Integrand size = 22, antiderivative size = 141 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^4} \, dx=\frac {b e n}{6 g (e f-d g) (f+g x)^2}+\frac {b e^2 n}{3 g (e f-d g)^2 (f+g x)}+\frac {b e^3 n \log (d+e x)}{3 g (e f-d g)^3}-\frac {a+b \log \left (c (d+e x)^n\right )}{3 g (f+g x)^3}-\frac {b e^3 n \log (f+g x)}{3 g (e f-d g)^3} \]
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Time = 0.06 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2442, 46} \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^4} \, dx=-\frac {a+b \log \left (c (d+e x)^n\right )}{3 g (f+g x)^3}+\frac {b e^3 n \log (d+e x)}{3 g (e f-d g)^3}-\frac {b e^3 n \log (f+g x)}{3 g (e f-d g)^3}+\frac {b e^2 n}{3 g (f+g x) (e f-d g)^2}+\frac {b e n}{6 g (f+g x)^2 (e f-d g)} \]
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Rule 46
Rule 2442
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \log \left (c (d+e x)^n\right )}{3 g (f+g x)^3}+\frac {(b e n) \int \frac {1}{(d+e x) (f+g x)^3} \, dx}{3 g} \\ & = -\frac {a+b \log \left (c (d+e x)^n\right )}{3 g (f+g x)^3}+\frac {(b e n) \int \left (\frac {e^3}{(e f-d g)^3 (d+e x)}-\frac {g}{(e f-d g) (f+g x)^3}-\frac {e g}{(e f-d g)^2 (f+g x)^2}-\frac {e^2 g}{(e f-d g)^3 (f+g x)}\right ) \, dx}{3 g} \\ & = \frac {b e n}{6 g (e f-d g) (f+g x)^2}+\frac {b e^2 n}{3 g (e f-d g)^2 (f+g x)}+\frac {b e^3 n \log (d+e x)}{3 g (e f-d g)^3}-\frac {a+b \log \left (c (d+e x)^n\right )}{3 g (f+g x)^3}-\frac {b e^3 n \log (f+g x)}{3 g (e f-d g)^3} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.78 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^4} \, dx=\frac {-2 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {b e n (f+g x) \left ((e f-d g) (3 e f-d g+2 e g x)+2 e^2 (f+g x)^2 \log (d+e x)-2 e^2 (f+g x)^2 \log (f+g x)\right )}{(e f-d g)^3}}{6 g (f+g x)^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(454\) vs. \(2(134)=268\).
Time = 1.61 (sec) , antiderivative size = 455, normalized size of antiderivative = 3.23
method | result | size |
parallelrisch | \(-\frac {2 \ln \left (c \left (e x +d \right )^{n}\right ) b \,d^{3} e \,g^{5}-2 \ln \left (c \left (e x +d \right )^{n}\right ) b \,e^{4} f^{3} g^{2}+3 b \,e^{4} f^{3} g^{2} n -6 a \,d^{2} e^{2} f \,g^{4}+6 a d \,e^{3} f^{2} g^{3}+6 \ln \left (e x +d \right ) x^{2} b \,e^{4} f \,g^{4} n -6 \ln \left (g x +f \right ) x^{2} b \,e^{4} f \,g^{4} n +6 \ln \left (e x +d \right ) x b \,e^{4} f^{2} g^{3} n +2 x^{2} b \,e^{4} f \,g^{4} n +x b \,d^{2} e^{2} g^{5} n +5 x b \,e^{4} f^{2} g^{3} n -6 \ln \left (c \left (e x +d \right )^{n}\right ) b \,d^{2} e^{2} f \,g^{4}+6 \ln \left (c \left (e x +d \right )^{n}\right ) b d \,e^{3} f^{2} g^{3}+2 \ln \left (e x +d \right ) x^{3} b \,e^{4} g^{5} n -2 \ln \left (g x +f \right ) x^{3} b \,e^{4} g^{5} n +2 \ln \left (e x +d \right ) b \,e^{4} f^{3} g^{2} n -2 \ln \left (g x +f \right ) b \,e^{4} f^{3} g^{2} n +b \,d^{2} e^{2} f \,g^{4} n -4 b d \,e^{3} f^{2} g^{3} n +2 a \,d^{3} e \,g^{5}-2 a \,e^{4} f^{3} g^{2}-2 x^{2} b d \,e^{3} g^{5} n -6 \ln \left (g x +f \right ) x b \,e^{4} f^{2} g^{3} n -6 x b d \,e^{3} f \,g^{4} n}{6 \left (d^{3} g^{3}-3 d^{2} e f \,g^{2}+3 d \,e^{2} f^{2} g -e^{3} f^{3}\right ) \left (g x +f \right )^{3} g^{3} e}\) | \(455\) |
risch | \(\text {Expression too large to display}\) | \(950\) |
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Leaf count of result is larger than twice the leaf count of optimal. 507 vs. \(2 (131) = 262\).
