Integrand size = 18, antiderivative size = 133 \[ \int \frac {\log \left (c (a+b x)^p\right )}{(d+e x)^4} \, dx=\frac {b p}{6 e (b d-a e) (d+e x)^2}+\frac {b^2 p}{3 e (b d-a e)^2 (d+e x)}+\frac {b^3 p \log (a+b x)}{3 e (b d-a e)^3}-\frac {\log \left (c (a+b x)^p\right )}{3 e (d+e x)^3}-\frac {b^3 p \log (d+e x)}{3 e (b d-a e)^3} \]
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Time = 0.05 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2442, 46} \[ \int \frac {\log \left (c (a+b x)^p\right )}{(d+e x)^4} \, dx=\frac {b^3 p \log (a+b x)}{3 e (b d-a e)^3}-\frac {b^3 p \log (d+e x)}{3 e (b d-a e)^3}+\frac {b^2 p}{3 e (d+e x) (b d-a e)^2}-\frac {\log \left (c (a+b x)^p\right )}{3 e (d+e x)^3}+\frac {b p}{6 e (d+e x)^2 (b d-a e)} \]
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Rule 46
Rule 2442
Rubi steps \begin{align*} \text {integral}& = -\frac {\log \left (c (a+b x)^p\right )}{3 e (d+e x)^3}+\frac {(b p) \int \frac {1}{(a+b x) (d+e x)^3} \, dx}{3 e} \\ & = -\frac {\log \left (c (a+b x)^p\right )}{3 e (d+e x)^3}+\frac {(b p) \int \left (\frac {b^3}{(b d-a e)^3 (a+b x)}-\frac {e}{(b d-a e) (d+e x)^3}-\frac {b e}{(b d-a e)^2 (d+e x)^2}-\frac {b^2 e}{(b d-a e)^3 (d+e x)}\right ) \, dx}{3 e} \\ & = \frac {b p}{6 e (b d-a e) (d+e x)^2}+\frac {b^2 p}{3 e (b d-a e)^2 (d+e x)}+\frac {b^3 p \log (a+b x)}{3 e (b d-a e)^3}-\frac {\log \left (c (a+b x)^p\right )}{3 e (d+e x)^3}-\frac {b^3 p \log (d+e x)}{3 e (b d-a e)^3} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.79 \[ \int \frac {\log \left (c (a+b x)^p\right )}{(d+e x)^4} \, dx=\frac {-2 \log \left (c (a+b x)^p\right )+\frac {b p (d+e x) \left ((b d-a e) (3 b d-a e+2 b e x)+2 b^2 (d+e x)^2 \log (a+b x)-2 b^2 (d+e x)^2 \log (d+e x)\right )}{(b d-a e)^3}}{6 e (d+e x)^3} \]
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Time = 1.78 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.83
method | result | size |
parts | \(-\frac {\ln \left (c \left (b x +a \right )^{p}\right )}{3 e \left (e x +d \right )^{3}}+\frac {p b \left (-\frac {1}{2 \left (a e -b d \right ) \left (e x +d \right )^{2}}+\frac {b^{2} \ln \left (e x +d \right )}{\left (a e -b d \right )^{3}}+\frac {b}{\left (a e -b d \right )^{2} \left (e x +d \right )}-\frac {b^{2} \ln \left (b x +a \right )}{\left (a e -b d \right )^{3}}\right )}{3 e}\) | \(111\) |
parallelrisch | \(-\frac {-6 x a \,b^{3} d \,e^{4} p +6 \ln \left (b x +a \right ) x^{2} b^{4} d \,e^{4} p -6 \ln \left (e x +d \right ) x^{2} b^{4} d \,e^{4} p +6 \ln \left (b x +a \right ) x \,b^{4} d^{2} e^{3} p -6 \ln \left (e x +d \right ) x \,b^{4} d^{2} e^{3} p +2 \ln \left (b x +a \right ) x^{3} b^{4} e^{5} p -2 \ln \left (e x +d \right ) x^{3} b^{4} e^{5} p +2 \ln \left (b x +a \right ) b^{4} d^{3} e^{2} p -2 \ln \left (e x +d \right ) b^{4} d^{3} e^{2} p -2 x^{2} a \,b^{3} e^{5} p +2 x^{2} b^{4} d \,e^{4} p +x \,a^{2} b^{2} e^{5} p +5 x \,b^{4} d^{2} e^{3} p -6 \ln \left (c \left (b x +a \right )^{p}\right ) a^{2} b^{2} d \,e^{4}+6 \ln \left (c \left (b x +a \right )^{p}\right ) a \,b^{3} d^{2} e^{3}-4 a \,b^{3} d^{2} e^{3} p +a^{2} b^{2} d \,e^{4} p +3 b^{4} d^{3} e^{2} p +2 \ln \left (c \left (b x +a \right )^{p}\right ) a^{3} b \,e^{5}-2 \ln \left (c \left (b x +a \right )^{p}\right ) b^{4} d^{3} e^{2}}{6 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \left (e x +d \right )^{3} e^{3} b}\) | \(387\) |
