Integrand size = 20, antiderivative size = 127 \[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{(d+e x)^3} \, dx=\frac {b p}{2 d (a d-b e) (d+e x)}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 e (d+e x)^2}-\frac {p \log (x)}{2 d^2 e}+\frac {a^2 p \log (b+a x)}{2 e (a d-b e)^2}-\frac {b (2 a d-b e) p \log (d+e x)}{2 d^2 (a d-b e)^2} \]
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Time = 0.08 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2513, 528, 84} \[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{(d+e x)^3} \, dx=\frac {a^2 p \log (a x+b)}{2 e (a d-b e)^2}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 e (d+e x)^2}-\frac {b p (2 a d-b e) \log (d+e x)}{2 d^2 (a d-b e)^2}+\frac {b p}{2 d (d+e x) (a d-b e)}-\frac {p \log (x)}{2 d^2 e} \]
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Rule 84
Rule 528
Rule 2513
Rubi steps \begin{align*} \text {integral}& = -\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 e (d+e x)^2}-\frac {(b p) \int \frac {1}{\left (a+\frac {b}{x}\right ) x^2 (d+e x)^2} \, dx}{2 e} \\ & = -\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 e (d+e x)^2}-\frac {(b p) \int \frac {1}{x (b+a x) (d+e x)^2} \, dx}{2 e} \\ & = -\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 e (d+e x)^2}-\frac {(b p) \int \left (\frac {1}{b d^2 x}-\frac {a^3}{b (-a d+b e)^2 (b+a x)}+\frac {e^2}{d (a d-b e) (d+e x)^2}+\frac {e^2 (2 a d-b e)}{d^2 (a d-b e)^2 (d+e x)}\right ) \, dx}{2 e} \\ & = \frac {b p}{2 d (a d-b e) (d+e x)}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 e (d+e x)^2}-\frac {p \log (x)}{2 d^2 e}+\frac {a^2 p \log (b+a x)}{2 e (a d-b e)^2}-\frac {b (2 a d-b e) p \log (d+e x)}{2 d^2 (a d-b e)^2} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.89 \[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{(d+e x)^3} \, dx=\frac {\frac {b e p}{d (a d-b e) (d+e x)}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{(d+e x)^2}-\frac {p \log (x)}{d^2}+\frac {a^2 p \log (b+a x)}{(a d-b e)^2}+\frac {b e (-2 a d+b e) p \log (d+e x)}{d^2 (a d-b e)^2}}{2 e} \]
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Time = 1.33 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.94
method | result | size |
parts | \(-\frac {\ln \left (c \left (a +\frac {b}{x}\right )^{p}\right )}{2 e \left (e x +d \right )^{2}}-\frac {p b \left (\frac {\ln \left (x \right )}{b \,d^{2}}-\frac {e}{d \left (a d -b e \right ) \left (e x +d \right )}+\frac {e \left (2 a d -b e \right ) \ln \left (e x +d \right )}{d^{2} \left (a d -b e \right )^{2}}-\frac {a^{2} \ln \left (a x +b \right )}{b \left (a d -b e \right )^{2}}\right )}{2 e}\) | \(120\) |
parallelrisch | \(-\frac {-2 \ln \left (x \right ) x^{2} a^{2} b d \,e^{4} p^{2}+2 \ln \left (e x +d \right ) x^{2} a^{2} b d \,e^{4} p^{2}-4 \ln \left (x \right ) x \,a^{2} b \,d^{2} e^{3} p^{2}+2 \ln \left (x \right ) x a \,b^{2} d \,e^{4} p^{2}+4 \ln \left (e x +d \right ) x \,a^{2} b \,d^{2} e^{3} p^{2}-2 \ln \left (e x +d \right ) x a \,b^{2} d \,e^{4} p^{2}-a^{2} b \,d^{3} e^{2} p^{2}+a \,b^{2} d^{2} e^{3} p^{2}+\ln \left (x \right ) x^{2} a \,b^{2} e^{5} p^{2}-\ln \left (e x +d \right ) x^{2} a \,b^{2} e^{5} p^{2}-2 \ln \left (x \right ) a^{2} b \,d^{3} e^{2} p^{2}+\ln \left (x \right ) a \,b^{2} d^{2} e^{3} p^{2}+2 \ln \left (e x +d \right ) a^{2} b \,d^{3} e^{2} p^{2}-\ln \left (e x +d \right ) a \,b^{2} d^{2} e^{3} p^{2}+x a \,b^{2} d \,e^{4} p^{2}-2 \ln \left (c \left (\frac {a x +b}{x}\right )^{p}\right ) a^{2} b \,d^{3} e^{2} p -x \,a^{2} b \,d^{2} e^{3} p^{2}-2 x \ln \left (c \left (\frac {a x +b}{x}\right )^{p}\right ) a^{3} d^{3} e^{2} p -x^{2} \ln \left (c \left (\frac {a x +b}{x}\right )^{p}\right ) a^{3} d^{2} e^{3} p +\ln \left (c \left (\frac {a x +b}{x}\right )^{p}\right ) a \,b^{2} d^{2} e^{3} p}{2 \left (e x +d \right )^{2} d^{2} \left (a d -b e \right )^{2} a \,e^{2} p}\) | \(428\) |
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Leaf count of result is larger than twice the leaf count of optimal. 428 vs. \(2 (117) = 234\).
Time = 0.78 (sec) , antiderivative size = 428, normalized size of antiderivative = 3.37 \[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{(d+e x)^3} \, dx=\frac {{\left (a b d^{2} e^{2} - b^{2} d e^{3}\right )} p x - {\left (a^{2} d^{4} - 2 \, a b d^{3} e + b^{2} d^{2} e^{2}\right )} p \log \left (\frac {a x + b}{x}\right ) + {\left (a b d^{3} e - b^{2} d^{2} e^{2}\right )} p + {\left (a^{2} d^{2} e^{2} p x^{2} + 2 \, a^{2} d^{3} e p x + a^{2} d^{4} p\right )} \log \left (a x + b\right ) - {\left ({\left (2 \, a b d e^{3} - b^{2} e^{4}\right )} p x^{2} + 2 \, {\left (2 \, a b d^{2} e^{2} - b^{2} d e^{3}\right )} p x + {\left (2 \, a b d^{3} e - b^{2} d^{2} e^{2}\right )} p\right )} \log \left (e x + d\right ) - {\left (a^{2} d^{4} - 2 \, a b d^{3} e + b^{2} d^{2} e^{2}\right )} \log \left (c\right ) - {\left ({\left (a^{2} d^{2} e^{2} - 2 \, a b d e^{3} + b^{2} e^{4}\right )} p x^{2} + 2 \, {\left (a^{2} d^{3} e - 2 \, a b d^{2} e^{2} + b^{2} d e^{3}\right )} p x + {\left (a^{2} d^{4} - 2 \, a b d^{3} e + b^{2} d^{2} e^{2}\right )} p\right )} \log \left (x\right )}{2 \, {\left (a^{2} d^{6} e - 2 \, a b d^{5} e^{2} + b^{2} d^{4} e^{3} + {\left (a^{2} d^{4} e^{3} - 2 \, a b d^{3} e^{4} + b^{2} d^{2} e^{5}\right )} x^{2} + 2 \, {\left (a^{2} d^{5} e^{2} - 2 \, a b d^{4} e^{3} + b^{2} d^{3} e^{4}\right )} x\right )}} \]
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Timed out. \[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{(d+e x)^3} \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.26 \[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{(d+e x)^3} \, dx=\frac {{\left (\frac {a^{2} \log \left (a x + b\right )}{a^{2} b d^{2} - 2 \, a b^{2} d e + b^{3} e^{2}} - \frac {{\left (2 \, a d e - b e^{2}\right )} \log \left (e x + d\right )}{a^{2} d^{4} - 2 \, a b d^{3} e + b^{2} d^{2} e^{2}} + \frac {e}{a d^{3} - b d^{2} e + {\left (a d^{2} e - b d e^{2}\right )} x} - \frac {\log \left (x\right )}{b d^{2}}\right )} b p}{2 \, e} - \frac {\log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right )}{2 \, {\left (e x + d\right )}^{2} e} \]
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Leaf count of result is larger than twice the leaf count of optimal. 470 vs. \(2 (117) = 234\).
