Integrand size = 22, antiderivative size = 178 \[ \int (f+g x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=-\frac {b (e f-d g)^4 n x}{5 e^4}-\frac {b (e f-d g)^3 n (f+g x)^2}{10 e^3 g}-\frac {b (e f-d g)^2 n (f+g x)^3}{15 e^2 g}-\frac {b (e f-d g) n (f+g x)^4}{20 e g}-\frac {b n (f+g x)^5}{25 g}-\frac {b (e f-d g)^5 n \log (d+e x)}{5 e^5 g}+\frac {(f+g x)^5 \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g} \]
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Time = 0.07 (sec) , antiderivative size = 178, 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, 45} \[ \int (f+g x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=\frac {(f+g x)^5 \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g}-\frac {b n (e f-d g)^5 \log (d+e x)}{5 e^5 g}-\frac {b n x (e f-d g)^4}{5 e^4}-\frac {b n (f+g x)^2 (e f-d g)^3}{10 e^3 g}-\frac {b n (f+g x)^3 (e f-d g)^2}{15 e^2 g}-\frac {b n (f+g x)^4 (e f-d g)}{20 e g}-\frac {b n (f+g x)^5}{25 g} \]
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Rule 45
Rule 2442
Rubi steps \begin{align*} \text {integral}& = \frac {(f+g x)^5 \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g}-\frac {(b e n) \int \frac {(f+g x)^5}{d+e x} \, dx}{5 g} \\ & = \frac {(f+g x)^5 \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g}-\frac {(b e n) \int \left (\frac {g (e f-d g)^4}{e^5}+\frac {(e f-d g)^5}{e^5 (d+e x)}+\frac {g (e f-d g)^3 (f+g x)}{e^4}+\frac {g (e f-d g)^2 (f+g x)^2}{e^3}+\frac {g (e f-d g) (f+g x)^3}{e^2}+\frac {g (f+g x)^4}{e}\right ) \, dx}{5 g} \\ & = -\frac {b (e f-d g)^4 n x}{5 e^4}-\frac {b (e f-d g)^3 n (f+g x)^2}{10 e^3 g}-\frac {b (e f-d g)^2 n (f+g x)^3}{15 e^2 g}-\frac {b (e f-d g) n (f+g x)^4}{20 e g}-\frac {b n (f+g x)^5}{25 g}-\frac {b (e f-d g)^5 n \log (d+e x)}{5 e^5 g}+\frac {(f+g x)^5 \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.77 \[ \int (f+g x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=\frac {e x \left (60 a e^4 \left (5 f^4+10 f^3 g x+10 f^2 g^2 x^2+5 f g^3 x^3+g^4 x^4\right )-b n \left (60 d^4 g^4-30 d^3 e g^3 (10 f+g x)+10 d^2 e^2 g^2 \left (60 f^2+15 f g x+2 g^2 x^2\right )-5 d e^3 g \left (120 f^3+60 f^2 g x+20 f g^2 x^2+3 g^3 x^3\right )+e^4 \left (300 f^4+300 f^3 g x+200 f^2 g^2 x^2+75 f g^3 x^3+12 g^4 x^4\right )\right )\right )+60 b d^2 g \left (-10 e^3 f^3+10 d e^2 f^2 g-5 d^2 e f g^2+d^3 g^3\right ) n \log (d+e x)+60 b e^4 \left (5 d f^4+e x \left (5 f^4+10 f^3 g x+10 f^2 g^2 x^2+5 f g^3 x^3+g^4 x^4\right )\right ) \log \left (c (d+e x)^n\right )}{300 e^5} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(588\) vs. \(2(164)=328\).
