {e}{\sqrt {x}}\right )}{d^8}-\frac {e^7 \log (x)}{d^8}\right ) \]
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\[\int x^{3} \left (a +b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{n}\right )\right )d x\]
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Time = 0.34 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.05 \[ \int x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx=\frac {420 \, b d^{8} x^{4} \log \left (c\right ) - 70 \, b d^{6} e^{2} n x^{3} + 420 \, a d^{8} x^{4} - 105 \, b d^{4} e^{4} n x^{2} - 210 \, b d^{2} e^{6} n x - 420 \, b d^{8} n \log \left (\sqrt {x}\right ) + 420 \, {\left (b d^{8} - b e^{8}\right )} n \log \left (d \sqrt {x} + e\right ) + 420 \, {\left (b d^{8} n x^{4} - b d^{8} n\right )} \log \left (\frac {d x + e \sqrt {x}}{x}\right ) + 4 \, {\left (15 \, b d^{7} e n x^{3} + 21 \, b d^{5} e^{3} n x^{2} + 35 \, b d^{3} e^{5} n x + 105 \, b d e^{7} n\right )} \sqrt {x}}{1680 \, d^{8}} \]
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Time = 58.18 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.84 \[ \int x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx=\frac {a x^{4}}{4} + b \left (\frac {e n \left (\frac {2 x^{\frac {7}{2}}}{7 d} - \frac {e x^{3}}{3 d^{2}} + \frac {2 e^{2} x^{\frac {5}{2}}}{5 d^{3}} - \frac {e^{3} x^{2}}{2 d^{4}} + \frac {2 e^{4} x^{\frac {3}{2}}}{3 d^{5}} - \frac {e^{5} x}{d^{6}} - \frac {2 e^{7} \left (\begin {cases} \frac {\sqrt {x}}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d \sqrt {x} + e \right )}}{d} & \text {otherwise} \end {cases}\right )}{d^{7}} + \frac {2 e^{6} \sqrt {x}}{d^{7}}\right )}{8} + \frac {x^{4} \log {\left (c \left (d + \frac {e}{\sqrt {x}}\right )^{n} \right )}}{4}\right ) \]
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Time = 0.21 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.69 \[ \int x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx=\frac {1}{4} \, b x^{4} \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) + \frac {1}{4} \, a x^{4} - \frac {1}{1680} \, b e n {\left (\frac {420 \, e^{7} \log \left (d \sqrt {x} + e\right )}{d^{8}} - \frac {60 \, d^{6} x^{\frac {7}{2}} - 70 \, d^{5} e x^{3} + 84 \, d^{4} e^{2} x^{\frac {5}{2}} - 105 \, d^{3} e^{3} x^{2} + 140 \, d^{2} e^{4} x^{\frac {3}{2}} - 210 \, d e^{5} x + 420 \, e^{6} \sqrt {x}}{d^{7}}\right )} \]
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Time = 0.36 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.56 \[ \int x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx=\frac {1}{4} \, b x^{4} \log \left (c\right ) + \frac {1}{4} \, a x^{4} - \frac {{\left (e^{9} {\left (\frac {420 \, \log \left (\frac {{\left | d \sqrt {x} + e \right |}}{\sqrt {{\left | x \right |}}}\right )}{d^{8}} - \frac {420 \, \log \left ({\left | -d + \frac {d \sqrt {x} + e}{\sqrt {x}} \right |}\right )}{d^{8}} + \frac {1089 \, d^{7} - \frac {4683 \, {\left (d \sqrt {x} + e\right )} d^{6}}{\sqrt {x}} + \frac {9639 \, {\left (d \sqrt {x} + e\right )}^{2} d^{5}}{x} - \frac {11165 \, {\left (d \sqrt {x} + e\right )}^{3} d^{4}}{x^{\frac {3}{2}}} + \frac {7490 \, {\left (d \sqrt {x} + e\right )}^{4} d^{3}}{x^{2}} - \frac {2730 \, {\left (d \sqrt {x} + e\right )}^{5} d^{2}}{x^{\frac {5}{2}}} + \frac {420 \, {\left (d \sqrt {x} + e\right )}^{6} d}{x^{3}}}{{\left (d - \frac {d \sqrt {x} + e}{\sqrt {x}}\right )}^{7} d^{8}}\right )} - \frac {420 \, e^{9} \log \left (-{\left (e - \frac {d}{\frac {d}{e} - \frac {d \sqrt {x} + e}{e \sqrt {x}}}\right )} {\left (\frac {d}{e} - \frac {d \sqrt {x} + e}{e \sqrt {x}}\right )}\right )}{{\left (d - \frac {d \sqrt {x} + e}{\sqrt {x}}\right )}^{8}}\right )} b n}{1680 \, e} \]
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Time = 2.21 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.82 \[ \int x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx=\frac {\frac {b\,d\,e^7\,n\,\sqrt {x}}{4}-\frac {b\,d^2\,e^6\,n\,x}{8}+\frac {b\,d^7\,e\,n\,x^{7/2}}{28}-\frac {b\,d^4\,e^4\,n\,x^2}{16}-\frac {b\,d^6\,e^2\,n\,x^3}{24}+\frac {b\,d^3\,e^5\,n\,x^{3/2}}{12}+\frac {b\,d^5\,e^3\,n\,x^{5/2}}{20}+\frac {b\,e^8\,n\,\mathrm {atan}\left (\frac {d\,1{}\mathrm {i}+\frac {e\,2{}\mathrm {i}}{\sqrt {x}}}{d}\right )\,1{}\mathrm {i}}{2}}{d^8}+\frac {a\,x^4}{4}+\frac {b\,x^4\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}{4} \]
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