Integrand size = 21, antiderivative size = 90 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=-\frac {b d^3 n}{16 x^4}-\frac {b d^2 e n}{3 x^3}-\frac {3 b d e^2 n}{4 x^2}-\frac {b e^3 n}{x}+\frac {b e^4 n \log (x)}{4 d}-\frac {(d+e x)^4 \left (a+b \log \left (c x^n\right )\right )}{4 d x^4} \]
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Time = 0.05 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {37, 2372, 12, 45} \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=-\frac {(d+e x)^4 \left (a+b \log \left (c x^n\right )\right )}{4 d x^4}-\frac {b d^3 n}{16 x^4}-\frac {b d^2 e n}{3 x^3}+\frac {b e^4 n \log (x)}{4 d}-\frac {3 b d e^2 n}{4 x^2}-\frac {b e^3 n}{x} \]
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Rule 12
Rule 37
Rule 45
Rule 2372
Rubi steps \begin{align*} \text {integral}& = -\frac {(d+e x)^4 \left (a+b \log \left (c x^n\right )\right )}{4 d x^4}-(b n) \int -\frac {(d+e x)^4}{4 d x^5} \, dx \\ & = -\frac {(d+e x)^4 \left (a+b \log \left (c x^n\right )\right )}{4 d x^4}+\frac {(b n) \int \frac {(d+e x)^4}{x^5} \, dx}{4 d} \\ & = -\frac {(d+e x)^4 \left (a+b \log \left (c x^n\right )\right )}{4 d x^4}+\frac {(b n) \int \left (\frac {d^4}{x^5}+\frac {4 d^3 e}{x^4}+\frac {6 d^2 e^2}{x^3}+\frac {4 d e^3}{x^2}+\frac {e^4}{x}\right ) \, dx}{4 d} \\ & = -\frac {b d^3 n}{16 x^4}-\frac {b d^2 e n}{3 x^3}-\frac {3 b d e^2 n}{4 x^2}-\frac {b e^3 n}{x}+\frac {b e^4 n \log (x)}{4 d}-\frac {(d+e x)^4 \left (a+b \log \left (c x^n\right )\right )}{4 d x^4} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.21 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=-\frac {12 a \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )+b n \left (3 d^3+16 d^2 e x+36 d e^2 x^2+48 e^3 x^3\right )+12 b \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right ) \log \left (c x^n\right )}{48 x^4} \]
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Time = 0.42 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.49
method | result | size |
parallelrisch | \(-\frac {48 b \ln \left (c \,x^{n}\right ) e^{3} x^{3}+48 b \,e^{3} n \,x^{3}+48 a \,e^{3} x^{3}+72 b \ln \left (c \,x^{n}\right ) d \,e^{2} x^{2}+36 b d \,e^{2} n \,x^{2}+72 a d \,e^{2} x^{2}+48 b \ln \left (c \,x^{n}\right ) d^{2} e x +16 b \,d^{2} e n x +48 a \,d^{2} e x +12 b \ln \left (c \,x^{n}\right ) d^{3}+3 b \,d^{3} n +12 a \,d^{3}}{48 x^{4}}\) | \(134\) |
risch | \(-\frac {b \left (4 e^{3} x^{3}+6 d \,e^{2} x^{2}+4 d^{2} e x +d^{3}\right ) \ln \left (x^{n}\right )}{4 x^{4}}-\frac {48 \ln \left (c \right ) b \,e^{3} x^{3}-36 i \pi b d \,e^{2} x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-24 i \pi b \,d^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) e x -24 i \pi b \,e^{3} x^{3} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+36 i \pi b d \,e^{2} x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+36 i \pi b d \,e^{2} x^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+24 i \pi b \,d^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} e x +24 i \pi b \,d^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} e x -6 i \pi b \,d^{3} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+12 a \,d^{3}+12 d^{3} b \ln \left (c \right )+48 a \,e^{3} x^{3}-24 i \pi b \,d^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3} e x +24 i \pi b \,e^{3} x^{3} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+24 i \pi b \,e^{3} x^{3} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-36 i \pi b d \,e^{2} x^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+3 b \,d^{3} n -6 i \pi b \,d^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+72 a d \,e^{2} x^{2}+48 a \,d^{2} e x +72 \ln \left (c \right ) b d \,e^{2} x^{2}+48 \ln \left (c \right ) b \,d^{2} e x +16 b \,d^{2} e n x +36 b d \,e^{2} n \,x^{2}+48 b \,e^{3} n \,x^{3}+6 i \pi b \,d^{3} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-24 i \pi b \,e^{3} x^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+6 i \pi b \,d^{3} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{48 x^{4}}\) | \(569\) |