Time = 1.62 (sec) , antiderivative size = 589, normalized size of antiderivative = 3.31
method | result | size |
parallelrisch | \(\frac {300 b d \,e^{4} f^{4} n +300 x \ln \left (c \left (e x +d \right )^{n}\right ) b \,e^{5} f^{4}-300 x b \,e^{5} f^{4} n -300 \ln \left (c \left (e x +d \right )^{n}\right ) b d \,e^{4} f^{4}+60 \ln \left (e x +d \right ) b \,d^{5} g^{4} n +60 x^{5} \ln \left (c \left (e x +d \right )^{n}\right ) b \,e^{5} g^{4}-12 x^{5} b \,e^{5} g^{4} n +300 x^{4} a \,e^{5} f \,g^{3}+600 x^{3} a \,e^{5} f^{2} g^{2}+600 x^{2} a \,e^{5} f^{3} g +60 x^{5} a \,e^{5} g^{4}+600 \ln \left (e x +d \right ) b d \,e^{4} f^{4} n +15 x^{4} b d \,e^{4} g^{4} n -75 x^{4} b \,e^{5} f \,g^{3} n +600 x^{3} \ln \left (c \left (e x +d \right )^{n}\right ) b \,e^{5} f^{2} g^{2}-20 x^{3} b \,d^{2} e^{3} g^{4} n -200 x^{3} b \,e^{5} f^{2} g^{2} n +600 x^{2} \ln \left (c \left (e x +d \right )^{n}\right ) b \,e^{5} f^{3} g +30 x^{2} b \,d^{3} e^{2} g^{4} n -300 x^{2} b \,e^{5} f^{3} g n -60 x b \,d^{4} e \,g^{4} n +300 x^{4} \ln \left (c \left (e x +d \right )^{n}\right ) b \,e^{5} f \,g^{3}-300 b \,d^{4} e f \,g^{3} n +600 b \,d^{3} e^{2} f^{2} g^{2} n -300 a d \,e^{4} f^{4}-600 b \,d^{2} e^{3} f^{3} g n +300 x a \,e^{5} f^{4}-600 x b \,d^{2} e^{3} f^{2} g^{2} n +600 x b d \,e^{4} f^{3} g n -300 \ln \left (e x +d \right ) b \,d^{4} e f \,g^{3} n +600 \ln \left (e x +d \right ) b \,d^{3} e^{2} f^{2} g^{2} n +100 x^{3} b d \,e^{4} f \,g^{3} n -150 x^{2} b \,d^{2} e^{3} f \,g^{3} n +300 x^{2} b d \,e^{4} f^{2} g^{2} n +300 x b \,d^{3} e^{2} f \,g^{3} n -600 \ln \left (e x +d \right ) b \,d^{2} e^{3} f^{3} g n +60 b \,d^{5} g^{4} n}{300 e^{5}}\) | \(589\) |
risch | \(\text {Expression too large to display}\) | \(1105\) |
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Leaf count of result is larger than twice the leaf count of optimal. 471 vs. \(2 (164) = 328\).
Time = 0.32 (sec) , antiderivative size = 471, normalized size of antiderivative = 2.65 \[ \int (f+g x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=-\frac {12 \, {\left (b e^{5} g^{4} n - 5 \, a e^{5} g^{4}\right )} x^{5} - 15 \, {\left (20 \, a e^{5} f g^{3} - {\left (5 \, b e^{5} f g^{3} - b d e^{4} g^{4}\right )} n\right )} x^{4} - 20 \, {\left (30 \, a e^{5} f^{2} g^{2} - {\left (10 \, b e^{5} f^{2} g^{2} - 5 \, b d e^{4} f g^{3} + b d^{2} e^{3} g^{4}\right )} n\right )} x^{3} - 30 \, {\left (20 \, a e^{5} f^{3} g - {\left (10 \, b e^{5} f^{3} g - 10 \, b d e^{4} f^{2} g^{2} + 5 \, b d^{2} e^{3} f g^{3} - b d^{3} e^{2} g^{4}\right )} n\right )} x^{2} - 60 \, {\left (5 \, a e^{5} f^{4} - {\left (5 \, b e^{5} f^{4} - 10 \, b d e^{4} f^{3} g + 10 \, b d^{2} e^{3} f^{2} g^{2} - 5 \, b d^{3} e^{2} f g^{3} + b d^{4} e g^{4}\right )} n\right )} x - 60 \, {\left (b e^{5} g^{4} n x^{5} + 5 \, b e^{5} f g^{3} n x^{4} + 10 \, b e^{5} f^{2} g^{2} n x^{3} + 10 \, b e^{5} f^{3} g n x^{2} + 5 \, b e^{5} f^{4} n x + {\left (5 \, b d e^{4} f^{4} - 10 \, b d^{2} e^{3} f^{3} g + 10 \, b d^{3} e^{2} f^{2} g^{2} - 5 \, b d^{4} e f g^{3} + b d^{5} g^{4}\right )} n\right )} \log \left (e x + d\right ) - 60 \, {\left (b e^{5} g^{4} x^{5} + 5 \, b e^{5} f g^{3} x^{4} + 10 \, b e^{5} f^{2} g^{2} x^{3} + 10 \, b e^{5} f^{3} g x^{2} + 5 \, b e^{5} f^{4} x\right )} \log \left (c\right )}{300 \, e^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 568 vs. \(2 (153) = 306\).
