Integrand size = 23, antiderivative size = 168 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^4} \, dx=-\frac {2 b^2 d^2 n^2}{27 x^3}-\frac {b^2 d e n^2}{2 x^2}-\frac {2 b^2 e^2 n^2}{x}-\frac {2 b d^2 n \left (a+b \log \left (c x^n\right )\right )}{9 x^3}-\frac {b d e n \left (a+b \log \left (c x^n\right )\right )}{x^2}-\frac {2 b e^2 n \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}-\frac {d e \left (a+b \log \left (c x^n\right )\right )^2}{x^2}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{x} \]
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Time = 0.14 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2395, 2342, 2341} \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^4} \, dx=-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}-\frac {2 b d^2 n \left (a+b \log \left (c x^n\right )\right )}{9 x^3}-\frac {d e \left (a+b \log \left (c x^n\right )\right )^2}{x^2}-\frac {b d e n \left (a+b \log \left (c x^n\right )\right )}{x^2}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {2 b e^2 n \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {2 b^2 d^2 n^2}{27 x^3}-\frac {b^2 d e n^2}{2 x^2}-\frac {2 b^2 e^2 n^2}{x} \]
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Rule 2341
Rule 2342
Rule 2395
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^4}+\frac {2 d e \left (a+b \log \left (c x^n\right )\right )^2}{x^3}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^2}\right ) \, dx \\ & = d^2 \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^4} \, dx+(2 d e) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx+e^2 \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx \\ & = -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}-\frac {d e \left (a+b \log \left (c x^n\right )\right )^2}{x^2}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac {1}{3} \left (2 b d^2 n\right ) \int \frac {a+b \log \left (c x^n\right )}{x^4} \, dx+(2 b d e n) \int \frac {a+b \log \left (c x^n\right )}{x^3} \, dx+\left (2 b e^2 n\right ) \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx \\ & = -\frac {2 b^2 d^2 n^2}{27 x^3}-\frac {b^2 d e n^2}{2 x^2}-\frac {2 b^2 e^2 n^2}{x}-\frac {2 b d^2 n \left (a+b \log \left (c x^n\right )\right )}{9 x^3}-\frac {b d e n \left (a+b \log \left (c x^n\right )\right )}{x^2}-\frac {2 b e^2 n \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}-\frac {d e \left (a+b \log \left (c x^n\right )\right )^2}{x^2}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{x} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.78 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^4} \, dx=-\frac {18 d^2 \left (a+b \log \left (c x^n\right )\right )^2+54 d e x \left (a+b \log \left (c x^n\right )\right )^2+54 e^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2+108 b e^2 n x^2 \left (a+b n+b \log \left (c x^n\right )\right )+27 b d e n x \left (2 a+b n+2 b \log \left (c x^n\right )\right )+4 b d^2 n \left (3 a+b n+3 b \log \left (c x^n\right )\right )}{54 x^3} \]
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Time = 0.72 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.42
method | result | size |
parallelrisch | \(-\frac {54 b^{2} \ln \left (c \,x^{n}\right )^{2} e^{2} x^{2}+108 x^{2} \ln \left (c \,x^{n}\right ) b^{2} e^{2} n +108 b^{2} e^{2} n^{2} x^{2}+108 a b \ln \left (c \,x^{n}\right ) e^{2} x^{2}+108 b n \,x^{2} a \,e^{2}+54 b^{2} \ln \left (c \,x^{n}\right )^{2} d e x +54 b^{2} d e n x \ln \left (c \,x^{n}\right )+27 b^{2} d e \,n^{2} x +54 a^{2} e^{2} x^{2}+108 a b \ln \left (c \,x^{n}\right ) d e x +54 a b d e n x +18 b^{2} \ln \left (c \,x^{n}\right )^{2} d^{2}+12 \ln \left (c \,x^{n}\right ) n \,b^{2} d^{2}+4 b^{2} d^{2} n^{2}+54 a^{2} d e x +36 a b \ln \left (c \,x^{n}\right ) d^{2}+12 b \,d^{2} n a +18 a^{2} d^{2}}{54 x^{3}}\) | \(238\) |
risch | \(\text {Expression too large to display}\) | \(2473\) |
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Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (160) = 320\).