Time = 0.32 (sec) , antiderivative size = 470, normalized size of antiderivative = 3.70 \[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{(d+e x)^3} \, dx=-\frac {\frac {{\left (2 \, a b^{2} d p - b^{3} e p\right )} \log \left (-a d + b e + \frac {{\left (a x + b\right )} d}{x}\right )}{a^{2} d^{4} - 2 \, a b d^{3} e + b^{2} d^{2} e^{2}} + \frac {{\left (2 \, a b^{2} d p - b^{3} e p - \frac {2 \, {\left (a x + b\right )} b^{2} d p}{x}\right )} \log \left (\frac {a x + b}{x}\right )}{a^{2} d^{4} - 2 \, a b d^{3} e + b^{2} d^{2} e^{2} - \frac {2 \, {\left (a x + b\right )} a d^{4}}{x} + \frac {2 \, {\left (a x + b\right )} b d^{3} e}{x} + \frac {{\left (a x + b\right )}^{2} d^{4}}{x^{2}}} - \frac {{\left (2 \, a b^{2} d p - b^{3} e p\right )} \log \left (\frac {a x + b}{x}\right )}{a^{2} d^{4} - 2 \, a b d^{3} e + b^{2} d^{2} e^{2}} - \frac {a b^{3} d e p - b^{4} e^{2} p - 2 \, a^{2} b^{2} d^{2} \log \left (c\right ) + 3 \, a b^{3} d e \log \left (c\right ) - b^{4} e^{2} \log \left (c\right ) - \frac {{\left (a x + b\right )} b^{3} d e p}{x} + \frac {2 \, {\left (a x + b\right )} a b^{2} d^{2} \log \left (c\right )}{x} - \frac {2 \, {\left (a x + b\right )} b^{3} d e \log \left (c\right )}{x}}{a^{3} d^{5} - 3 \, a^{2} b d^{4} e + 3 \, a b^{2} d^{3} e^{2} - b^{3} d^{2} e^{3} - \frac {2 \, {\left (a x + b\right )} a^{2} d^{5}}{x} + \frac {4 \, {\left (a x + b\right )} a b d^{4} e}{x} - \frac {2 \, {\left (a x + b\right )} b^{2} d^{3} e^{2}}{x} + \frac {{\left (a x + b\right )}^{2} a d^{5}}{x^{2}} - \frac {{\left (a x + b\right )}^{2} b d^{4} e}{x^{2}}}}{2 \, b} \]
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Time = 2.11 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.71 \[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{(d+e x)^3} \, dx=\frac {a^2\,p\,\ln \left (b+a\,x\right )}{2\,a^2\,d^2\,e-4\,a\,b\,d\,e^2+2\,b^2\,e^3}-\frac {\ln \left (c\,{\left (\frac {b+a\,x}{x}\right )}^p\right )}{2\,\left (d^2\,e+2\,d\,e^2\,x+e^3\,x^2\right )}-\frac {p\,\ln \left (x\right )}{2\,d^2\,e}-\frac {b\,e\,p}{2\,b\,d^2\,e^2-2\,a\,d^3\,e+2\,b\,d\,e^3\,x-2\,a\,d^2\,e^2\,x}+\frac {b^2\,e\,p\,\ln \left (d+e\,x\right )}{2\,a^2\,d^4-4\,a\,b\,d^3\,e+2\,b^2\,d^2\,e^2}-\frac {2\,a\,b\,d\,p\,\ln \left (d+e\,x\right )}{2\,a^2\,d^4-4\,a\,b\,d^3\,e+2\,b^2\,d^2\,e^2} \]
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