Integrand size = 21, antiderivative size = 79 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=\frac {b d n}{6 e^3 (d+e x)^2}-\frac {2 b n}{3 e^3 (d+e x)}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 d (d+e x)^3}-\frac {b n \log (d+e x)}{3 d e^3} \]
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Time = 0.05 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2373, 45} \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 d (d+e x)^3}-\frac {2 b n}{3 e^3 (d+e x)}+\frac {b d n}{6 e^3 (d+e x)^2}-\frac {b n \log (d+e x)}{3 d e^3} \]
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Rule 45
Rule 2373
Rubi steps \begin{align*} \text {integral}& = \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 d (d+e x)^3}-\frac {(b n) \int \frac {x^2}{(d+e x)^3} \, dx}{3 d} \\ & = \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 d (d+e x)^3}-\frac {(b n) \int \left (\frac {d^2}{e^2 (d+e x)^3}-\frac {2 d}{e^2 (d+e x)^2}+\frac {1}{e^2 (d+e x)}\right ) \, dx}{3 d} \\ & = \frac {b d n}{6 e^3 (d+e x)^2}-\frac {2 b n}{3 e^3 (d+e x)}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{3 d (d+e x)^3}-\frac {b n \log (d+e x)}{3 d e^3} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(172\) vs. \(2(79)=158\).
Time = 0.08 (sec) , antiderivative size = 172, normalized size of antiderivative = 2.18 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=-\frac {a d^2}{3 e^3 (d+e x)^3}+\frac {a d}{e^3 (d+e x)^2}+\frac {b d n}{6 e^3 (d+e x)^2}-\frac {a}{e^3 (d+e x)}-\frac {2 b n}{3 e^3 (d+e x)}+\frac {b n \log (x)}{3 d e^3}-\frac {b d^2 \log \left (c x^n\right )}{3 e^3 (d+e x)^3}+\frac {b d \log \left (c x^n\right )}{e^3 (d+e x)^2}-\frac {b \log \left (c x^n\right )}{e^3 (d+e x)}-\frac {b n \log (d+e x)}{3 d e^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(164\) vs. \(2(71)=142\).
Time = 0.51 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.09
method | result | size |
parallelrisch | \(\frac {-2 \ln \left (e x +d \right ) x^{3} b \,e^{3} n^{2}-6 \ln \left (e x +d \right ) x^{2} b d \,e^{2} n^{2}+2 x^{3} \ln \left (c \,x^{n}\right ) b \,e^{3} n -6 \ln \left (e x +d \right ) x b \,d^{2} e \,n^{2}-4 x^{2} b d \,e^{2} n^{2}-2 \ln \left (e x +d \right ) b \,d^{3} n^{2}-6 x^{2} a d \,e^{2} n -7 x b \,d^{2} e \,n^{2}-6 x a \,d^{2} e n -3 b \,d^{3} n^{2}-2 a \,d^{3} n}{6 n d \,e^{3} \left (e x +d \right )^{3}}\) | \(165\) |
risch | \(-\frac {b \left (3 e^{2} x^{2}+3 d e x +d^{2}\right ) \ln \left (x^{n}\right )}{3 \left (e x +d \right )^{3} e^{3}}-\frac {-6 \ln \left (-x \right ) b d \,e^{2} n \,x^{2}-6 \ln \left (-x \right ) b \,d^{2} e n x -3 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 )-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 ) e x +3 i \pi b d \,e^{2} x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+3 i \pi b d \,e^{2} 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 c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} e x +3 i \pi b \,d^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} e x -2 \ln \left (-x \right ) b \,e^{3} n \,x^{3}-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 )+2 a \,d^{3}+2 d^{3} b \ln \left (c \right )+6 \ln \left (e x +d \right ) b \,d^{2} e n x -3 i \pi b \,d^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3} e x -3 i \pi b d \,e^{2} x^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+3 b \,d^{3} n -i \pi b \,d^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-2 \ln \left (-x \right ) b \,d^{3} n +2 \ln \left (e x +d \right ) b \,d^{3} n +6 a d \,e^{2} x^{2}+6 a \,d^{2} e x +6 \ln \left (c \right ) b d \,e^{2} x^{2}+6 \ln \left (c \right ) b \,d^{2} e x +2 \ln \left (e x +d \right ) b \,e^{3} n \,x^{3}+6 \ln \left (e x +d \right ) b d \,e^{2} n \,x^{2}+7 b \,d^{2} e n x +4 b d \,e^{2} n \,x^{2}+i \pi b \,d^{3} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b \,d^{3} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{6 d \,e^{3} \left (e x +d \right )^{3}}\) | \(553\) |
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Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (71) = 142\).
Time = 0.30 (sec) , antiderivative size = 178, normalized size of antiderivative = 2.25 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=\frac {2 \, b e^{3} n x^{3} \log \left (x\right ) - 3 \, b d^{3} n - 2 \, a d^{3} - 2 \, {\left (2 \, b d e^{2} n + 3 \, a d e^{2}\right )} x^{2} - {\left (7 \, b d^{2} e n + 6 \, a d^{2} e\right )} x - 2 \, {\left (b e^{3} n x^{3} + 3 \, b d e^{2} n x^{2} + 3 \, b d^{2} e n x + b d^{3} n\right )} \log \left (e x + d\right ) - 2 \, {\left (3 \, b d e^{2} x^{2} + 3 \, b d^{2} e x + b d^{3}\right )} \log \left (c\right )}{6 \, {\left (d e^{6} x^{3} + 3 \, d^{2} e^{5} x^{2} + 3 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 677 vs. \(2 (71) = 142\).