Time = 0.31 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.94 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^4} \, dx=-\frac {4 \, b^{2} d^{2} n^{2} + 12 \, a b d^{2} n + 18 \, a^{2} d^{2} + 54 \, {\left (2 \, b^{2} e^{2} n^{2} + 2 \, a b e^{2} n + a^{2} e^{2}\right )} x^{2} + 18 \, {\left (3 \, b^{2} e^{2} x^{2} + 3 \, b^{2} d e x + b^{2} d^{2}\right )} \log \left (c\right )^{2} + 18 \, {\left (3 \, b^{2} e^{2} n^{2} x^{2} + 3 \, b^{2} d e n^{2} x + b^{2} d^{2} n^{2}\right )} \log \left (x\right )^{2} + 27 \, {\left (b^{2} d e n^{2} + 2 \, a b d e n + 2 \, a^{2} d e\right )} x + 6 \, {\left (2 \, b^{2} d^{2} n + 6 \, a b d^{2} + 18 \, {\left (b^{2} e^{2} n + a b e^{2}\right )} x^{2} + 9 \, {\left (b^{2} d e n + 2 \, a b d e\right )} x\right )} \log \left (c\right ) + 6 \, {\left (2 \, b^{2} d^{2} n^{2} + 6 \, a b d^{2} n + 18 \, {\left (b^{2} e^{2} n^{2} + a b e^{2} n\right )} x^{2} + 9 \, {\left (b^{2} d e n^{2} + 2 \, a b d e n\right )} x + 6 \, {\left (3 \, b^{2} e^{2} n x^{2} + 3 \, b^{2} d e n x + b^{2} d^{2} n\right )} \log \left (c\right )\right )} \log \left (x\right )}{54 \, x^{3}} \]
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Time = 0.33 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.71 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^4} \, dx=- \frac {a^{2} d^{2}}{3 x^{3}} - \frac {a^{2} d e}{x^{2}} - \frac {a^{2} e^{2}}{x} - \frac {2 a b d^{2} n}{9 x^{3}} - \frac {2 a b d^{2} \log {\left (c x^{n} \right )}}{3 x^{3}} - \frac {a b d e n}{x^{2}} - \frac {2 a b d e \log {\left (c x^{n} \right )}}{x^{2}} - \frac {2 a b e^{2} n}{x} - \frac {2 a b e^{2} \log {\left (c x^{n} \right )}}{x} - \frac {2 b^{2} d^{2} n^{2}}{27 x^{3}} - \frac {2 b^{2} d^{2} n \log {\left (c x^{n} \right )}}{9 x^{3}} - \frac {b^{2} d^{2} \log {\left (c x^{n} \right )}^{2}}{3 x^{3}} - \frac {b^{2} d e n^{2}}{2 x^{2}} - \frac {b^{2} d e n \log {\left (c x^{n} \right )}}{x^{2}} - \frac {b^{2} d e \log {\left (c x^{n} \right )}^{2}}{x^{2}} - \frac {2 b^{2} e^{2} n^{2}}{x} - \frac {2 b^{2} e^{2} n \log {\left (c x^{n} \right )}}{x} - \frac {b^{2} e^{2} \log {\left (c x^{n} \right )}^{2}}{x} \]
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Time = 0.21 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.49 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^4} \, dx=-2 \, b^{2} e^{2} {\left (\frac {n^{2}}{x} + \frac {n \log \left (c x^{n}\right )}{x}\right )} - \frac {1}{2} \, b^{2} d e {\left (\frac {n^{2}}{x^{2}} + \frac {2 \, n \log \left (c x^{n}\right )}{x^{2}}\right )} - \frac {2}{27} \, b^{2} d^{2} {\left (\frac {n^{2}}{x^{3}} + \frac {3 \, n \log \left (c x^{n}\right )}{x^{3}}\right )} - \frac {b^{2} e^{2} \log \left (c x^{n}\right )^{2}}{x} - \frac {2 \, a b e^{2} n}{x} - \frac {2 \, a b e^{2} \log \left (c x^{n}\right )}{x} - \frac {b^{2} d e \log \left (c x^{n}\right )^{2}}{x^{2}} - \frac {a b d e n}{x^{2}} - \frac {a^{2} e^{2}}{x} - \frac {2 \, a b d e \log \left (c x^{n}\right )}{x^{2}} - \frac {b^{2} d^{2} \log \left (c x^{n}\right )^{2}}{3 \, x^{3}} - \frac {2 \, a b d^{2} n}{9 \, x^{3}} - \frac {a^{2} d e}{x^{2}} - \frac {2 \, a b d^{2} \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac {a^{2} d^{2}}{3 \, x^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 354 vs. \(2 (160) = 320\).
