Integrand size = 35, antiderivative size = 131 \[ \int \frac {(d+e x)^5}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {e \left (c d^2-a e^2\right )^3 x}{c^4 d^4}+\frac {\left (c d^2-a e^2\right )^2 (d+e x)^2}{2 c^3 d^3}+\frac {\left (c d^2-a e^2\right ) (d+e x)^3}{3 c^2 d^2}+\frac {(d+e x)^4}{4 c d}+\frac {\left (c d^2-a e^2\right )^4 \log (a e+c d x)}{c^5 d^5} \]
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Time = 0.05 (sec) , antiderivative size = 131, 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} \[ \int \frac {(d+e x)^5}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {\left (c d^2-a e^2\right )^4 \log (a e+c d x)}{c^5 d^5}+\frac {e x \left (c d^2-a e^2\right )^3}{c^4 d^4}+\frac {(d+e x)^2 \left (c d^2-a e^2\right )^2}{2 c^3 d^3}+\frac {(d+e x)^3 \left (c d^2-a e^2\right )}{3 c^2 d^2}+\frac {(d+e x)^4}{4 c d} \]
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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} \, dx \\ & = \int \left (\frac {e \left (c d^2-a e^2\right )^3}{c^4 d^4}+\frac {\left (c d^2-a e^2\right )^4}{c^4 d^4 (a e+c d x)}+\frac {e \left (c d^2-a e^2\right )^2 (d+e x)}{c^3 d^3}+\frac {e \left (c d^2-a e^2\right ) (d+e x)^2}{c^2 d^2}+\frac {e (d+e x)^3}{c d}\right ) \, dx \\ & = \frac {e \left (c d^2-a e^2\right )^3 x}{c^4 d^4}+\frac {\left (c d^2-a e^2\right )^2 (d+e x)^2}{2 c^3 d^3}+\frac {\left (c d^2-a e^2\right ) (d+e x)^3}{3 c^2 d^2}+\frac {(d+e x)^4}{4 c d}+\frac {\left (c d^2-a e^2\right )^4 \log (a e+c d x)}{c^5 d^5} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.02 \[ \int \frac {(d+e x)^5}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {c d e x \left (-12 a^3 e^6+6 a^2 c d e^4 (8 d+e x)-4 a c^2 d^2 e^2 \left (18 d^2+6 d e x+e^2 x^2\right )+c^3 d^3 \left (48 d^3+36 d^2 e x+16 d e^2 x^2+3 e^3 x^3\right )\right )+12 \left (c d^2-a e^2\right )^4 \log (a e+c d x)}{12 c^5 d^5} \]
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Time = 2.81 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.53
method | result | size |
norman | \(\frac {e^{4} x^{4}}{4 c d}+\frac {e^{2} \left (a^{2} e^{4}-4 a c \,d^{2} e^{2}+6 c^{2} d^{4}\right ) x^{2}}{2 c^{3} d^{3}}-\frac {e^{3} \left (e^{2} a -4 c \,d^{2}\right ) x^{3}}{3 c^{2} d^{2}}-\frac {e \left (e^{6} a^{3}-4 d^{2} e^{4} a^{2} c +6 d^{4} e^{2} c^{2} a -4 c^{3} d^{6}\right ) x}{c^{4} d^{4}}+\frac {\left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \ln \left (c d x +a e \right )}{c^{5} d^{5}}\) | \(201\) |
default | \(-\frac {e \left (-\frac {x^{4} c^{3} d^{3} e^{3}}{4}+\frac {\left (\left (e^{2} a -2 c \,d^{2}\right ) c^{2} d^{2} e^{2}-2 c^{3} d^{4} e^{2}\right ) x^{3}}{3}+\frac {\left (2 \left (e^{2} a -2 c \,d^{2}\right ) d^{3} e \,c^{2}-c d e \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+2 c^{2} d^{4}\right )\right ) x^{2}}{2}+\left (e^{2} a -2 c \,d^{2}\right ) \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+2 c^{2} d^{4}\right ) x \right )}{c^{4} d^{4}}+\frac {\left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \ln \left (c d x +a e \right )}{c^{5} d^{5}}\) | \(232\) |
risch | \(\frac {e^{4} x^{4}}{4 c d}-\frac {e^{5} x^{3} a}{3 c^{2} d^{2}}+\frac {4 e^{3} x^{3}}{3 c}-\frac {2 e^{4} x^{2} a}{c^{2} d}+\frac {3 e^{2} d \,x^{2}}{c}+\frac {e^{6} x^{2} a^{2}}{2 c^{3} d^{3}}-\frac {e^{7} a^{3} x}{c^{4} d^{4}}+\frac {4 e^{5} a^{2} x}{c^{3} d^{2}}-\frac {6 e^{3} a x}{c^{2}}+\frac {4 e \,d^{2} x}{c}+\frac {\ln \left (c d x +a e \right ) a^{4} e^{8}}{c^{5} d^{5}}-\frac {4 \ln \left (c d x +a e \right ) a^{3} e^{6}}{c^{4} d^{3}}+\frac {6 \ln \left (c d x +a e \right ) a^{2} e^{4}}{c^{3} d}-\frac {4 d \ln \left (c d x +a e \right ) a \,e^{2}}{c^{2}}+\frac {d^{3} \ln \left (c d x +a e \right )}{c}\) | \(239\) |
