Integrand size = 35, antiderivative size = 139 \[ \int \frac {(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {1}{3 \left (c d^2-a e^2\right ) (a e+c d x)^3}+\frac {e}{2 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}-\frac {e^2}{\left (c d^2-a e^2\right )^3 (a e+c d x)}-\frac {e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}+\frac {e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^4} \]
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Time = 0.06 (sec) , antiderivative size = 139, 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, 46} \[ \int \frac {(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {e^2}{\left (c d^2-a e^2\right )^3 (a e+c d x)}+\frac {e}{2 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}-\frac {1}{3 \left (c d^2-a e^2\right ) (a e+c d x)^3}-\frac {e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}+\frac {e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^4} \]
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Rule 46
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(a e+c d x)^4 (d+e x)} \, dx \\ & = \int \left (\frac {c d}{\left (c d^2-a e^2\right ) (a e+c d x)^4}-\frac {c d e}{\left (c d^2-a e^2\right )^2 (a e+c d x)^3}+\frac {c d e^2}{\left (c d^2-a e^2\right )^3 (a e+c d x)^2}-\frac {c d e^3}{\left (c d^2-a e^2\right )^4 (a e+c d x)}+\frac {e^4}{\left (c d^2-a e^2\right )^4 (d+e x)}\right ) \, dx \\ & = -\frac {1}{3 \left (c d^2-a e^2\right ) (a e+c d x)^3}+\frac {e}{2 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}-\frac {e^2}{\left (c d^2-a e^2\right )^3 (a e+c d x)}-\frac {e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}+\frac {e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^4} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.84 \[ \int \frac {(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {\frac {\left (c d^2-a e^2\right ) \left (11 a^2 e^4+a c d e^2 (-7 d+15 e x)+c^2 d^2 \left (2 d^2-3 d e x+6 e^2 x^2\right )\right )}{(a e+c d x)^3}+6 e^3 \log (a e+c d x)-6 e^3 \log (d+e x)}{6 \left (c d^2-a e^2\right )^4} \]
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Time = 3.07 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.97
method | result | size |
default | \(\frac {1}{3 \left (e^{2} a -c \,d^{2}\right ) \left (c d x +a e \right )^{3}}-\frac {e^{3} \ln \left (c d x +a e \right )}{\left (e^{2} a -c \,d^{2}\right )^{4}}+\frac {e}{2 \left (e^{2} a -c \,d^{2}\right )^{2} \left (c d x +a e \right )^{2}}+\frac {e^{2}}{\left (e^{2} a -c \,d^{2}\right )^{3} \left (c d x +a e \right )}+\frac {e^{3} \ln \left (e x +d \right )}{\left (e^{2} a -c \,d^{2}\right )^{4}}\) | \(135\) |
risch | \(\frac {\frac {c^{2} d^{2} e^{2} x^{2}}{e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}}+\frac {c d \left (5 e^{2} a -c \,d^{2}\right ) e x}{2 e^{6} a^{3}-6 d^{2} e^{4} a^{2} c +6 d^{4} e^{2} c^{2} a -2 c^{3} d^{6}}+\frac {11 a^{2} e^{4}-7 a c \,d^{2} e^{2}+2 c^{2} d^{4}}{6 e^{6} a^{3}-18 d^{2} e^{4} a^{2} c +18 d^{4} e^{2} c^{2} a -6 c^{3} d^{6}}}{\left (c d x +a e \right )^{3}}-\frac {e^{3} \ln \left (c d x +a e \right )}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}+\frac {e^{3} \ln \left (-e x -d \right )}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}\) | \(338\) |