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Time = 0.27 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.69 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=-\frac {3 \, b d^{3} n + 12 \, a d^{3} + 48 \, {\left (b e^{3} n + a e^{3}\right )} x^{3} + 36 \, {\left (b d e^{2} n + 2 \, a d e^{2}\right )} x^{2} + 16 \, {\left (b d^{2} e n + 3 \, a d^{2} e\right )} x + 12 \, {\left (4 \, b e^{3} x^{3} + 6 \, b d e^{2} x^{2} + 4 \, b d^{2} e x + b d^{3}\right )} \log \left (c\right ) + 12 \, {\left (4 \, b e^{3} n x^{3} + 6 \, b d e^{2} n x^{2} + 4 \, b d^{2} e n x + b d^{3} n\right )} \log \left (x\right )}{48 \, x^{4}} \]
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Time = 0.44 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.76 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=- \frac {a d^{3}}{4 x^{4}} - \frac {a d^{2} e}{x^{3}} - \frac {3 a d e^{2}}{2 x^{2}} - \frac {a e^{3}}{x} - \frac {b d^{3} n}{16 x^{4}} - \frac {b d^{3} \log {\left (c x^{n} \right )}}{4 x^{4}} - \frac {b d^{2} e n}{3 x^{3}} - \frac {b d^{2} e \log {\left (c x^{n} \right )}}{x^{3}} - \frac {3 b d e^{2} n}{4 x^{2}} - \frac {3 b d e^{2} \log {\left (c x^{n} \right )}}{2 x^{2}} - \frac {b e^{3} n}{x} - \frac {b e^{3} \log {\left (c x^{n} \right )}}{x} \]
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Time = 0.20 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.59 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=-\frac {b e^{3} n}{x} - \frac {b e^{3} \log \left (c x^{n}\right )}{x} - \frac {3 \, b d e^{2} n}{4 \, x^{2}} - \frac {a e^{3}}{x} - \frac {3 \, b d e^{2} \log \left (c x^{n}\right )}{2 \, x^{2}} - \frac {b d^{2} e n}{3 \, x^{3}} - \frac {3 \, a d e^{2}}{2 \, x^{2}} - \frac {b d^{2} e \log \left (c x^{n}\right )}{x^{3}} - \frac {b d^{3} n}{16 \, x^{4}} - \frac {a d^{2} e}{x^{3}} - \frac {b d^{3} \log \left (c x^{n}\right )}{4 \, x^{4}} - \frac {a d^{3}}{4 \, x^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (80) = 160\).
Time = 0.36 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.80 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=-\frac {{\left (4 \, b e^{3} n x^{3} + 6 \, b d e^{2} n x^{2} + 4 \, b d^{2} e n x + b d^{3} n\right )} \log \left (x\right )}{4 \, x^{4}} - \frac {48 \, b e^{3} n x^{3} + 48 \, b e^{3} x^{3} \log \left (c\right ) + 36 \, b d e^{2} n x^{2} + 48 \, a e^{3} x^{3} + 72 \, b d e^{2} x^{2} \log \left (c\right ) + 16 \, b d^{2} e n x + 72 \, a d e^{2} x^{2} + 48 \, b d^{2} e x \log \left (c\right ) + 3 \, b d^{3} n + 48 \, a d^{2} e x + 12 \, b d^{3} \log \left (c\right ) + 12 \, a d^{3}}{48 \, x^{4}} \]
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Time = 0.41 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.31 \[ \int \frac {(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=-\frac {x^3\,\left (4\,a\,e^3+4\,b\,e^3\,n\right )+x\,\left (4\,a\,d^2\,e+\frac {4\,b\,d^2\,e\,n}{3}\right )+a\,d^3+x^2\,\left (6\,a\,d\,e^2+3\,b\,d\,e^2\,n\right )+\frac {b\,d^3\,n}{4}}{4\,x^4}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d^3}{4}+b\,d^2\,e\,x+\frac {3\,b\,d\,e^2\,x^2}{2}+b\,e^3\,x^3\right )}{x^4} \]
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