Integrand size = 23, antiderivative size = 82 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=-\frac {b d^2 n}{9 x^3}-\frac {2 b d e n}{x}-b e^2 n x-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {2 d e \left (a+b \log \left (c x^n\right )\right )}{x}+e^2 x \left (a+b \log \left (c x^n\right )\right ) \]
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Time = 0.06 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {276, 2372} \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {2 d e \left (a+b \log \left (c x^n\right )\right )}{x}+e^2 x \left (a+b \log \left (c x^n\right )\right )-\frac {b d^2 n}{9 x^3}-\frac {2 b d e n}{x}-b e^2 n x \]
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Rule 276
Rule 2372
Rubi steps \begin{align*} \text {integral}& = -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {2 d e \left (a+b \log \left (c x^n\right )\right )}{x}+e^2 x \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (e^2-\frac {d^2}{3 x^4}-\frac {2 d e}{x^2}\right ) \, dx \\ & = -\frac {b d^2 n}{9 x^3}-\frac {2 b d e n}{x}-b e^2 n x-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {2 d e \left (a+b \log \left (c x^n\right )\right )}{x}+e^2 x \left (a+b \log \left (c x^n\right )\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.98 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=-\frac {3 a \left (d^2+6 d e x^2-3 e^2 x^4\right )+b n \left (d^2+18 d e x^2+9 e^2 x^4\right )+3 b \left (d^2+6 d e x^2-3 e^2 x^4\right ) \log \left (c x^n\right )}{9 x^3} \]
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Time = 0.47 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.17
method | result | size |
parallelrisch | \(-\frac {-9 x^{4} b \ln \left (c \,x^{n}\right ) e^{2}+9 b \,e^{2} n \,x^{4}-9 x^{4} a \,e^{2}+18 b \ln \left (c \,x^{n}\right ) d e \,x^{2}+18 b d e n \,x^{2}+18 a d e \,x^{2}+3 b \ln \left (c \,x^{n}\right ) d^{2}+b \,d^{2} n +3 a \,d^{2}}{9 x^{3}}\) | \(96\) |
risch | \(-\frac {b \left (-3 e^{2} x^{4}+6 d e \,x^{2}+d^{2}\right ) \ln \left (x^{n}\right )}{3 x^{3}}-\frac {-9 i \pi b \,e^{2} x^{4} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+3 i \pi b \,d^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-18 i \pi b d e \,x^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+9 i \pi b \,e^{2} x^{4} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-3 i \pi b \,d^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-9 i \pi b \,e^{2} x^{4} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+18 i \pi b d e \,x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+9 i \pi b \,e^{2} x^{4} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-3 i \pi b \,d^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+18 i \pi b d e \,x^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+3 i \pi b \,d^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-18 i \pi b d e \,x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-18 \ln \left (c \right ) b \,e^{2} x^{4}+18 b \,e^{2} n \,x^{4}-18 x^{4} a \,e^{2}+36 e \ln \left (c \right ) b d \,x^{2}+36 b d e n \,x^{2}+36 a d e \,x^{2}+6 d^{2} b \ln \left (c \right )+2 b \,d^{2} n +6 a \,d^{2}}{18 x^{3}}\) | \(417\) |
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Time = 0.32 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.34 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=-\frac {9 \, {\left (b e^{2} n - a e^{2}\right )} x^{4} + b d^{2} n + 3 \, a d^{2} + 18 \, {\left (b d e n + a d e\right )} x^{2} - 3 \, {\left (3 \, b e^{2} x^{4} - 6 \, b d e x^{2} - b d^{2}\right )} \log \left (c\right ) - 3 \, {\left (3 \, b e^{2} n x^{4} - 6 \, b d e n x^{2} - b d^{2} n\right )} \log \left (x\right )}{9 \, x^{3}} \]
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Time = 0.41 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.22 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=- \frac {a d^{2}}{3 x^{3}} - \frac {2 a d e}{x} + a e^{2} x - \frac {b d^{2} n}{9 x^{3}} - \frac {b d^{2} \log {\left (c x^{n} \right )}}{3 x^{3}} - \frac {2 b d e n}{x} - \frac {2 b d e \log {\left (c x^{n} \right )}}{x} - b e^{2} n x + b e^{2} x \log {\left (c x^{n} \right )} \]
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Time = 0.19 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.12 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=-b e^{2} n x + b e^{2} x \log \left (c x^{n}\right ) + a e^{2} x - \frac {2 \, b d e n}{x} - \frac {2 \, b d e \log \left (c x^{n}\right )}{x} - \frac {2 \, a d e}{x} - \frac {b d^{2} n}{9 \, x^{3}} - \frac {b d^{2} \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac {a d^{2}}{3 \, x^{3}} \]
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Time = 0.29 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.37 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=-{\left (b e^{2} n - b e^{2} \log \left (c\right ) - a e^{2}\right )} x + \frac {1}{3} \, {\left (3 \, b e^{2} n x - \frac {6 \, b d e n x^{2} + b d^{2} n}{x^{3}}\right )} \log \left (x\right ) - \frac {18 \, b d e n x^{2} + 18 \, b d e x^{2} \log \left (c\right ) + 18 \, a d e x^{2} + b d^{2} n + 3 \, b d^{2} \log \left (c\right ) + 3 \, a d^{2}}{9 \, x^{3}} \]
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Time = 0.39 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.10 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx=e^2\,x\,\left (a-b\,n\right )-\frac {x^2\,\left (6\,a\,d\,e+6\,b\,d\,e\,n\right )+a\,d^2+\frac {b\,d^2\,n}{3}}{3\,x^3}-\ln \left (c\,x^n\right )\,\left (\frac {\frac {b\,d^2}{3}+2\,b\,d\,e\,x^2+\frac {5\,b\,e^2\,x^4}{3}}{x^3}-\frac {8\,b\,e^2\,x}{3}\right ) \]
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