Integrand size = 27, antiderivative size = 112 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (f+\frac {g}{x}\right )^3 x^3} \, dx=-\frac {b e n}{2 f (d f-e g) (g+f x)}+\frac {b e^2 n \log (d+e x)}{2 f (d f-e g)^2}-\frac {a+b \log \left (c (d+e x)^n\right )}{2 f (g+f x)^2}-\frac {b e^2 n \log (g+f x)}{2 f (d f-e g)^2} \]
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Time = 0.08 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2459, 2442, 46} \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (f+\frac {g}{x}\right )^3 x^3} \, dx=-\frac {a+b \log \left (c (d+e x)^n\right )}{2 f (f x+g)^2}+\frac {b e^2 n \log (d+e x)}{2 f (d f-e g)^2}-\frac {b e^2 n \log (f x+g)}{2 f (d f-e g)^2}-\frac {b e n}{2 f (f x+g) (d f-e g)} \]
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Rule 46
Rule 2442
Rule 2459
Rubi steps \begin{align*} \text {integral}& = \int \frac {a+b \log \left (c (d+e x)^n\right )}{(g+f x)^3} \, dx \\ & = -\frac {a+b \log \left (c (d+e x)^n\right )}{2 f (g+f x)^2}+\frac {(b e n) \int \frac {1}{(d+e x) (g+f x)^2} \, dx}{2 f} \\ & = -\frac {a+b \log \left (c (d+e x)^n\right )}{2 f (g+f x)^2}+\frac {(b e n) \int \left (\frac {e^2}{(d f-e g)^2 (d+e x)}+\frac {f}{(d f-e g) (g+f x)^2}-\frac {e f}{(d f-e g)^2 (g+f x)}\right ) \, dx}{2 f} \\ & = -\frac {b e n}{2 f (d f-e g) (g+f x)}+\frac {b e^2 n \log (d+e x)}{2 f (d f-e g)^2}-\frac {a+b \log \left (c (d+e x)^n\right )}{2 f (g+f x)^2}-\frac {b e^2 n \log (g+f x)}{2 f (d f-e g)^2} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.74 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (f+\frac {g}{x}\right )^3 x^3} \, dx=-\frac {a+b \log \left (c (d+e x)^n\right )-\frac {b e n (g+f x) (-d f+e g+e (g+f x) \log (d+e x)-e (g+f x) \log (g+f x))}{(d f-e g)^2}}{2 f (g+f x)^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(282\) vs. \(2(107)=214\).
Time = 0.61 (sec) , antiderivative size = 283, normalized size of antiderivative = 2.53
method | result | size |
parallelrisch | \(\frac {-a \,d^{2} e \,f^{3}-a \,e^{3} f \,g^{2}-\ln \left (c \left (e x +d \right )^{n}\right ) b \,d^{2} e \,f^{3}-\ln \left (c \left (e x +d \right )^{n}\right ) b \,e^{3} f \,g^{2}-\ln \left (f x +g \right ) b \,e^{3} f \,g^{2} n -x b d \,e^{2} f^{3} n +2 \ln \left (c \left (e x +d \right )^{n}\right ) b d \,e^{2} f^{2} g -b d \,e^{2} f^{2} n g +b \,e^{3} f^{2} g n x +2 a d \,e^{2} f^{2} g +b \,e^{3} f \,g^{2} n +2 \ln \left (e x +d \right ) b \,e^{3} f^{2} g n x +\ln \left (e x +d \right ) x^{2} b \,e^{3} f^{3} n -\ln \left (f x +g \right ) x^{2} b \,e^{3} f^{3} n -2 \ln \left (f x +g \right ) x b \,e^{3} f^{2} g n +\ln \left (e x +d \right ) b \,e^{3} f \,g^{2} n}{2 \left (f^{2} d^{2}-2 d e f g +e^{2} g^{2}\right ) \left (f x +g \right )^{2} e \,f^{2}}\) | \(283\) |
risch | \(-\frac {b \ln \left (\left (e x +d \right )^{n}\right )}{2 f \left (f x +g \right )^{2}}-\frac {2 \ln \left (c \right ) b \,d^{2} f^{2}+2 \ln \left (c \right ) b \,e^{2} g^{2}-4 \ln \left (c \right ) b d e f g +2 i \pi b d e f g \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}-i \pi b \,e^{2} g^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )-i \pi b \,d^{2} f^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )+2 g b d e f n -i \pi b \,d^{2} f^{2} \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}-i \pi b \,e^{2} g^{2} \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+2 b d e \,f^{2} n x +2 i \pi b d e f g \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )+2 a \,e^{2} g^{2}-2 i \pi b d e f g \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-2 i \pi b d e f g \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-2 b \,e^{2} g^{2} n -4 \ln \left (-e x -d \right ) b \,e^{2} f g n x -4 a d e f g -2 \ln \left (-e x -d \right ) b \,e^{2} g^{2} n +2 \ln \left (f x +g \right ) b \,e^{2} g^{2} n +4 \ln \left (f x +g \right ) b \,e^{2} f g n x +2 a \,d^{2} f^{2}-2 \ln \left (-e x -d \right ) b \,e^{2} f^{2} n \,x^{2}+2 \ln \left (f x +g \right ) b \,e^{2} f^{2} n \,x^{2}-2 b \,e^{2} f g n x +i \pi b \,e^{2} g^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i \pi b \,e^{2} g^{2} \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i \pi b \,d^{2} f^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i \pi b \,d^{2} f^{2} \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{4 \left (f x +g \right )^{2} \left (d f -e g \right )^{2} f}\) | \(633\) |
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Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (104) = 208\).
