Integrand size = 35, antiderivative size = 146 \[ \int \frac {(d+e x)^8}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {e^4 x}{c^4 d^4}-\frac {\left (c d^2-a e^2\right )^4}{3 c^5 d^5 (a e+c d x)^3}-\frac {2 e \left (c d^2-a e^2\right )^3}{c^5 d^5 (a e+c d x)^2}-\frac {6 e^2 \left (c d^2-a e^2\right )^2}{c^5 d^5 (a e+c d x)}+\frac {4 e^3 \left (c d^2-a e^2\right ) \log (a e+c d x)}{c^5 d^5} \]
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Time = 0.09 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 45} \[ \int \frac {(d+e x)^8}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {6 e^2 \left (c d^2-a e^2\right )^2}{c^5 d^5 (a e+c d x)}-\frac {2 e \left (c d^2-a e^2\right )^3}{c^5 d^5 (a e+c d x)^2}-\frac {\left (c d^2-a e^2\right )^4}{3 c^5 d^5 (a e+c d x)^3}+\frac {4 e^3 \left (c d^2-a e^2\right ) \log (a e+c d x)}{c^5 d^5}+\frac {e^4 x}{c^4 d^4} \]
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Rule 45
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int \frac {(d+e x)^4}{(a e+c d x)^4} \, dx \\ & = \int \left (\frac {e^4}{c^4 d^4}+\frac {\left (c d^2-a e^2\right )^4}{c^4 d^4 (a e+c d x)^4}+\frac {4 e \left (c d^2-a e^2\right )^3}{c^4 d^4 (a e+c d x)^3}+\frac {6 \left (c d^2 e-a e^3\right )^2}{c^4 d^4 (a e+c d x)^2}+\frac {4 \left (c d^2 e^3-a e^5\right )}{c^4 d^4 (a e+c d x)}\right ) \, dx \\ & = \frac {e^4 x}{c^4 d^4}-\frac {\left (c d^2-a e^2\right )^4}{3 c^5 d^5 (a e+c d x)^3}-\frac {2 e \left (c d^2-a e^2\right )^3}{c^5 d^5 (a e+c d x)^2}-\frac {6 e^2 \left (c d^2-a e^2\right )^2}{c^5 d^5 (a e+c d x)}+\frac {4 e^3 \left (c d^2-a e^2\right ) \log (a e+c d x)}{c^5 d^5} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.33 \[ \int \frac {(d+e x)^8}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {-13 a^4 e^8+a^3 c d e^6 (22 d-27 e x)-3 a^2 c^2 d^2 e^4 \left (2 d^2-18 d e x+3 e^2 x^2\right )+a c^3 d^3 e^2 \left (-2 d^3-18 d^2 e x+36 d e^2 x^2+9 e^3 x^3\right )-c^4 \left (d^8+6 d^7 e x+18 d^6 e^2 x^2-3 d^4 e^4 x^4\right )-12 e^3 \left (-c d^2+a e^2\right ) (a e+c d x)^3 \log (a e+c d x)}{3 c^5 d^5 (a e+c d x)^3} \]
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Time = 2.48 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.48
method | result | size |
risch | \(\frac {e^{4} x}{c^{4} d^{4}}+\frac {\left (-6 d \,e^{6} a^{2} c +12 a \,c^{2} d^{3} e^{4}-6 c^{3} d^{5} e^{2}\right ) x^{2}-2 e \left (5 e^{6} a^{3}-9 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a +c^{3} d^{6}\right ) x -\frac {13 a^{4} e^{8}-22 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}+2 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}{3 c d}}{c^{4} d^{4} \left (c d x +a e \right )^{3}}-\frac {4 e^{5} \ln \left (c d x +a e \right ) a}{c^{5} d^{5}}+\frac {4 e^{3} \ln \left (c d x +a e \right )}{c^{4} d^{3}}\) | \(216\) |