Time = 5.38 (sec) , antiderivative size = 677, normalized size of antiderivative = 8.57 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {a}{x} - \frac {b n}{x} - \frac {b \log {\left (c x^{n} \right )}}{x}\right ) & \text {for}\: d = 0 \wedge e = 0 \\\frac {\frac {a x^{3}}{3} - \frac {b n x^{3}}{9} + \frac {b x^{3} \log {\left (c x^{n} \right )}}{3}}{d^{4}} & \text {for}\: e = 0 \\\frac {- \frac {a}{x} - \frac {b n}{x} - \frac {b \log {\left (c x^{n} \right )}}{x}}{e^{4}} & \text {for}\: d = 0 \\- \frac {2 a d^{3}}{6 d^{4} e^{3} + 18 d^{3} e^{4} x + 18 d^{2} e^{5} x^{2} + 6 d e^{6} x^{3}} - \frac {6 a d^{2} e x}{6 d^{4} e^{3} + 18 d^{3} e^{4} x + 18 d^{2} e^{5} x^{2} + 6 d e^{6} x^{3}} - \frac {6 a d e^{2} x^{2}}{6 d^{4} e^{3} + 18 d^{3} e^{4} x + 18 d^{2} e^{5} x^{2} + 6 d e^{6} x^{3}} - \frac {2 b d^{3} n \log {\left (\frac {d}{e} + x \right )}}{6 d^{4} e^{3} + 18 d^{3} e^{4} x + 18 d^{2} e^{5} x^{2} + 6 d e^{6} x^{3}} - \frac {3 b d^{3} n}{6 d^{4} e^{3} + 18 d^{3} e^{4} x + 18 d^{2} e^{5} x^{2} + 6 d e^{6} x^{3}} - \frac {6 b d^{2} e n x \log {\left (\frac {d}{e} + x \right )}}{6 d^{4} e^{3} + 18 d^{3} e^{4} x + 18 d^{2} e^{5} x^{2} + 6 d e^{6} x^{3}} - \frac {7 b d^{2} e n x}{6 d^{4} e^{3} + 18 d^{3} e^{4} x + 18 d^{2} e^{5} x^{2} + 6 d e^{6} x^{3}} - \frac {6 b d e^{2} n x^{2} \log {\left (\frac {d}{e} + x \right )}}{6 d^{4} e^{3} + 18 d^{3} e^{4} x + 18 d^{2} e^{5} x^{2} + 6 d e^{6} x^{3}} - \frac {4 b d e^{2} n x^{2}}{6 d^{4} e^{3} + 18 d^{3} e^{4} x + 18 d^{2} e^{5} x^{2} + 6 d e^{6} x^{3}} - \frac {2 b e^{3} n x^{3} \log {\left (\frac {d}{e} + x \right )}}{6 d^{4} e^{3} + 18 d^{3} e^{4} x + 18 d^{2} e^{5} x^{2} + 6 d e^{6} x^{3}} + \frac {2 b e^{3} x^{3} \log {\left (c x^{n} \right )}}{6 d^{4} e^{3} + 18 d^{3} e^{4} x + 18 d^{2} e^{5} x^{2} + 6 d e^{6} x^{3}} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 179 vs. \(2 (71) = 142\).
Time = 0.20 (sec) , antiderivative size = 179, normalized size of antiderivative = 2.27 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=-\frac {1}{6} \, b n {\left (\frac {4 \, e x + 3 \, d}{e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}} + \frac {2 \, \log \left (e x + d\right )}{d e^{3}} - \frac {2 \, \log \left (x\right )}{d e^{3}}\right )} - \frac {{\left (3 \, e^{2} x^{2} + 3 \, d e x + d^{2}\right )} b \log \left (c x^{n}\right )}{3 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} - \frac {{\left (3 \, e^{2} x^{2} + 3 \, d e x + d^{2}\right )} a}{3 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (71) = 142\).
Time = 0.38 (sec) , antiderivative size = 202, normalized size of antiderivative = 2.56 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=-\frac {{\left (3 \, b e^{2} n x^{2} + 3 \, b d e n x + b d^{2} n\right )} \log \left (x\right )}{3 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} - \frac {4 \, b e^{2} n x^{2} + 6 \, b e^{2} x^{2} \log \left (c\right ) + 7 \, b d e n x + 6 \, a e^{2} x^{2} + 6 \, b d e x \log \left (c\right ) + 3 \, b d^{2} n + 6 \, a d e x + 2 \, b d^{2} \log \left (c\right ) + 2 \, a d^{2}}{6 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} - \frac {b n \log \left (e x + d\right )}{3 \, d e^{3}} + \frac {b n \log \left (x\right )}{3 \, d e^{3}} \]
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Time = 0.78 (sec) , antiderivative size = 167, normalized size of antiderivative = 2.11 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=-\frac {x^2\,\left (3\,a\,e^2+2\,b\,e^2\,n\right )+a\,d^2+x\,\left (3\,a\,d\,e+\frac {7\,b\,d\,e\,n}{2}\right )+\frac {3\,b\,d^2\,n}{2}}{3\,d^3\,e^3+9\,d^2\,e^4\,x+9\,d\,e^5\,x^2+3\,e^6\,x^3}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d^2}{3\,e^3}+\frac {b\,x^2}{e}+\frac {b\,d\,x}{e^2}\right )}{d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3}-\frac {2\,b\,n\,\mathrm {atanh}\left (\frac {2\,e\,x}{d}+1\right )}{3\,d\,e^3} \]
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