Integrand size = 21, antiderivative size = 226 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=-\frac {b d^2 n}{30 e^4 (d+e x)^5}+\frac {13 b d n}{120 e^4 (d+e x)^4}-\frac {19 b n}{180 e^4 (d+e x)^3}+\frac {b n}{120 d e^4 (d+e x)^2}+\frac {b n}{60 d^2 e^4 (d+e x)}+\frac {b n \log (x)}{60 d^3 e^4}+\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{6 e^4 (d+e x)^6}-\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )}{5 e^4 (d+e x)^5}+\frac {3 d \left (a+b \log \left (c x^n\right )\right )}{4 e^4 (d+e x)^4}-\frac {a+b \log \left (c x^n\right )}{3 e^4 (d+e x)^3}-\frac {b n \log (d+e x)}{60 d^3 e^4} \]
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Time = 0.14 (sec) , antiderivative size = 226, 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, 2382, 12, 1634} \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{6 e^4 (d+e x)^6}-\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )}{5 e^4 (d+e x)^5}+\frac {3 d \left (a+b \log \left (c x^n\right )\right )}{4 e^4 (d+e x)^4}-\frac {a+b \log \left (c x^n\right )}{3 e^4 (d+e x)^3}+\frac {b n \log (x)}{60 d^3 e^4}-\frac {b n \log (d+e x)}{60 d^3 e^4}-\frac {b d^2 n}{30 e^4 (d+e x)^5}+\frac {b n}{60 d^2 e^4 (d+e x)}+\frac {13 b d n}{120 e^4 (d+e x)^4}-\frac {19 b n}{180 e^4 (d+e x)^3}+\frac {b n}{120 d e^4 (d+e x)^2} \]
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Rule 12
Rule 45
Rule 1634
Rule 2382
Rubi steps \begin{align*} \text {integral}& = \frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{6 e^4 (d+e x)^6}-\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )}{5 e^4 (d+e x)^5}+\frac {3 d \left (a+b \log \left (c x^n\right )\right )}{4 e^4 (d+e x)^4}-\frac {a+b \log \left (c x^n\right )}{3 e^4 (d+e x)^3}-(b n) \int \frac {-d^3-6 d^2 e x-15 d e^2 x^2-20 e^3 x^3}{60 e^4 x (d+e x)^6} \, dx \\ & = \frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{6 e^4 (d+e x)^6}-\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )}{5 e^4 (d+e x)^5}+\frac {3 d \left (a+b \log \left (c x^n\right )\right )}{4 e^4 (d+e x)^4}-\frac {a+b \log \left (c x^n\right )}{3 e^4 (d+e x)^3}-\frac {(b n) \int \frac {-d^3-6 d^2 e x-15 d e^2 x^2-20 e^3 x^3}{x (d+e x)^6} \, dx}{60 e^4} \\ & = \frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{6 e^4 (d+e x)^6}-\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )}{5 e^4 (d+e x)^5}+\frac {3 d \left (a+b \log \left (c x^n\right )\right )}{4 e^4 (d+e x)^4}-\frac {a+b \log \left (c x^n\right )}{3 e^4 (d+e x)^3}-\frac {(b n) \int \left (-\frac {1}{d^3 x}-\frac {10 d^2 e}{(d+e x)^6}+\frac {26 d e}{(d+e x)^5}-\frac {19 e}{(d+e x)^4}+\frac {e}{d (d+e x)^3}+\frac {e}{d^2 (d+e x)^2}+\frac {e}{d^3 (d+e x)}\right ) \, dx}{60 e^4} \\ & = -\frac {b d^2 n}{30 e^4 (d+e x)^5}+\frac {13 b d n}{120 e^4 (d+e x)^4}-\frac {19 b n}{180 e^4 (d+e x)^3}+\frac {b n}{120 d e^4 (d+e x)^2}+\frac {b n}{60 d^2 e^4 (d+e x)}+\frac {b n \log (x)}{60 d^3 e^4}+\frac {d^3 \left (a+b \log \left (c x^n\right )\right )}{6 e^4 (d+e x)^6}-\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )}{5 