risch | \(-\frac {\ln \left (\left (b x +a \right )^{p}\right )}{3 e \left (e x +d \right )^{3}}+\frac {-6 \ln \left (b x +a \right ) b^{3} d \,e^{2} p \,x^{2}+6 \ln \left (-e x -d \right ) b^{3} d \,e^{2} p \,x^{2}-6 \ln \left (b x +a \right ) b^{3} d^{2} e p x +6 \ln \left (-e x -d \right ) b^{3} d^{2} e p x -3 b^{3} d^{3} p -2 \ln \left (b x +a \right ) b^{3} d^{3} p +2 a \,b^{2} e^{3} p \,x^{2}-2 \ln \left (b x +a \right ) b^{3} e^{3} p \,x^{3}+2 \ln \left (-e x -d \right ) b^{3} e^{3} p \,x^{3}-a^{2} b d p \,e^{2}+4 a \,b^{2} d^{2} p e +3 i \pi \,a^{2} b d \,e^{2} \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \operatorname {csgn}\left (i c \right )-3 i \pi a \,b^{2} d^{2} e \,\operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}+3 i \pi \,a^{2} b d \,e^{2} \operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}-3 i \pi a \,b^{2} d^{2} e \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \operatorname {csgn}\left (i c \right )+i \pi \,a^{3} e^{3} \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}-i \pi \,b^{3} d^{3} \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}-2 b^{3} d \,e^{2} p \,x^{2}-a^{2} b \,e^{3} p x -5 b^{3} d^{2} e p x -2 \ln \left (c \right ) a^{3} e^{3}+2 \ln \left (c \right ) b^{3} d^{3}+6 a \,b^{2} d \,e^{2} p x +2 \ln \left (-e x -d \right ) b^{3} d^{3} p -3 i \pi \,a^{2} b d \,e^{2} \operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )+3 i \pi a \,b^{2} d^{2} e \,\operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )+6 \ln \left (c \right ) a^{2} b d \,e^{2}-6 \ln \left (c \right ) a \,b^{2} d^{2} e +i \pi \,a^{3} e^{3} \operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-3 i \pi \,a^{2} b d \,e^{2} \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}+3 i \pi a \,b^{2} d^{2} e \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}-i \pi \,b^{3} d^{3} \operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-i \pi \,a^{3} e^{3} \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \operatorname {csgn}\left (i c \right )-i \pi \,a^{3} e^{3} \operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}+i \pi \,b^{3} d^{3} \operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}+i \pi \,b^{3} d^{3} \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \operatorname {csgn}\left (i c \right )}{6 \left (e x +d \right )^{3} \left (a^{2} e^{2}-2 a d e b +b^{2} d^{2}\right ) \left (a e -b d \right ) e}\) | \(873\) |
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Leaf count of result is larger than twice the leaf count of optimal. 443 vs. \(2 (123) = 246\).
