Integrand size = 21, antiderivative size = 100 \[ \int x^3 (d+e x)^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{16} b d^3 n x^4-\frac {3}{25} b d^2 e n x^5-\frac {1}{12} b d e^2 n x^6-\frac {1}{49} b e^3 n x^7+\frac {1}{140} \left (35 d^3 x^4+84 d^2 e x^5+70 d e^2 x^6+20 e^3 x^7\right ) \left (a+b \log \left (c x^n\right )\right ) \]
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Time = 0.08 (sec) , antiderivative size = 100, 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, 2371, 12, 14} \[ \int x^3 (d+e x)^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{140} \left (35 d^3 x^4+84 d^2 e x^5+70 d e^2 x^6+20 e^3 x^7\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{16} b d^3 n x^4-\frac {3}{25} b d^2 e n x^5-\frac {1}{12} b d e^2 n x^6-\frac {1}{49} b e^3 n x^7 \]
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Rule 12
Rule 14
Rule 45
Rule 2371
Rubi steps \begin{align*} \text {integral}& = \frac {1}{140} \left (35 d^3 x^4+84 d^2 e x^5+70 d e^2 x^6+20 e^3 x^7\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {1}{140} x^3 \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right ) \, dx \\ & = \frac {1}{140} \left (35 d^3 x^4+84 d^2 e x^5+70 d e^2 x^6+20 e^3 x^7\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{140} (b n) \int x^3 \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right ) \, dx \\ & = \frac {1}{140} \left (35 d^3 x^4+84 d^2 e x^5+70 d e^2 x^6+20 e^3 x^7\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{140} (b n) \int \left (35 d^3 x^3+84 d^2 e x^4+70 d e^2 x^5+20 e^3 x^6\right ) \, dx \\ & = -\frac {1}{16} b d^3 n x^4-\frac {3}{25} b d^2 e n x^5-\frac {1}{12} b d e^2 n x^6-\frac {1}{49} b e^3 n x^7+\frac {1}{140} \left (35 d^3 x^4+84 d^2 e x^5+70 d e^2 x^6+20 e^3 x^7\right ) \left (a+b \log \left (c x^n\right )\right ) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.33 \[ \int x^3 (d+e x)^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{16} b d^3 n x^4-\frac {3}{25} b d^2 e n x^5-\frac {1}{12} b d e^2 n x^6-\frac {1}{49} b e^3 n x^7+\frac {1}{4} d^3 x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {3}{5} d^2 e x^5 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{2} d e^2 x^6 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{7} e^3 x^7 \left (a+b \log \left (c x^n\right )\right ) \]
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Time = 11.00 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.44
method | result | size |
parallelrisch | \(\frac {x^{7} \ln \left (c \,x^{n}\right ) b \,e^{3}}{7}-\frac {b \,e^{3} n \,x^{7}}{49}+\frac {a \,e^{3} x^{7}}{7}+\frac {x^{6} \ln \left (c \,x^{n}\right ) b d \,e^{2}}{2}-\frac {b d \,e^{2} n \,x^{6}}{12}+\frac {a d \,e^{2} x^{6}}{2}+\frac {3 x^{5} \ln \left (c \,x^{n}\right ) b \,d^{2} e}{5}-\frac {3 b \,d^{2} e n \,x^{5}}{25}+\frac {3 a \,d^{2} e \,x^{5}}{5}+\frac {x^{4} \ln \left (c \,x^{n}\right ) b \,d^{3}}{4}-\frac {b \,d^{3} n \,x^{4}}{16}+\frac {a \,d^{3} x^{4}}{4}\) | \(144\) |
risch | \(\frac {a \,e^{3} x^{7}}{7}+\frac {a \,d^{3} x^{4}}{4}+\frac {3 a \,d^{2} e \,x^{5}}{5}+\frac {a d \,e^{2} x^{6}}{2}+\frac {i \pi b \,e^{3} x^{7} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{14}+\frac {i \pi b \,e^{3} x^{7} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{14}-\frac {i \pi b d \,e^{2} x^{6} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{4}+\frac {b \,x^{4} \left (20 e^{3} x^{3}+70 d \,e^{2} x^{2}+84 d^{2} e x +35 d^{3}\right ) \ln \left (x^{n}\right )}{140}+\frac {\ln \left (c \right ) b \,e^{3} x^{7}}{7}+\frac {\ln \left (c \right ) b \,d^{3} x^{4}}{4}-\frac {i \pi b \,d^{3} x^{4} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{8}-\frac {i \pi b \,e^{3} x^{7} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{14}-\frac {3 i \pi b \,d^{2} e \,x^{5} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{10}+\frac {i \pi b \,d^{3} x^{4} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{8}+\frac {i \pi b \,d^{3} x^{4} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{8}-\frac {i \pi b d \,e^{2} x^{6} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{4}-\frac {3 i \pi b \,d^{2} e \,x^{5} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{10}-\frac {3 b \,d^{2} e n \,x^{5}}{25}-\frac {b d \,e^{2} n \,x^{6}}{12}-\frac {b \,d^{3} n \,x^{4}}{16}-\frac {b \,e^{3} n \,x^{7}}{49}+\frac {3 \ln \left (c \right ) b \,d^{2} e \,x^{5}}{5}+\frac {\ln \left (c \right ) b d \,e^{2} x^{6}}{2}+\frac {i \pi b d \,e^{2} x^{6} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{4}-\frac {i \pi b \,e^{3} x^{7} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{14}-\frac {i \pi b \,d^{3} x^{4} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{8}+\frac {3 i \pi b \,d^{2} e \,x^{5} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{10}+\frac {i \pi b d \,e^{2} x^{6} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{4}+\frac {3 i \pi b \,d^{2} e \,x^{5} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{10}\) | \(600\) |
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Time = 0.29 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.67 \[ \int x^3 (d+e x)^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{49} \, {\left (b e^{3} n - 7 \, a e^{3}\right )} x^{7} - \frac {1}{12} \, {\left (b d e^{2} n - 6 \, a d e^{2}\right )} x^{6} - \frac {3}{25} \, {\left (b d^{2} e n - 5 \, a d^{2} e\right )} x^{5} - \frac {1}{16} \, {\left (b d^{3} n - 4 \, a d^{3}\right )} x^{4} + \frac {1}{140} \, {\left (20 \, b e^{3} x^{7} + 70 \, b d e^{2} x^{6} + 84 \, b d^{2} e x^{5} + 35 \, b d^{3} x^{4}\right )} \log \left (c\right ) + \frac {1}{140} \, {\left (20 \, b e^{3} n x^{7} + 70 \, b d e^{2} n x^{6} + 84 \, b d^{2} e n x^{5} + 35 \, b d^{3} n x^{4}\right )} \log \left (x\right ) \]
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Time = 0.66 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.70 \[ \int x^3 (d+e x)^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {a d^{3} x^{4}}{4} + \frac {3 a d^{2} e x^{5}}{5} + \frac {a d e^{2} x^{6}}{2} + \frac {a e^{3} x^{7}}{7} - \frac {b d^{3} n x^{4}}{16} + \frac {b d^{3} x^{4} \log {\left (c x^{n} \right )}}{4} - \frac {3 b d^{2} e n x^{5}}{25} + \frac {3 b d^{2} e x^{5} \log {\left (c x^{n} \right )}}{5} - \frac {b d e^{2} n x^{6}}{12} + \frac {b d e^{2} x^{6} \log {\left (c x^{n} \right )}}{2} - \frac {b e^{3} n x^{7}}{49} + \frac {b e^{3} x^{7} \log {\left (c x^{n} \right )}}{7} \]
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Time = 0.20 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.43 \[ \int x^3 (d+e x)^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{49} \, b e^{3} n x^{7} + \frac {1}{7} \, b e^{3} x^{7} \log \left (c x^{n}\right ) - \frac {1}{12} \, b d e^{2} n x^{6} + \frac {1}{7} \, a e^{3} x^{7} + \frac {1}{2} \, b d e^{2} x^{6} \log \left (c x^{n}\right ) - \frac {3}{25} \, b d^{2} e n x^{5} + \frac {1}{2} \, a d e^{2} x^{6} + \frac {3}{5} \, b d^{2} e x^{5} \log \left (c x^{n}\right ) - \frac {1}{16} \, b d^{3} n x^{4} + \frac {3}{5} \, a d^{2} e x^{5} + \frac {1}{4} \, b d^{3} x^{4} \log \left (c x^{n}\right ) + \frac {1}{4} \, a d^{3} x^{4} \]
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Time = 0.36 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.77 \[ \int x^3 (d+e x)^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{7} \, b e^{3} n x^{7} \log \left (x\right ) - \frac {1}{49} \, b e^{3} n x^{7} + \frac {1}{7} \, b e^{3} x^{7} \log \left (c\right ) + \frac {1}{2} \, b d e^{2} n x^{6} \log \left (x\right ) - \frac {1}{12} \, b d e^{2} n x^{6} + \frac {1}{7} \, a e^{3} x^{7} + \frac {1}{2} \, b d e^{2} x^{6} \log \left (c\right ) + \frac {3}{5} \, b d^{2} e n x^{5} \log \left (x\right ) - \frac {3}{25} \, b d^{2} e n x^{5} + \frac {1}{2} \, a d e^{2} x^{6} + \frac {3}{5} \, b d^{2} e x^{5} \log \left (c\right ) + \frac {1}{4} \, b d^{3} n x^{4} \log \left (x\right ) - \frac {1}{16} \, b d^{3} n x^{4} + \frac {3}{5} \, a d^{2} e x^{5} + \frac {1}{4} \, b d^{3} x^{4} \log \left (c\right ) + \frac {1}{4} \, a d^{3} x^{4} \]
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Time = 0.37 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.13 \[ \int x^3 (d+e x)^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\ln \left (c\,x^n\right )\,\left (\frac {b\,d^3\,x^4}{4}+\frac {3\,b\,d^2\,e\,x^5}{5}+\frac {b\,d\,e^2\,x^6}{2}+\frac {b\,e^3\,x^7}{7}\right )+\frac {d^3\,x^4\,\left (4\,a-b\,n\right )}{16}+\frac {e^3\,x^7\,\left (7\,a-b\,n\right )}{49}+\frac {3\,d^2\,e\,x^5\,\left (5\,a-b\,n\right )}{25}+\frac {d\,e^2\,x^6\,\left (6\,a-b\,n\right )}{12} \]
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