default | \(\frac {e^{4} x}{c^{4} d^{4}}+\frac {2 e \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}\right )}{c^{5} d^{5} \left (c d x +a e \right )^{2}}-\frac {a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}{3 c^{5} d^{5} \left (c d x +a e \right )^{3}}-\frac {4 e^{3} \left (e^{2} a -c \,d^{2}\right ) \ln \left (c d x +a e \right )}{c^{5} d^{5}}-\frac {6 e^{2} \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{c^{5} d^{5} \left (c d x +a e \right )}\) | \(221\) |
parallelrisch | \(-\frac {-12 \ln \left (c d x +a e \right ) x^{3} c^{4} d^{5} e^{3}+36 a^{2} c^{2} d^{2} e^{6} x^{2}-36 a \,c^{3} d^{4} e^{4} x^{2}+54 a^{3} c d \,e^{7} x -54 a^{2} c^{2} d^{3} e^{5} x +18 a \,c^{3} d^{5} e^{3} x -3 c^{4} d^{4} e^{4} x^{4}+18 c^{4} d^{6} e^{2} x^{2}+6 c^{4} d^{7} e x -22 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}+2 a \,c^{3} d^{6} e^{2}+12 \ln \left (c d x +a e \right ) x^{3} a \,c^{3} d^{3} e^{5}-36 \ln \left (c d x +a e \right ) x^{2} a \,c^{3} d^{4} e^{4}+36 \ln \left (c d x +a e \right ) x^{2} a^{2} c^{2} d^{2} e^{6}-36 \ln \left (c d x +a e \right ) x \,a^{2} c^{2} d^{3} e^{5}+36 \ln \left (c d x +a e \right ) x \,a^{3} c d \,e^{7}+c^{4} d^{8}+22 a^{4} e^{8}+12 \ln \left (c d x +a e \right ) a^{4} e^{8}-12 \ln \left (c d x +a e \right ) a^{3} c \,d^{2} e^{6}}{3 c^{5} d^{5} \left (c d x +a e \right )^{3}}\) | \(361\) |
norman | \(\frac {\frac {e^{7} x^{7}}{c d}-\frac {22 a^{4} e^{8}-13 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}+2 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}{3 d^{2} c^{5}}-\frac {\left (22 a^{4} e^{14}+149 a^{3} c \,d^{2} e^{12}+33 d^{4} a^{2} c^{2} e^{10}+29 a \,c^{3} d^{6} e^{8}+82 c^{4} d^{8} e^{6}\right ) x^{3}}{3 c^{5} d^{5} e^{3}}-\frac {\left (22 a^{4} e^{12}+41 a^{3} c \,d^{2} e^{10}-9 d^{4} a^{2} c^{2} e^{8}+17 a \,c^{3} d^{6} e^{6}+13 c^{4} d^{8} e^{4}\right ) x^{2}}{c^{5} d^{4} e^{2}}-\frac {\left (22 a^{4} e^{10}+5 a^{3} c \,d^{2} e^{8}-3 c^{2} d^{4} e^{6} a^{2}+8 c^{3} d^{6} e^{4} a +3 c^{4} d^{8} e^{2}\right ) x}{c^{5} d^{3} e}-\frac {\left (18 a^{3} e^{12}+27 c \,a^{2} d^{2} e^{10}-3 d^{4} c^{2} a \,e^{8}+28 d^{6} e^{6} c^{3}\right ) x^{4}}{c^{4} d^{4} e^{2}}-\frac {3 \left (4 a^{2} e^{10}-a c \,d^{2} e^{8}+4 c^{2} d^{4} e^{6}\right ) x^{5}}{c^{3} d^{3} e}}{\left (c d x +a e \right )^{3} \left (e x +d \right )^{3}}-\frac {4 e^{3} \left (e^{2} a -c \,d^{2}\right ) \ln \left (c d x +a e \right )}{c^{5} d^{5}}\) | \(444\) |
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Leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (144) = 288\).