Time = 2.17 (sec) , antiderivative size = 568, normalized size of antiderivative = 3.19 \[ \int (f+g x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=\begin {cases} a f^{4} x + 2 a f^{3} g x^{2} + 2 a f^{2} g^{2} x^{3} + a f g^{3} x^{4} + \frac {a g^{4} x^{5}}{5} + \frac {b d^{5} g^{4} \log {\left (c \left (d + e x\right )^{n} \right )}}{5 e^{5}} - \frac {b d^{4} f g^{3} \log {\left (c \left (d + e x\right )^{n} \right )}}{e^{4}} - \frac {b d^{4} g^{4} n x}{5 e^{4}} + \frac {2 b d^{3} f^{2} g^{2} \log {\left (c \left (d + e x\right )^{n} \right )}}{e^{3}} + \frac {b d^{3} f g^{3} n x}{e^{3}} + \frac {b d^{3} g^{4} n x^{2}}{10 e^{3}} - \frac {2 b d^{2} f^{3} g \log {\left (c \left (d + e x\right )^{n} \right )}}{e^{2}} - \frac {2 b d^{2} f^{2} g^{2} n x}{e^{2}} - \frac {b d^{2} f g^{3} n x^{2}}{2 e^{2}} - \frac {b d^{2} g^{4} n x^{3}}{15 e^{2}} + \frac {b d f^{4} \log {\left (c \left (d + e x\right )^{n} \right )}}{e} + \frac {2 b d f^{3} g n x}{e} + \frac {b d f^{2} g^{2} n x^{2}}{e} + \frac {b d f g^{3} n x^{3}}{3 e} + \frac {b d g^{4} n x^{4}}{20 e} - b f^{4} n x + b f^{4} x \log {\left (c \left (d + e x\right )^{n} \right )} - b f^{3} g n x^{2} + 2 b f^{3} g x^{2} \log {\left (c \left (d + e x\right )^{n} \right )} - \frac {2 b f^{2} g^{2} n x^{3}}{3} + 2 b f^{2} g^{2} x^{3} \log {\left (c \left (d + e x\right )^{n} \right )} - \frac {b f g^{3} n x^{4}}{4} + b f g^{3} x^{4} \log {\left (c \left (d + e x\right )^{n} \right )} - \frac {b g^{4} n x^{5}}{25} + \frac {b g^{4} x^{5} \log {\left (c \left (d + e x\right )^{n} \right )}}{5} & \text {for}\: e \neq 0 \\\left (a + b \log {\left (c d^{n} \right )}\right ) \left (f^{4} x + 2 f^{3} g x^{2} + 2 f^{2} g^{2} x^{3} + f g^{3} x^{4} + \frac {g^{4} x^{5}}{5}\right ) & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 393 vs. \(2 (164) = 328\).
