Integrand size = 21, antiderivative size = 95 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=-\frac {b d^2 n}{16 x^4}-\frac {2 b d e n}{9 x^3}-\frac {b e^2 n}{4 x^2}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {2 d e \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{2 x^2} \]
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Time = 0.05 (sec) , antiderivative size = 95, 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 = {45, 2372, 12, 14} \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {2 d e \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b d^2 n}{16 x^4}-\frac {2 b d e n}{9 x^3}-\frac {b e^2 n}{4 x^2} \]
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Rule 12
Rule 14
Rule 45
Rule 2372
Rubi steps \begin{align*} \text {integral}& = -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {2 d e \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-(b n) \int \frac {-3 d^2-8 d e x-6 e^2 x^2}{12 x^5} \, dx \\ & = -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {2 d e \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {1}{12} (b n) \int \frac {-3 d^2-8 d e x-6 e^2 x^2}{x^5} \, dx \\ & = -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {2 d e \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {1}{12} (b n) \int \left (-\frac {3 d^2}{x^5}-\frac {8 d e}{x^4}-\frac {6 e^2}{x^3}\right ) \, dx \\ & = -\frac {b d^2 n}{16 x^4}-\frac {2 b d e n}{9 x^3}-\frac {b e^2 n}{4 x^2}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac {2 d e \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{2 x^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.84 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=-\frac {12 a \left (3 d^2+8 d e x+6 e^2 x^2\right )+b n \left (9 d^2+32 d e x+36 e^2 x^2\right )+12 b \left (3 d^2+8 d e x+6 e^2 x^2\right ) \log \left (c x^n\right )}{144 x^4} \]
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Time = 0.58 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.96
method | result | size |
parallelrisch | \(-\frac {72 b \ln \left (c \,x^{n}\right ) e^{2} x^{2}+36 b \,e^{2} n \,x^{2}+72 a \,e^{2} x^{2}+96 b \ln \left (c \,x^{n}\right ) d e x +32 b d e n x +96 a d e x +36 b \ln \left (c \,x^{n}\right ) d^{2}+9 b \,d^{2} n +36 a \,d^{2}}{144 x^{4}}\) | \(91\) |
risch | \(-\frac {b \left (6 e^{2} x^{2}+8 d e x +3 d^{2}\right ) \ln \left (x^{n}\right )}{12 x^{4}}-\frac {18 i \pi b \,d^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-36 i \pi b \,e^{2} x^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-48 i \pi b d e x \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+48 i \pi b d e x \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+72 \ln \left (c \right ) b \,e^{2} x^{2}+36 b \,e^{2} n \,x^{2}+72 a \,e^{2} x^{2}+48 i \pi b d e x \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+36 i \pi b \,e^{2} x^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-18 i \pi b \,d^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-36 i \pi b \,e^{2} x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+96 \ln \left (c \right ) b d e x +32 b d e n x +96 a d e x -48 i \pi b d e x \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+36 i \pi b \,e^{2} x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+18 i \pi b \,d^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-18 i \pi b \,d^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+36 d^{2} b \ln \left (c \right )+9 b \,d^{2} n +36 a \,d^{2}}{144 x^{4}}\) | \(403\) |
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Time = 0.28 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.12 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=-\frac {9 \, b d^{2} n + 36 \, a d^{2} + 36 \, {\left (b e^{2} n + 2 \, a e^{2}\right )} x^{2} + 32 \, {\left (b d e n + 3 \, a d e\right )} x + 12 \, {\left (6 \, b e^{2} x^{2} + 8 \, b d e x + 3 \, b d^{2}\right )} \log \left (c\right ) + 12 \, {\left (6 \, b e^{2} n x^{2} + 8 \, b d e n x + 3 \, b d^{2} n\right )} \log \left (x\right )}{144 \, x^{4}} \]
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Time = 0.41 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.28 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=- \frac {a d^{2}}{4 x^{4}} - \frac {2 a d e}{3 x^{3}} - \frac {a e^{2}}{2 x^{2}} - \frac {b d^{2} n}{16 x^{4}} - \frac {b d^{2} \log {\left (c x^{n} \right )}}{4 x^{4}} - \frac {2 b d e n}{9 x^{3}} - \frac {2 b d e \log {\left (c x^{n} \right )}}{3 x^{3}} - \frac {b e^{2} n}{4 x^{2}} - \frac {b e^{2} \log {\left (c x^{n} \right )}}{2 x^{2}} \]
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Time = 0.19 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.05 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=-\frac {b e^{2} n}{4 \, x^{2}} - \frac {b e^{2} \log \left (c x^{n}\right )}{2 \, x^{2}} - \frac {2 \, b d e n}{9 \, x^{3}} - \frac {a e^{2}}{2 \, x^{2}} - \frac {2 \, b d e \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac {b d^{2} n}{16 \, x^{4}} - \frac {2 \, a d e}{3 \, x^{3}} - \frac {b d^{2} \log \left (c x^{n}\right )}{4 \, x^{4}} - \frac {a d^{2}}{4 \, x^{4}} \]
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Time = 0.31 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.17 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=-\frac {{\left (6 \, b e^{2} n x^{2} + 8 \, b d e n x + 3 \, b d^{2} n\right )} \log \left (x\right )}{12 \, x^{4}} - \frac {36 \, b e^{2} n x^{2} + 72 \, b e^{2} x^{2} \log \left (c\right ) + 32 \, b d e n x + 72 \, a e^{2} x^{2} + 96 \, b d e x \log \left (c\right ) + 9 \, b d^{2} n + 96 \, a d e x + 36 \, b d^{2} \log \left (c\right ) + 36 \, a d^{2}}{144 \, x^{4}} \]
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Time = 0.43 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.89 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx=-\frac {x^2\,\left (6\,a\,e^2+3\,b\,e^2\,n\right )+3\,a\,d^2+x\,\left (8\,a\,d\,e+\frac {8\,b\,d\,e\,n}{3}\right )+\frac {3\,b\,d^2\,n}{4}}{12\,x^4}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d^2}{4}+\frac {2\,b\,d\,e\,x}{3}+\frac {b\,e^2\,x^2}{2}\right )}{x^4} \]
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