Time = 0.34 (sec) , antiderivative size = 347, normalized size of antiderivative = 2.38 \[ \int \frac {(d+e x)^8}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {3 \, c^{4} d^{4} e^{4} x^{4} + 9 \, a c^{3} d^{3} e^{5} x^{3} - c^{4} d^{8} - 2 \, a c^{3} d^{6} e^{2} - 6 \, a^{2} c^{2} d^{4} e^{4} + 22 \, a^{3} c d^{2} e^{6} - 13 \, a^{4} e^{8} - 9 \, {\left (2 \, c^{4} d^{6} e^{2} - 4 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x^{2} - 3 \, {\left (2 \, c^{4} d^{7} e + 6 \, a c^{3} d^{5} e^{3} - 18 \, a^{2} c^{2} d^{3} e^{5} + 9 \, a^{3} c d e^{7}\right )} x + 12 \, {\left (a^{3} c d^{2} e^{6} - a^{4} e^{8} + {\left (c^{4} d^{5} e^{3} - a c^{3} d^{3} e^{5}\right )} x^{3} + 3 \, {\left (a c^{3} d^{4} e^{4} - a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 3 \, {\left (a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x\right )} \log \left (c d x + a e\right )}{3 \, {\left (c^{8} d^{8} x^{3} + 3 \, a c^{7} d^{7} e x^{2} + 3 \, a^{2} c^{6} d^{6} e^{2} x + a^{3} c^{5} d^{5} e^{3}\right )}} \]
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Time = 132.67 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.76 \[ \int \frac {(d+e x)^8}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {- 13 a^{4} e^{8} + 22 a^{3} c d^{2} e^{6} - 6 a^{2} c^{2} d^{4} e^{4} - 2 a c^{3} d^{6} e^{2} - c^{4} d^{8} + x^{2} \left (- 18 a^{2} c^{2} d^{2} e^{6} + 36 a c^{3} d^{4} e^{4} - 18 c^{4} d^{6} e^{2}\right ) + x \left (- 30 a^{3} c d e^{7} + 54 a^{2} c^{2} d^{3} e^{5} - 18 a c^{3} d^{5} e^{3} - 6 c^{4} d^{7} e\right )}{3 a^{3} c^{5} d^{5} e^{3} + 9 a^{2} c^{6} d^{6} e^{2} x + 9 a c^{7} d^{7} e x^{2} + 3 c^{8} d^{8} x^{3}} + \frac {e^{4} x}{c^{4} d^{4}} - \frac {4 e^{3} \left (a e^{2} - c d^{2}\right ) \log {\left (a e + c d x \right )}}{c^{5} d^{5}} \]
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Time = 0.21 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.66 \[ \int \frac {(d+e x)^8}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {c^{4} d^{8} + 2 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 22 \, a^{3} c d^{2} e^{6} + 13 \, a^{4} e^{8} + 18 \, {\left (c^{4} d^{6} e^{2} - 2 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 6 \, {\left (c^{4} d^{7} e + 3 \, a c^{3} d^{5} e^{3} - 9 \, a^{2} c^{2} d^{3} e^{5} + 5 \, a^{3} c d e^{7}\right )} x}{3 \, {\left (c^{8} d^{8} x^{3} + 3 \, a c^{7} d^{7} e x^{2} + 3 \, a^{2} c^{6} d^{6} e^{2} x + a^{3} c^{5} d^{5} e^{3}\right )}} + \frac {e^{4} x}{c^{4} d^{4}} + \frac {4 \, {\left (c d^{2} e^{3} - a e^{5}\right )} \log \left (c d x + a e\right )}{c^{5} d^{5}} \]
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Time = 0.27 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.41 \[ \int \frac {(d+e x)^8}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {e^{4} x}{c^{4} d^{4}} + \frac {4 \, {\left (c d^{2} e^{3} - a e^{5}\right )} \log \left ({\left | c d x + a e \right |}\right )}{c^{5} d^{5}} - \frac {c^{4} d^{8} + 2 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 22 \, a^{3} c d^{2} e^{6} + 13 \, a^{4} e^{8} + 18 \, {\left (c^{4} d^{6} e^{2} - 2 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 6 \, {\left (c^{4} d^{7} e + 3 \, a c^{3} d^{5} e^{3} - 9 \, a^{2} c^{2} d^{3} e^{5} + 5 \, a^{3} c d e^{7}\right )} x}{3 \, {\left (c d x + a e\right )}^{3} c^{5} d^{5}} \]
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Time = 0.17 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.68 \[ \int \frac {(d+e x)^8}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {e^4\,x}{c^4\,d^4}-\frac {x\,\left (10\,a^3\,e^7-18\,a^2\,c\,d^2\,e^5+6\,a\,c^2\,d^4\,e^3+2\,c^3\,d^6\,e\right )+x^2\,\left (6\,a^2\,c\,d\,e^6-12\,a\,c^2\,d^3\,e^4+6\,c^3\,d^5\,e^2\right )+\frac {13\,a^4\,e^8-22\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4+2\,a\,c^3\,d^6\,e^2+c^4\,d^8}{3\,c\,d}}{a^3\,c^4\,d^4\,e^3+3\,a^2\,c^5\,d^5\,e^2\,x+3\,a\,c^6\,d^6\,e\,x^2+c^7\,d^7\,x^3}-\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (4\,a\,e^5-4\,c\,d^2\,e^3\right )}{c^5\,d^5} \]
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