Time = 0.33 (sec) , antiderivative size = 272, normalized size of antiderivative = 2.43 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (f+\frac {g}{x}\right )^3 x^3} \, dx=-\frac {a d^{2} f^{2} - 2 \, a d e f g + a e^{2} g^{2} + {\left (b d e f^{2} - b e^{2} f g\right )} n x + {\left (b d e f g - b e^{2} g^{2}\right )} n - {\left (b e^{2} f^{2} n x^{2} + 2 \, b e^{2} f g n x - {\left (b d^{2} f^{2} - 2 \, b d e f g\right )} n\right )} \log \left (e x + d\right ) + {\left (b e^{2} f^{2} n x^{2} + 2 \, b e^{2} f g n x + b e^{2} g^{2} n\right )} \log \left (f x + g\right ) + {\left (b d^{2} f^{2} - 2 \, b d e f g + b e^{2} g^{2}\right )} \log \left (c\right )}{2 \, {\left (d^{2} f^{3} g^{2} - 2 \, d e f^{2} g^{3} + e^{2} f g^{4} + {\left (d^{2} f^{5} - 2 \, d e f^{4} g + e^{2} f^{3} g^{2}\right )} x^{2} + 2 \, {\left (d^{2} f^{4} g - 2 \, d e f^{3} g^{2} + e^{2} f^{2} g^{3}\right )} x\right )}} \]
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Timed out. \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (f+\frac {g}{x}\right )^3 x^3} \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.51 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (f+\frac {g}{x}\right )^3 x^3} \, dx=\frac {1}{2} \, b e n {\left (\frac {e \log \left (e x + d\right )}{d^{2} f^{3} - 2 \, d e f^{2} g + e^{2} f g^{2}} - \frac {e \log \left (f x + g\right )}{d^{2} f^{3} - 2 \, d e f^{2} g + e^{2} f g^{2}} - \frac {1}{d f^{2} g - e f g^{2} + {\left (d f^{3} - e f^{2} g\right )} x}\right )} - \frac {b \log \left ({\left (e x + d\right )}^{n} c\right )}{2 \, {\left (f^{3} x^{2} + 2 \, f^{2} g x + f g^{2}\right )}} - \frac {a}{2 \, {\left (f^{3} x^{2} + 2 \, f^{2} g x + f g^{2}\right )}} \]
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Time = 0.34 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.79 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (f+\frac {g}{x}\right )^3 x^3} \, dx=\frac {b e^{2} n \log \left (e x + d\right )}{2 \, {\left (d^{2} f^{3} - 2 \, d e f^{2} g + e^{2} f g^{2}\right )}} - \frac {b e^{2} n \log \left (f x + g\right )}{2 \, {\left (d^{2} f^{3} - 2 \, d e f^{2} g + e^{2} f g^{2}\right )}} - \frac {b n \log \left (e x + d\right )}{2 \, {\left (f^{3} x^{2} + 2 \, f^{2} g x + f g^{2}\right )}} - \frac {b e f n x + b e g n + b d f \log \left (c\right ) - b e g \log \left (c\right ) + a d f - a e g}{2 \, {\left (d f^{4} x^{2} - e f^{3} g x^{2} + 2 \, d f^{3} g x - 2 \, e f^{2} g^{2} x + d f^{2} g^{2} - e f g^{3}\right )}} \]
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Time = 1.72 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.54 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (f+\frac {g}{x}\right )^3 x^3} \, dx=\frac {b\,e^2\,n\,\mathrm {atanh}\left (\frac {2\,d^2\,f^3-2\,e^2\,f\,g^2}{2\,f\,{\left (d\,f-e\,g\right )}^2}+\frac {2\,e\,f\,x}{d\,f-e\,g}\right )}{f\,{\left (d\,f-e\,g\right )}^2}-\frac {b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{2\,f\,\left (f^2\,x^2+2\,f\,g\,x+g^2\right )}-\frac {\frac {a\,d\,f-a\,e\,g+b\,e\,g\,n}{d\,f-e\,g}+\frac {b\,e\,f\,n\,x}{d\,f-e\,g}}{2\,f^3\,x^2+4\,f^2\,g\,x+2\,f\,g^2} \]
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