Time = 0.36 (sec) , antiderivative size = 354, normalized size of antiderivative = 2.11 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^4} \, dx=-\frac {{\left (3 \, b^{2} e^{2} n^{2} x^{2} + 3 \, b^{2} d e n^{2} x + b^{2} d^{2} n^{2}\right )} \log \left (x\right )^{2}}{3 \, x^{3}} - \frac {{\left (18 \, b^{2} e^{2} n^{2} x^{2} + 18 \, b^{2} e^{2} n x^{2} \log \left (c\right ) + 9 \, b^{2} d e n^{2} x + 18 \, a b e^{2} n x^{2} + 18 \, b^{2} d e n x \log \left (c\right ) + 2 \, b^{2} d^{2} n^{2} + 18 \, a b d e n x + 6 \, b^{2} d^{2} n \log \left (c\right ) + 6 \, a b d^{2} n\right )} \log \left (x\right )}{9 \, x^{3}} - \frac {108 \, b^{2} e^{2} n^{2} x^{2} + 108 \, b^{2} e^{2} n x^{2} \log \left (c\right ) + 54 \, b^{2} e^{2} x^{2} \log \left (c\right )^{2} + 27 \, b^{2} d e n^{2} x + 108 \, a b e^{2} n x^{2} + 54 \, b^{2} d e n x \log \left (c\right ) + 108 \, a b e^{2} x^{2} \log \left (c\right ) + 54 \, b^{2} d e x \log \left (c\right )^{2} + 4 \, b^{2} d^{2} n^{2} + 54 \, a b d e n x + 54 \, a^{2} e^{2} x^{2} + 12 \, b^{2} d^{2} n \log \left (c\right ) + 108 \, a b d e x \log \left (c\right ) + 18 \, b^{2} d^{2} \log \left (c\right )^{2} + 12 \, a b d^{2} n + 54 \, a^{2} d e x + 36 \, a b d^{2} \log \left (c\right ) + 18 \, a^{2} d^{2}}{54 \, x^{3}} \]
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Time = 0.60 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.10 \[ \int \frac {(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^4} \, dx=-\frac {x\,\left (9\,d\,e\,a^2+9\,d\,e\,a\,b\,n+\frac {9\,d\,e\,b^2\,n^2}{2}\right )+x^2\,\left (9\,a^2\,e^2+18\,a\,b\,e^2\,n+18\,b^2\,e^2\,n^2\right )+3\,a^2\,d^2+\frac {2\,b^2\,d^2\,n^2}{3}+2\,a\,b\,d^2\,n}{9\,x^3}-\frac {{\ln \left (c\,x^n\right )}^2\,\left (\frac {b^2\,d^2}{3}+b^2\,d\,e\,x+b^2\,e^2\,x^2\right )}{x^3}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {2\,b\,\left (3\,a+b\,n\right )\,d^2}{3}+3\,b\,\left (2\,a+b\,n\right )\,d\,e\,x+6\,b\,\left (a+b\,n\right )\,e^2\,x^2\right )}{3\,x^3} \]
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