parallelrisch | \(\frac {3 c^{4} d^{4} e^{4} x^{4}-4 a \,c^{3} d^{3} e^{5} x^{3}+16 c^{4} d^{5} e^{3} x^{3}+6 a^{2} c^{2} d^{2} e^{6} x^{2}-24 a \,c^{3} d^{4} e^{4} x^{2}+36 c^{4} d^{6} e^{2} x^{2}+12 \ln \left (c d x +a e \right ) a^{4} e^{8}-48 \ln \left (c d x +a e \right ) a^{3} c \,d^{2} e^{6}+72 \ln \left (c d x +a e \right ) a^{2} c^{2} d^{4} e^{4}-48 \ln \left (c d x +a e \right ) a \,c^{3} d^{6} e^{2}+12 \ln \left (c d x +a e \right ) c^{4} d^{8}-12 a^{3} c d \,e^{7} x +48 a^{2} c^{2} d^{3} e^{5} x -72 a \,c^{3} d^{5} e^{3} x +48 c^{4} d^{7} e x}{12 c^{5} d^{5}}\) | \(247\) |
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Time = 0.31 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.58 \[ \int \frac {(d+e x)^5}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {3 \, c^{4} d^{4} e^{4} x^{4} + 4 \, {\left (4 \, c^{4} d^{5} e^{3} - a c^{3} d^{3} e^{5}\right )} x^{3} + 6 \, {\left (6 \, c^{4} d^{6} e^{2} - 4 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 12 \, {\left (4 \, c^{4} d^{7} e - 6 \, a c^{3} d^{5} e^{3} + 4 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x + 12 \, {\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \log \left (c d x + a e\right )}{12 \, c^{5} d^{5}} \]
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Time = 0.26 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.17 \[ \int \frac {(d+e x)^5}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=x^{3} \left (- \frac {a e^{5}}{3 c^{2} d^{2}} + \frac {4 e^{3}}{3 c}\right ) + x^{2} \left (\frac {a^{2} e^{6}}{2 c^{3} d^{3}} - \frac {2 a e^{4}}{c^{2} d} + \frac {3 d e^{2}}{c}\right ) + x \left (- \frac {a^{3} e^{7}}{c^{4} d^{4}} + \frac {4 a^{2} e^{5}}{c^{3} d^{2}} - \frac {6 a e^{3}}{c^{2}} + \frac {4 d^{2} e}{c}\right ) + \frac {e^{4} x^{4}}{4 c d} + \frac {\left (a e^{2} - c d^{2}\right )^{4} \log {\left (a e + c d x \right )}}{c^{5} d^{5}} \]
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Time = 0.23 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.56 \[ \int \frac {(d+e x)^5}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {3 \, c^{3} d^{3} e^{4} x^{4} + 4 \, {\left (4 \, c^{3} d^{4} e^{3} - a c^{2} d^{2} e^{5}\right )} x^{3} + 6 \, {\left (6 \, c^{3} d^{5} e^{2} - 4 \, a c^{2} d^{3} e^{4} + a^{2} c d e^{6}\right )} x^{2} + 12 \, {\left (4 \, c^{3} d^{6} e - 6 \, a c^{2} d^{4} e^{3} + 4 \, a^{2} c d^{2} e^{5} - a^{3} e^{7}\right )} x}{12 \, c^{4} d^{4}} + \frac {{\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \log \left (c d x + a e\right )}{c^{5} d^{5}} \]
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Time = 0.28 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.60 \[ \int \frac {(d+e x)^5}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {3 \, c^{3} d^{3} e^{4} x^{4} + 16 \, c^{3} d^{4} e^{3} x^{3} - 4 \, a c^{2} d^{2} e^{5} x^{3} + 36 \, c^{3} d^{5} e^{2} x^{2} - 24 \, a c^{2} d^{3} e^{4} x^{2} + 6 \, a^{2} c d e^{6} x^{2} + 48 \, c^{3} d^{6} e x - 72 \, a c^{2} d^{4} e^{3} x + 48 \, a^{2} c d^{2} e^{5} x - 12 \, a^{3} e^{7} x}{12 \, c^{4} d^{4}} + \frac {{\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \log \left ({\left | c d x + a e \right |}\right )}{c^{5} d^{5}} \]
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Time = 9.64 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.66 \[ \int \frac {(d+e x)^5}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=x^3\,\left (\frac {4\,e^3}{3\,c}-\frac {a\,e^5}{3\,c^2\,d^2}\right )+x\,\left (\frac {4\,d^2\,e}{c}-\frac {a\,e\,\left (\frac {6\,d\,e^2}{c}-\frac {a\,e\,\left (\frac {4\,e^3}{c}-\frac {a\,e^5}{c^2\,d^2}\right )}{c\,d}\right )}{c\,d}\right )+x^2\,\left (\frac {3\,d\,e^2}{c}-\frac {a\,e\,\left (\frac {4\,e^3}{c}-\frac {a\,e^5}{c^2\,d^2}\right )}{2\,c\,d}\right )+\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8\right )}{c^5\,d^5}+\frac {e^4\,x^4}{4\,c\,d} \]
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