Time = 0.39 (sec) , antiderivative size = 443, normalized size of antiderivative = 3.33 \[ \int \frac {\log \left (c (a+b x)^p\right )}{(d+e x)^4} \, dx=\frac {2 \, {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} p x^{2} + {\left (5 \, b^{3} d^{2} e - 6 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} p x + {\left (3 \, b^{3} d^{3} - 4 \, a b^{2} d^{2} e + a^{2} b d e^{2}\right )} p + 2 \, {\left (b^{3} e^{3} p x^{3} + 3 \, b^{3} d e^{2} p x^{2} + 3 \, b^{3} d^{2} e p x + {\left (3 \, a b^{2} d^{2} e - 3 \, a^{2} b d e^{2} + a^{3} e^{3}\right )} p\right )} \log \left (b x + a\right ) - 2 \, {\left (b^{3} e^{3} p x^{3} + 3 \, b^{3} d e^{2} p x^{2} + 3 \, b^{3} d^{2} e p x + b^{3} d^{3} p\right )} \log \left (e x + d\right ) - 2 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \log \left (c\right )}{6 \, {\left (b^{3} d^{6} e - 3 \, a b^{2} d^{5} e^{2} + 3 \, a^{2} b d^{4} e^{3} - a^{3} d^{3} e^{4} + {\left (b^{3} d^{3} e^{4} - 3 \, a b^{2} d^{2} e^{5} + 3 \, a^{2} b d e^{6} - a^{3} e^{7}\right )} x^{3} + 3 \, {\left (b^{3} d^{4} e^{3} - 3 \, a b^{2} d^{3} e^{4} + 3 \, a^{2} b d^{2} e^{5} - a^{3} d e^{6}\right )} x^{2} + 3 \, {\left (b^{3} d^{5} e^{2} - 3 \, a b^{2} d^{4} e^{3} + 3 \, a^{2} b d^{3} e^{4} - a^{3} d^{2} e^{5}\right )} x\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 4571 vs. \(2 (109) = 218\).
Time = 17.52 (sec) , antiderivative size = 4571, normalized size of antiderivative = 34.37 \[ \int \frac {\log \left (c (a+b x)^p\right )}{(d+e x)^4} \, dx=\text {Too large to display} \]
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Time = 0.21 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.74 \[ \int \frac {\log \left (c (a+b x)^p\right )}{(d+e x)^4} \, dx=\frac {{\left (\frac {2 \, b^{2} \log \left (b x + a\right )}{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}} - \frac {2 \, b^{2} \log \left (e x + d\right )}{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}} + \frac {2 \, b e x + 3 \, b d - a e}{b^{2} d^{4} - 2 \, a b d^{3} e + a^{2} d^{2} e^{2} + {\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} x^{2} + 2 \, {\left (b^{2} d^{3} e - 2 \, a b d^{2} e^{2} + a^{2} d e^{3}\right )} x}\right )} b p}{6 \, e} - \frac {\log \left ({\left (b x + a\right )}^{p} c\right )}{3 \, {\left (e x + d\right )}^{3} e} \]
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Leaf count of result is larger than twice the leaf count of optimal. 365 vs. \(2 (123) = 246\).
Time = 0.32 (sec) , antiderivative size = 365, normalized size of antiderivative = 2.74 \[ \int \frac {\log \left (c (a+b x)^p\right )}{(d+e x)^4} \, dx=\frac {b^{3} p \log \left (b x + a\right )}{3 \, {\left (b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}\right )}} - \frac {b^{3} p \log \left (e x + d\right )}{3 \, {\left (b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}\right )}} - \frac {p \log \left (b x + a\right )}{3 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} + \frac {2 \, b^{2} e^{2} p x^{2} + 5 \, b^{2} d e p x - a b e^{2} p x + 3 \, b^{2} d^{2} p - a b d e p - 2 \, b^{2} d^{2} \log \left (c\right ) + 4 \, a b d e \log \left (c\right ) - 2 \, a^{2} e^{2} \log \left (c\right )}{6 \, {\left (b^{2} d^{2} e^{4} x^{3} - 2 \, a b d e^{5} x^{3} + a^{2} e^{6} x^{3} + 3 \, b^{2} d^{3} e^{3} x^{2} - 6 \, a b d^{2} e^{4} x^{2} + 3 \, a^{2} d e^{5} x^{2} + 3 \, b^{2} d^{4} e^{2} x - 6 \, a b d^{3} e^{3} x + 3 \, a^{2} d^{2} e^{4} x + b^{2} d^{5} e - 2 \, a b d^{4} e^{2} + a^{2} d^{3} e^{3}\right )}} \]
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Time = 1.86 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.09 \[ \int \frac {\log \left (c (a+b x)^p\right )}{(d+e x)^4} \, dx=\frac {b^2\,p\,x}{3\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^2}-\frac {\ln \left (c\,{\left (a+b\,x\right )}^p\right )}{3\,e\,{\left (d+e\,x\right )}^3}-\frac {a\,b\,p}{6\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^2}+\frac {b^2\,d\,p}{2\,e\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^2}+\frac {b^3\,p\,\mathrm {atan}\left (\frac {a\,e\,1{}\mathrm {i}+b\,d\,1{}\mathrm {i}+b\,e\,x\,2{}\mathrm {i}}{a\,e-b\,d}\right )\,2{}\mathrm {i}}{3\,e\,{\left (a\,e-b\,d\right )}^3} \]
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