Time = 0.21 (sec) , antiderivative size = 393, normalized size of antiderivative = 2.21 \[ \int (f+g x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=\frac {1}{5} \, b g^{4} x^{5} \log \left ({\left (e x + d\right )}^{n} c\right ) + \frac {1}{5} \, a g^{4} x^{5} + b f g^{3} x^{4} \log \left ({\left (e x + d\right )}^{n} c\right ) + a f g^{3} x^{4} + 2 \, b f^{2} g^{2} x^{3} \log \left ({\left (e x + d\right )}^{n} c\right ) + 2 \, a f^{2} g^{2} x^{3} - b e f^{4} n {\left (\frac {x}{e} - \frac {d \log \left (e x + d\right )}{e^{2}}\right )} + \frac {1}{300} \, b e g^{4} n {\left (\frac {60 \, d^{5} \log \left (e x + d\right )}{e^{6}} - \frac {12 \, e^{4} x^{5} - 15 \, d e^{3} x^{4} + 20 \, d^{2} e^{2} x^{3} - 30 \, d^{3} e x^{2} + 60 \, d^{4} x}{e^{5}}\right )} - \frac {1}{12} \, b e f g^{3} n {\left (\frac {12 \, d^{4} \log \left (e x + d\right )}{e^{5}} + \frac {3 \, e^{3} x^{4} - 4 \, d e^{2} x^{3} + 6 \, d^{2} e x^{2} - 12 \, d^{3} x}{e^{4}}\right )} + \frac {1}{3} \, b e f^{2} g^{2} n {\left (\frac {6 \, d^{3} \log \left (e x + d\right )}{e^{4}} - \frac {2 \, e^{2} x^{3} - 3 \, d e x^{2} + 6 \, d^{2} x}{e^{3}}\right )} - b e f^{3} g n {\left (\frac {2 \, d^{2} \log \left (e x + d\right )}{e^{3}} + \frac {e x^{2} - 2 \, d x}{e^{2}}\right )} + 2 \, b f^{3} g x^{2} \log \left ({\left (e x + d\right )}^{n} c\right ) + 2 \, a f^{3} g x^{2} + b f^{4} x \log \left ({\left (e x + d\right )}^{n} c\right ) + a f^{4} x \]
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Leaf count of result is larger than twice the leaf count of optimal. 1209 vs. \(2 (164) = 328\).
Time = 0.32 (sec) , antiderivative size = 1209, normalized size of antiderivative = 6.79 \[ \int (f+g x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=\text {Too large to display} \]
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Time = 1.47 (sec) , antiderivative size = 526, normalized size of antiderivative = 2.96 \[ \int (f+g x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=x\,\left (\frac {5\,a\,e\,f^4+20\,a\,d\,f^3\,g-5\,b\,e\,f^4\,n}{5\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {g^3\,\left (a\,d\,g+4\,a\,e\,f-b\,e\,f\,n\right )}{e}-\frac {d\,g^4\,\left (5\,a-b\,n\right )}{5\,e}\right )}{e}-\frac {2\,f\,g^2\,\left (2\,a\,d\,g+3\,a\,e\,f-b\,e\,f\,n\right )}{e}\right )}{e}+\frac {2\,f^2\,g\,\left (3\,a\,d\,g+2\,a\,e\,f-b\,e\,f\,n\right )}{e}\right )}{e}\right )-x^3\,\left (\frac {d\,\left (\frac {g^3\,\left (a\,d\,g+4\,a\,e\,f-b\,e\,f\,n\right )}{e}-\frac {d\,g^4\,\left (5\,a-b\,n\right )}{5\,e}\right )}{3\,e}-\frac {2\,f\,g^2\,\left (2\,a\,d\,g+3\,a\,e\,f-b\,e\,f\,n\right )}{3\,e}\right )+x^4\,\left (\frac {g^3\,\left (a\,d\,g+4\,a\,e\,f-b\,e\,f\,n\right )}{4\,e}-\frac {d\,g^4\,\left (5\,a-b\,n\right )}{20\,e}\right )+\ln \left (c\,{\left (d+e\,x\right )}^n\right )\,\left (b\,f^4\,x+2\,b\,f^3\,g\,x^2+2\,b\,f^2\,g^2\,x^3+b\,f\,g^3\,x^4+\frac {b\,g^4\,x^5}{5}\right )+x^2\,\left (\frac {d\,\left (\frac {d\,\left (\frac {g^3\,\left (a\,d\,g+4\,a\,e\,f-b\,e\,f\,n\right )}{e}-\frac {d\,g^4\,\left (5\,a-b\,n\right )}{5\,e}\right )}{e}-\frac {2\,f\,g^2\,\left (2\,a\,d\,g+3\,a\,e\,f-b\,e\,f\,n\right )}{e}\right )}{2\,e}+\frac {f^2\,g\,\left (3\,a\,d\,g+2\,a\,e\,f-b\,e\,f\,n\right )}{e}\right )+\frac {g^4\,x^5\,\left (5\,a-b\,n\right )}{25}+\frac {\ln \left (d+e\,x\right )\,\left (b\,n\,d^5\,g^4-5\,b\,n\,d^4\,e\,f\,g^3+10\,b\,n\,d^3\,e^2\,f^2\,g^2-10\,b\,n\,d^2\,e^3\,f^3\,g+5\,b\,n\,d\,e^4\,f^4\right )}{5\,e^5} \]
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