e^4 (d+e x)^5}+\frac {3 d \left (a+b \log \left (c x^n\right )\right )}{4 e^4 (d+e x)^4}-\frac {a+b \log \left (c x^n\right )}{3 e^4 (d+e x)^3}-\frac {b n \log (d+e x)}{60 d^3 e^4} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.24 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\frac {a d^3}{6 e^4 (d+e x)^6}-\frac {3 a d^2}{5 e^4 (d+e x)^5}-\frac {b d^2 n}{30 e^4 (d+e x)^5}+\frac {3 a d}{4 e^4 (d+e x)^4}+\frac {13 b d n}{120 e^4 (d+e x)^4}-\frac {a}{3 e^4 (d+e x)^3}-\frac {19 b n}{180 e^4 (d+e x)^3}+\frac {b n}{120 d e^4 (d+e x)^2}+\frac {b n}{60 d^2 e^4 (d+e x)}+\frac {b n \log (x)}{60 d^3 e^4}+\frac {b d^3 \log \left (c x^n\right )}{6 e^4 (d+e x)^6}-\frac {3 b d^2 \log \left (c x^n\right )}{5 e^4 (d+e x)^5}+\frac {3 b d \log \left (c x^n\right )}{4 e^4 (d+e x)^4}-\frac {b \log \left (c x^n\right )}{3 e^4 (d+e x)^3}-\frac {b n \log (d+e x)}{60 d^3 e^4} \]
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Time = 1.34 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.83
method | result | size |
parallelrisch | \(\frac {-450 x^{2} \ln \left (c \,x^{n}\right ) b \,d^{4} e^{4}-600 x^{3} \ln \left (c \,x^{n}\right ) b \,d^{3} e^{5}-36 x^{5} b d \,e^{7} n -96 x b \,d^{5} e^{3} n -150 x^{2} b \,d^{4} e^{4} n -50 x^{3} b \,d^{3} e^{5} n +30 \ln \left (x \right ) x^{6} b \,e^{8} n -30 \ln \left (e x +d \right ) x^{6} b \,e^{8} n +30 \ln \left (x \right ) b \,d^{6} e^{2} n -30 \ln \left (e x +d \right ) b \,d^{6} e^{2} n -180 x \ln \left (c \,x^{n}\right ) b \,d^{5} e^{3}-30 a \,d^{6} e^{2}+180 \ln \left (x \right ) x^{5} b d \,e^{7} n -180 \ln \left (e x +d \right ) x^{5} b d \,e^{7} n +450 \ln \left (x \right ) x^{4} b \,d^{2} e^{6} n -450 \ln \left (e x +d \right ) x^{4} b \,d^{2} e^{6} n +600 \ln \left (x \right ) x^{3} b \,d^{3} e^{5} n -600 \ln \left (e x +d \right ) x^{3} b \,d^{3} e^{5} n +450 \ln \left (x \right ) x^{2} b \,d^{4} e^{4} n -450 \ln \left (e x +d \right ) x^{2} b \,d^{4} e^{4} n +180 \ln \left (x \right ) x b \,d^{5} e^{3} n -180 \ln \left (e x +d \right ) x b \,d^{5} e^{3} n -21 b \,d^{6} e^{2} n -30 \ln \left (c \,x^{n}\right ) b \,d^{6} e^{2}-11 x^{6} b \,e^{8} n -180 x a \,d^{5} e^{3}-450 x^{2} a \,d^{4} e^{4}-600 x^{3} a \,d^{3} e^{5}}{1800 d^{3} e^{6} \left (e x +d \right )^{6}}\) | \(413\) |
risch | \(-\frac {b \left (20 e^{3} x^{3}+15 d \,e^{2} x^{2}+6 d^{2} e x +d^{3}\right ) \ln \left (x^{n}\right )}{60 \left (e x +d \right )^{6} e^{4}}+\frac {-90 \ln \left (e x +d \right ) b \,d^{2} e^{4} n \,x^{4}-120 \ln \left (c \right ) b \,d^{3} e^{3} x^{3}-90 \ln \left (c \right ) b \,d^{4} e^{2} x^{2}-36 \ln \left (c \right ) b \,d^{5} e x +18 i \pi b \,d^{5} e x \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+60 i \pi b \,d^{3} 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 )+45 i \pi b \,d^{4} 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 )+36 \ln \left (-x \right ) b d \,e^{5} n \,x^{5}+3 i \pi b \,d^{6} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+18 