Time = 0.32 (sec) , antiderivative size = 507, normalized size of antiderivative = 3.60 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^4} \, dx=-\frac {2 \, a e^{3} f^{3} - 6 \, a d e^{2} f^{2} g + 6 \, a d^{2} e f g^{2} - 2 \, a d^{3} g^{3} - 2 \, {\left (b e^{3} f g^{2} - b d e^{2} g^{3}\right )} n x^{2} - {\left (5 \, b e^{3} f^{2} g - 6 \, b d e^{2} f g^{2} + b d^{2} e g^{3}\right )} n x - {\left (3 \, b e^{3} f^{3} - 4 \, b d e^{2} f^{2} g + b d^{2} e f g^{2}\right )} n - 2 \, {\left (b e^{3} g^{3} n x^{3} + 3 \, b e^{3} f g^{2} n x^{2} + 3 \, b e^{3} f^{2} g n x + {\left (3 \, b d e^{2} f^{2} g - 3 \, b d^{2} e f g^{2} + b d^{3} g^{3}\right )} n\right )} \log \left (e x + d\right ) + 2 \, {\left (b e^{3} g^{3} n x^{3} + 3 \, b e^{3} f g^{2} n x^{2} + 3 \, b e^{3} f^{2} g n x + b e^{3} f^{3} n\right )} \log \left (g x + f\right ) + 2 \, {\left (b e^{3} f^{3} - 3 \, b d e^{2} f^{2} g + 3 \, b d^{2} e f g^{2} - b d^{3} g^{3}\right )} \log \left (c\right )}{6 \, {\left (e^{3} f^{6} g - 3 \, d e^{2} f^{5} g^{2} + 3 \, d^{2} e f^{4} g^{3} - d^{3} f^{3} g^{4} + {\left (e^{3} f^{3} g^{4} - 3 \, d e^{2} f^{2} g^{5} + 3 \, d^{2} e f g^{6} - d^{3} g^{7}\right )} x^{3} + 3 \, {\left (e^{3} f^{4} g^{3} - 3 \, d e^{2} f^{3} g^{4} + 3 \, d^{2} e f^{2} g^{5} - d^{3} f g^{6}\right )} x^{2} + 3 \, {\left (e^{3} f^{5} g^{2} - 3 \, d e^{2} f^{4} g^{3} + 3 \, d^{2} e f^{3} g^{4} - d^{3} f^{2} g^{5}\right )} x\right )}} \]
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Exception generated. \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^4} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (131) = 262\).
Time = 0.21 (sec) , antiderivative size = 301, normalized size of antiderivative = 2.13 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^4} \, dx=\frac {1}{6} \, {\left (\frac {2 \, e^{2} \log \left (e x + d\right )}{e^{3} f^{3} g - 3 \, d e^{2} f^{2} g^{2} + 3 \, d^{2} e f g^{3} - d^{3} g^{4}} - \frac {2 \, e^{2} \log \left (g x + f\right )}{e^{3} f^{3} g - 3 \, d e^{2} f^{2} g^{2} + 3 \, d^{2} e f g^{3} - d^{3} g^{4}} + \frac {2 \, e g x + 3 \, e f - d g}{e^{2} f^{4} g - 2 \, d e f^{3} g^{2} + d^{2} f^{2} g^{3} + {\left (e^{2} f^{2} g^{3} - 2 \, d e f g^{4} + d^{2} g^{5}\right )} x^{2} + 2 \, {\left (e^{2} f^{3} g^{2} - 2 \, d e f^{2} g^{3} + d^{2} f g^{4}\right )} x}\right )} b e n - \frac {b \log \left ({\left (e x + d\right )}^{n} c\right )}{3 \, {\left (g^{4} x^{3} + 3 \, f g^{3} x^{2} + 3 \, f^{2} g^{2} x + f^{3} g\right )}} - \frac {a}{3 \, {\left (g^{4} x^{3} + 3 \, f g^{3} x^{2} + 3 \, f^{2} g^{2} x + f^{3} g\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 401 vs. \(2 (131) = 262\).