parallelrisch | \(\frac {-x^{3} c^{6} d^{8} e +6 a^{5} c d \,e^{8}-12 a^{4} c^{2} d^{3} e^{6}+8 a^{3} c^{3} d^{5} e^{4}-2 a^{2} c^{4} d^{7} e^{2}-5 x^{3} a^{2} c^{4} d^{4} e^{5}+6 x^{3} a \,c^{5} d^{6} e^{3}-9 x^{2} a^{3} c^{3} d^{3} e^{6}+12 x^{2} a^{2} c^{4} d^{5} e^{4}-3 x^{2} a \,c^{5} d^{7} e^{2}+6 \ln \left (e x +d \right ) a^{5} c d \,e^{8}-6 \ln \left (c d x +a e \right ) a^{5} c d \,e^{8}+6 \ln \left (e x +d \right ) x^{3} a^{2} c^{4} d^{4} e^{5}-6 \ln \left (c d x +a e \right ) x^{3} a^{2} c^{4} d^{4} e^{5}+18 \ln \left (e x +d \right ) x^{2} a^{3} c^{3} d^{3} e^{6}-18 \ln \left (c d x +a e \right ) x^{2} a^{3} c^{3} d^{3} e^{6}+18 \ln \left (e x +d \right ) x \,a^{4} c^{2} d^{2} e^{7}-18 \ln \left (c d x +a e \right ) x \,a^{4} c^{2} d^{2} e^{7}}{6 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (c d x +a e \right )^{3} d \,e^{2} a^{2} c}\) | \(404\) |
norman | \(\frac {\frac {c^{2} d^{2} e^{5} x^{5}}{e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}}+\frac {11 a^{2} c^{3} d^{3} e^{4}-7 a \,c^{4} d^{5} e^{2}+2 c^{5} d^{7}}{6 c^{3} \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}\right )}+\frac {\left (5 a \,c^{3} d^{3} e^{8}+5 e^{6} d^{5} c^{4}\right ) x^{4}}{2 e^{2} d^{2} c^{2} \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}\right )}+\frac {\left (11 a^{2} c^{3} d^{3} e^{6}-2 a \,c^{4} d^{5} e^{4}+e^{2} d^{7} c^{5}\right ) x}{2 e d \,c^{3} \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}\right )}+\frac {\left (11 a^{2} c^{3} d^{3} e^{8}+8 a \,c^{4} d^{5} e^{6}+e^{4} d^{7} c^{5}\right ) x^{2}}{2 e^{2} d^{2} c^{3} \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}\right )}+\frac {\left (11 a^{2} c^{3} d^{3} e^{10}+38 a \,c^{4} d^{5} e^{8}+11 c^{5} d^{7} e^{6}\right ) x^{3}}{6 e^{3} d^{3} c^{3} \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}\right )}}{\left (c d x +a e \right )^{3} \left (e x +d \right )^{3}}+\frac {e^{3} \ln \left (e x +d \right )}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}-\frac {e^{3} \ln \left (c d x +a e \right )}{a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}\) | \(650\) |
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Leaf count of result is larger than twice the leaf count of optimal. 484 vs. \(2 (135) = 270\).
Time = 0.28 (sec) , antiderivative size = 484, normalized size of antiderivative = 3.48 \[ \int \frac {(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {2 \, c^{3} d^{6} - 9 \, a c^{2} d^{4} e^{2} + 18 \, a^{2} c d^{2} e^{4} - 11 \, a^{3} e^{6} + 6 \, {\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} - 3 \, {\left (c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} + 5 \, a^{2} c d e^{5}\right )} x + 6 \, {\left (c^{3} d^{3} e^{3} x^{3} + 3 \, a c^{2} d^{2} e^{4} x^{2} + 3 \, a^{2} c d e^{5} x + a^{3} e^{6}\right )} \log \left (c d x + a e\right ) - 6 \, {\left (c^{3} d^{3} e^{3} x^{3} + 3 \, a c^{2} d^{2} e^{4} x^{2} + 3 \, a^{2} c d e^{5} x + a^{3} e^{6}\right )} \log \left (e x + d\right )}{6 \, {\left (a^{3} c^{4} d^{8} e^{3} - 4 \, a^{4} c^{3} d^{6} e^{5} + 6 \, a^{5} c^{2} d^{4} e^{7} - 4 \, a^{6} c d^{2} e^{9} + a^{7} e^{11} + {\left (c^{7} d^{11} - 4 \, a c^{6} d^{9} e^{2} + 6 \, a^{2} c^{5} d^{7} e^{4} - 4 \, a^{3} c^{4} d^{5} e^{6} + a^{4} c^{3} d^{3} e^{8}\right )} x^{3} + 3 \, {\left (a c^{6} d^{10} e - 4 \, a^{2} c^{5} d^{8} e^{3} + 6 \, a^{3} c^{4} d^{6} e^{5} - 4 \, a^{4} c^{3} d^{4} e^{7} + a^{5} c^{2} d^{2} e^{9}\right )} x^{2} + 3 \, {\left (a^{2} c^{5} d^{9} e^{2} - 4 \, a^{3} c^{4} d^{7} e^{4} + 6 \, a^{4} c^{3} d^{5} e^{6} - 4 \, a^{5} c^{2} d^{3} e^{8} + a^{6} c d e^{10}\right )} x\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 668 vs. \(2 (119) = 238\).