i \pi b \,d^{5} e x \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+60 i \pi b \,d^{3} e^{3} x^{3} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-60 i \pi b \,d^{3} e^{3} x^{3} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-60 i \pi b \,d^{3} e^{3} x^{3} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-45 i \pi b \,d^{4} e^{2} x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-45 i \pi b \,d^{4} 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^{5} e x \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-90 \ln \left (e x +d \right ) b \,d^{4} e^{2} n \,x^{2}-2 b \,d^{6} n +90 \ln \left (-x \right ) b \,d^{2} e^{4} n \,x^{4}+120 \ln \left (-x \right ) b \,d^{3} e^{3} n \,x^{3}+90 \ln \left (-x \right ) b \,d^{4} e^{2} n \,x^{2}+36 \ln \left (-x \right ) b \,d^{5} e n x +6 \ln \left (-x \right ) b \,e^{6} n \,x^{6}-120 \ln \left (e x +d \right ) b \,d^{3} e^{3} n \,x^{3}-18 i \pi b \,d^{5} e x \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-36 \ln \left (e x +d \right ) b d \,e^{5} n \,x^{5}-120 a \,d^{3} e^{3} x^{3}+3 b \,d^{4} e^{2} n \,x^{2}+6 b d \,e^{5} n \,x^{5}+33 b \,d^{2} e^{4} n \,x^{4}+34 b \,d^{3} e^{3} n \,x^{3}-6 b \,d^{5} e n x -36 \ln \left (e x +d \right ) b \,d^{5} e n x +6 \ln \left (-x \right ) b \,d^{6} n -6 \ln \left (e x +d \right ) b \,e^{6} n \,x^{6}+45 i \pi b \,d^{4} e^{2} x^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+3 i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) b \,d^{6}-3 i \pi b \,d^{6} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-90 a \,d^{4} e^{2} x^{2}-36 a \,d^{5} e x -6 \ln \left (e x +d \right ) b \,d^{6} n -6 a \,d^{6}-3 i \pi b \,d^{6} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-6 \ln \left (c \right ) b \,d^{6}}{360 e^{4} d^{3} \left (e x +d \right )^{6}}\) | \(867\) |
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Time = 0.30 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.52 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\frac {6 \, b d e^{5} n x^{5} + 33 \, b d^{2} e^{4} n x^{4} - 2 \, b d^{6} n - 6 \, a d^{6} + 2 \, {\left (17 \, b d^{3} e^{3} n - 60 \, a d^{3} e^{3}\right )} x^{3} + 3 \, {\left (b d^{4} e^{2} n - 30 \, a d^{4} e^{2}\right )} x^{2} - 6 \, {\left (b d^{5} e n + 6 \, a d^{5} e\right )} x - 6 \, {\left (b e^{6} n x^{6} + 6 \, b d e^{5} n x^{5} + 15 \, b d^{2} e^{4} n x^{4} + 20 \, b d^{3} e^{3} n x^{3} + 15 \, b d^{4} e^{2} n x^{2} + 6 \, b d^{5} e n x + b d^{6} n\right )} \log \left (e x + d\right ) - 6 \, {\left (20 \, b d^{3} e^{3} x^{3} + 15 \, b d^{4} e^{2} x^{2} + 6 \, b d^{5} e x + b d^{6}\right )} \log \left (c\right ) + 6 \, {\left (b e^{6} n x^{6} + 6 \, b d e^{5} n x^{5} + 15 \, b d^{2} e^{4} n x^{4}\right )} \log \left (x\right )}{360 \, {\left (d^{3} e^{10} x^{6} + 6 \, d^{4} e^{9} x^{5} + 15 \, d^{5} e^{8} x^{4} + 20 \, d^{6} e^{7} x^{3} + 15 \, d^{7} e^{6} x^{2} + 6 \, d^{8} e^{5} x + d^{9} e^{4}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1979 vs. \(2 (224) = 448\).