Time = 0.31 (sec) , antiderivative size = 401, normalized size of antiderivative = 2.84 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^4} \, dx=\frac {b e^{3} n \log \left (e x + d\right )}{3 \, {\left (e^{3} f^{3} g - 3 \, d e^{2} f^{2} g^{2} + 3 \, d^{2} e f g^{3} - d^{3} g^{4}\right )}} - \frac {b e^{3} n \log \left (g x + f\right )}{3 \, {\left (e^{3} f^{3} g - 3 \, d e^{2} f^{2} g^{2} + 3 \, d^{2} e f g^{3} - d^{3} g^{4}\right )}} - \frac {b n \log \left (e x + d\right )}{3 \, {\left (g^{4} x^{3} + 3 \, f g^{3} x^{2} + 3 \, f^{2} g^{2} x + f^{3} g\right )}} + \frac {2 \, b e^{2} g^{2} n x^{2} + 5 \, b e^{2} f g n x - b d e g^{2} n x + 3 \, b e^{2} f^{2} n - b d e f g n - 2 \, b e^{2} f^{2} \log \left (c\right ) + 4 \, b d e f g \log \left (c\right ) - 2 \, b d^{2} g^{2} \log \left (c\right ) - 2 \, a e^{2} f^{2} + 4 \, a d e f g - 2 \, a d^{2} g^{2}}{6 \, {\left (e^{2} f^{2} g^{4} x^{3} - 2 \, d e f g^{5} x^{3} + d^{2} g^{6} x^{3} + 3 \, e^{2} f^{3} g^{3} x^{2} - 6 \, d e f^{2} g^{4} x^{2} + 3 \, d^{2} f g^{5} x^{2} + 3 \, e^{2} f^{4} g^{2} x - 6 \, d e f^{3} g^{3} x + 3 \, d^{2} f^{2} g^{4} x + e^{2} f^{5} g - 2 \, d e f^{4} g^{2} + d^{2} f^{3} g^{3}\right )}} \]
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Time = 1.19 (sec) , antiderivative size = 283, normalized size of antiderivative = 2.01 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^4} \, dx=\frac {2\,a\,d\,e\,f}{3\,{\left (f+g\,x\right )}^3\,{\left (d\,g-e\,f\right )}^2}-\frac {a\,d^2\,g}{3\,{\left (f+g\,x\right )}^3\,{\left (d\,g-e\,f\right )}^2}-\frac {b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{3\,g\,{\left (f+g\,x\right )}^3}-\frac {a\,e^2\,f^2}{3\,g\,{\left (f+g\,x\right )}^3\,{\left (d\,g-e\,f\right )}^2}+\frac {5\,b\,e^2\,f\,n\,x}{6\,{\left (f+g\,x\right )}^3\,{\left (d\,g-e\,f\right )}^2}+\frac {b\,e^2\,g\,n\,x^2}{3\,{\left (f+g\,x\right )}^3\,{\left (d\,g-e\,f\right )}^2}-\frac {b\,d\,e\,f\,n}{6\,{\left (f+g\,x\right )}^3\,{\left (d\,g-e\,f\right )}^2}+\frac {b\,e^2\,f^2\,n}{2\,g\,{\left (f+g\,x\right )}^3\,{\left (d\,g-e\,f\right )}^2}-\frac {b\,d\,e\,g\,n\,x}{6\,{\left (f+g\,x\right )}^3\,{\left (d\,g-e\,f\right )}^2}+\frac {b\,e^3\,n\,\mathrm {atan}\left (\frac {d\,g\,1{}\mathrm {i}+e\,f\,1{}\mathrm {i}+e\,g\,x\,2{}\mathrm {i}}{d\,g-e\,f}\right )\,2{}\mathrm {i}}{3\,g\,{\left (d\,g-e\,f\right )}^3} \]
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