Integrand size = 18, antiderivative size = 99 \[ \int x^4 \log \left (d \left (b x+c x^2\right )^n\right ) \, dx=-\frac {b^4 n x}{5 c^4}+\frac {b^3 n x^2}{10 c^3}-\frac {b^2 n x^3}{15 c^2}+\frac {b n x^4}{20 c}-\frac {2 n x^5}{25}+\frac {b^5 n \log (b+c x)}{5 c^5}+\frac {1}{5} x^5 \log \left (d \left (b x+c x^2\right )^n\right ) \]
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Time = 0.05 (sec) , antiderivative size = 99, 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 = {2605, 78} \[ \int x^4 \log \left (d \left (b x+c x^2\right )^n\right ) \, dx=\frac {b^5 n \log (b+c x)}{5 c^5}-\frac {b^4 n x}{5 c^4}+\frac {b^3 n x^2}{10 c^3}-\frac {b^2 n x^3}{15 c^2}+\frac {1}{5} x^5 \log \left (d \left (b x+c x^2\right )^n\right )+\frac {b n x^4}{20 c}-\frac {2 n x^5}{25} \]
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Rule 78
Rule 2605
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} x^5 \log \left (d \left (b x+c x^2\right )^n\right )-\frac {1}{5} n \int \frac {x^4 (b+2 c x)}{b+c x} \, dx \\ & = \frac {1}{5} x^5 \log \left (d \left (b x+c x^2\right )^n\right )-\frac {1}{5} n \int \left (\frac {b^4}{c^4}-\frac {b^3 x}{c^3}+\frac {b^2 x^2}{c^2}-\frac {b x^3}{c}+2 x^4-\frac {b^5}{c^4 (b+c x)}\right ) \, dx \\ & = -\frac {b^4 n x}{5 c^4}+\frac {b^3 n x^2}{10 c^3}-\frac {b^2 n x^3}{15 c^2}+\frac {b n x^4}{20 c}-\frac {2 n x^5}{25}+\frac {b^5 n \log (b+c x)}{5 c^5}+\frac {1}{5} x^5 \log \left (d \left (b x+c x^2\right )^n\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.86 \[ \int x^4 \log \left (d \left (b x+c x^2\right )^n\right ) \, dx=\frac {c n x \left (-60 b^4+30 b^3 c x-20 b^2 c^2 x^2+15 b c^3 x^3-24 c^4 x^4\right )+60 b^5 n \log (b+c x)+60 c^5 x^5 \log \left (d (x (b+c x))^n\right )}{300 c^5} \]
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Time = 0.35 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.87
method | result | size |
parts | \(\frac {x^{5} \ln \left (d \left (c \,x^{2}+b x \right )^{n}\right )}{5}-\frac {n \left (\frac {\frac {2}{5} c^{4} x^{5}-\frac {1}{4} b \,x^{4} c^{3}+\frac {1}{3} b^{2} c^{2} x^{3}-\frac {1}{2} b^{3} c \,x^{2}+b^{4} x}{c^{4}}-\frac {b^{5} \ln \left (x c +b \right )}{c^{5}}\right )}{5}\) | \(86\) |
parallelrisch | \(-\frac {-60 x^{5} \ln \left (d \left (x \left (x c +b \right )\right )^{n}\right ) c^{5} n +24 x^{5} c^{5} n^{2}-15 x^{4} b \,c^{4} n^{2}+20 x^{3} b^{2} c^{3} n^{2}-30 x^{2} b^{3} c^{2} n^{2}+60 \ln \left (x \right ) b^{5} n^{2}+60 x \,b^{4} c \,n^{2}-60 \ln \left (d \left (x \left (x c +b \right )\right )^{n}\right ) b^{5} n -60 b^{5} n^{2}}{300 c^{5} n}\) | \(128\) |
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Time = 0.34 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.99 \[ \int x^4 \log \left (d \left (b x+c x^2\right )^n\right ) \, dx=\frac {60 \, c^{5} n x^{5} \log \left (c x^{2} + b x\right ) - 24 \, c^{5} n x^{5} + 60 \, c^{5} x^{5} \log \left (d\right ) + 15 \, b c^{4} n x^{4} - 20 \, b^{2} c^{3} n x^{3} + 30 \, b^{3} c^{2} n x^{2} - 60 \, b^{4} c n x + 60 \, b^{5} n \log \left (c x + b\right )}{300 \, c^{5}} \]
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Time = 4.45 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.13 \[ \int x^4 \log \left (d \left (b x+c x^2\right )^n\right ) \, dx=\begin {cases} \frac {b^{5} n \log {\left (b + c x \right )}}{5 c^{5}} - \frac {b^{4} n x}{5 c^{4}} + \frac {b^{3} n x^{2}}{10 c^{3}} - \frac {b^{2} n x^{3}}{15 c^{2}} + \frac {b n x^{4}}{20 c} - \frac {2 n x^{5}}{25} + \frac {x^{5} \log {\left (d \left (b x + c x^{2}\right )^{n} \right )}}{5} & \text {for}\: c \neq 0 \\- \frac {n x^{5}}{25} + \frac {x^{5} \log {\left (d \left (b x\right )^{n} \right )}}{5} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.88 \[ \int x^4 \log \left (d \left (b x+c x^2\right )^n\right ) \, dx=\frac {1}{5} \, x^{5} \log \left ({\left (c x^{2} + b x\right )}^{n} d\right ) + \frac {1}{300} \, n {\left (\frac {60 \, b^{5} \log \left (c x + b\right )}{c^{5}} - \frac {24 \, c^{4} x^{5} - 15 \, b c^{3} x^{4} + 20 \, b^{2} c^{2} x^{3} - 30 \, b^{3} c x^{2} + 60 \, b^{4} x}{c^{4}}\right )} \]
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Time = 0.33 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.90 \[ \int x^4 \log \left (d \left (b x+c x^2\right )^n\right ) \, dx=\frac {1}{5} \, n x^{5} \log \left (c x^{2} + b x\right ) - \frac {1}{25} \, {\left (2 \, n - 5 \, \log \left (d\right )\right )} x^{5} + \frac {b n x^{4}}{20 \, c} - \frac {b^{2} n x^{3}}{15 \, c^{2}} + \frac {b^{3} n x^{2}}{10 \, c^{3}} - \frac {b^{4} n x}{5 \, c^{4}} + \frac {b^{5} n \log \left (c x + b\right )}{5 \, c^{5}} \]
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Time = 1.57 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.86 \[ \int x^4 \log \left (d \left (b x+c x^2\right )^n\right ) \, dx=\frac {x^5\,\ln \left (d\,{\left (c\,x^2+b\,x\right )}^n\right )}{5}-\frac {2\,n\,x^5}{25}-\frac {b^2\,n\,x^3}{15\,c^2}+\frac {b^3\,n\,x^2}{10\,c^3}+\frac {b^5\,n\,\ln \left (b+c\,x\right )}{5\,c^5}+\frac {b\,n\,x^4}{20\,c}-\frac {b^4\,n\,x}{5\,c^4} \]
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