Time = 75.70 (sec) , antiderivative size = 1979, normalized size of antiderivative = 8.76 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\text {Too large to display} \]
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Time = 0.22 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.50 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=\frac {1}{360} \, b n {\left (\frac {6 \, e^{4} x^{4} + 27 \, d e^{3} x^{3} + 7 \, d^{2} e^{2} x^{2} - 4 \, d^{3} e x - 2 \, d^{4}}{d^{2} e^{9} x^{5} + 5 \, d^{3} e^{8} x^{4} + 10 \, d^{4} e^{7} x^{3} + 10 \, d^{5} e^{6} x^{2} + 5 \, d^{6} e^{5} x + d^{7} e^{4}} - \frac {6 \, \log \left (e x + d\right )}{d^{3} e^{4}} + \frac {6 \, \log \left (x\right )}{d^{3} e^{4}}\right )} - \frac {{\left (20 \, e^{3} x^{3} + 15 \, d e^{2} x^{2} + 6 \, d^{2} e x + d^{3}\right )} b \log \left (c x^{n}\right )}{60 \, {\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} e^{4}\right )}} - \frac {{\left (20 \, e^{3} x^{3} + 15 \, d e^{2} x^{2} + 6 \, d^{2} e x + d^{3}\right )} a}{60 \, {\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} e^{4}\right )}} \]
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Time = 0.40 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.60 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=-\frac {{\left (20 \, b e^{3} n x^{3} + 15 \, b d e^{2} n x^{2} + 6 \, b d^{2} e n x + b d^{3} n\right )} \log \left (x\right )}{60 \, {\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} e^{4}\right )}} + \frac {6 \, b e^{5} n x^{5} + 33 \, b d e^{4} n x^{4} + 34 \, b d^{2} e^{3} n x^{3} - 120 \, b d^{2} e^{3} x^{3} \log \left (c\right ) + 3 \, b d^{3} e^{2} n x^{2} - 120 \, a d^{2} e^{3} x^{3} - 90 \, b d^{3} e^{2} x^{2} \log \left (c\right ) - 6 \, b d^{4} e n x - 90 \, a d^{3} e^{2} x^{2} - 36 \, b d^{4} e x \log \left (c\right ) - 2 \, b d^{5} n - 36 \, a d^{4} e x - 6 \, b d^{5} \log \left (c\right ) - 6 \, a d^{5}}{360 \, {\left (d^{2} e^{10} x^{6} + 6 \, d^{3} e^{9} x^{5} + 15 \, d^{4} e^{8} x^{4} + 20 \, d^{5} e^{7} x^{3} + 15 \, d^{6} e^{6} x^{2} + 6 \, d^{7} e^{5} x + d^{8} e^{4}\right )}} - \frac {b n \log \left (e x + d\right )}{60 \, d^{3} e^{4}} + \frac {b n \log \left (x\right )}{60 \, d^{3} e^{4}} \]
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Time = 1.01 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.31 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx=-\frac {x^3\,\left (20\,a\,e^3-\frac {17\,b\,e^3\,n}{3}\right )+x\,\left (6\,a\,d^2\,e+b\,d^2\,e\,n\right )+a\,d^3+x^2\,\left (15\,a\,d\,e^2-\frac {b\,d\,e^2\,n}{2}\right )+\frac {b\,d^3\,n}{3}-\frac {11\,b\,e^4\,n\,x^4}{2\,d}-\frac {b\,e^5\,n\,x^5}{d^2}}{60\,d^6\,e^4+360\,d^5\,e^5\,x+900\,d^4\,e^6\,x^2+1200\,d^3\,e^7\,x^3+900\,d^2\,e^8\,x^4+360\,d\,e^9\,x^5+60\,e^{10}\,x^6}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,d^3}{60\,e^4}+\frac {b\,x^3}{3\,e}+\frac {b\,d\,x^2}{4\,e^2}+\frac {b\,d^2\,x}{10\,e^3}\right )}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6}-\frac {b\,n\,\mathrm {atanh}\left (\frac {2\,e\,x}{d}+1\right )}{30\,d^3\,e^4} \]
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