Time = 1.12 (sec) , antiderivative size = 668, normalized size of antiderivative = 4.81 \[ \int \frac {(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {e^{3} \log {\left (x + \frac {- \frac {a^{5} e^{13}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {5 a^{4} c d^{2} e^{11}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {10 a^{3} c^{2} d^{4} e^{9}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {10 a^{2} c^{3} d^{6} e^{7}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {5 a c^{4} d^{8} e^{5}}{\left (a e^{2} - c d^{2}\right )^{4}} + a e^{5} + \frac {c^{5} d^{10} e^{3}}{\left (a e^{2} - c d^{2}\right )^{4}} + c d^{2} e^{3}}{2 c d e^{4}} \right )}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {e^{3} \log {\left (x + \frac {\frac {a^{5} e^{13}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {5 a^{4} c d^{2} e^{11}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {10 a^{3} c^{2} d^{4} e^{9}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac {10 a^{2} c^{3} d^{6} e^{7}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {5 a c^{4} d^{8} e^{5}}{\left (a e^{2} - c d^{2}\right )^{4}} + a e^{5} - \frac {c^{5} d^{10} e^{3}}{\left (a e^{2} - c d^{2}\right )^{4}} + c d^{2} e^{3}}{2 c d e^{4}} \right )}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac {11 a^{2} e^{4} - 7 a c d^{2} e^{2} + 2 c^{2} d^{4} + 6 c^{2} d^{2} e^{2} x^{2} + x \left (15 a c d e^{3} - 3 c^{2} d^{3} e\right )}{6 a^{6} e^{9} - 18 a^{5} c d^{2} e^{7} + 18 a^{4} c^{2} d^{4} e^{5} - 6 a^{3} c^{3} d^{6} e^{3} + x^{3} \cdot \left (6 a^{3} c^{3} d^{3} e^{6} - 18 a^{2} c^{4} d^{5} e^{4} + 18 a c^{5} d^{7} e^{2} - 6 c^{6} d^{9}\right ) + x^{2} \cdot \left (18 a^{4} c^{2} d^{2} e^{7} - 54 a^{3} c^{3} d^{4} e^{5} + 54 a^{2} c^{4} d^{6} e^{3} - 18 a c^{5} d^{8} e\right ) + x \left (18 a^{5} c d e^{8} - 54 a^{4} c^{2} d^{3} e^{6} + 54 a^{3} c^{3} d^{5} e^{4} - 18 a^{2} c^{4} d^{7} e^{2}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 412 vs. \(2 (135) = 270\).
Time = 0.20 (sec) , antiderivative size = 412, normalized size of antiderivative = 2.96 \[ \int \frac {(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {e^{3} \log \left (c d x + a e\right )}{c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}} + \frac {e^{3} \log \left (e x + d\right )}{c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}} - \frac {6 \, c^{2} d^{2} e^{2} x^{2} + 2 \, c^{2} d^{4} - 7 \, a c d^{2} e^{2} + 11 \, a^{2} e^{4} - 3 \, {\left (c^{2} d^{3} e - 5 \, a c d e^{3}\right )} x}{6 \, {\left (a^{3} c^{3} d^{6} e^{3} - 3 \, a^{4} c^{2} d^{4} e^{5} + 3 \, a^{5} c d^{2} e^{7} - a^{6} e^{9} + {\left (c^{6} d^{9} - 3 \, a c^{5} d^{7} e^{2} + 3 \, a^{2} c^{4} d^{5} e^{4} - a^{3} c^{3} d^{3} e^{6}\right )} x^{3} + 3 \, {\left (a c^{5} d^{8} e - 3 \, a^{2} c^{4} d^{6} e^{3} + 3 \, a^{3} c^{3} d^{4} e^{5} - a^{4} c^{2} d^{2} e^{7}\right )} x^{2} + 3 \, {\left (a^{2} c^{4} d^{7} e^{2} - 3 \, a^{3} c^{3} d^{5} e^{4} + 3 \, a^{4} c^{2} d^{3} e^{6} - a^{5} c d e^{8}\right )} x\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 273 vs. \(2 (135) = 270\).
Time = 0.28 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.96 \[ \int \frac {(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=-\frac {c d e^{3} \log \left ({\left | c d x + a e \right |}\right )}{c^{5} d^{9} - 4 \, a c^{4} d^{7} e^{2} + 6 \, a^{2} c^{3} d^{5} e^{4} - 4 \, a^{3} c^{2} d^{3} e^{6} + a^{4} c d e^{8}} + \frac {e^{4} \log \left ({\left | e x + d \right |}\right )}{c^{4} d^{8} e - 4 \, a c^{3} d^{6} e^{3} + 6 \, a^{2} c^{2} d^{4} e^{5} - 4 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}} - \frac {2 \, c^{3} d^{6} - 9 \, a c^{2} d^{4} e^{2} + 18 \, a^{2} c d^{2} e^{4} - 11 \, a^{3} e^{6} + 6 \, {\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} - 3 \, {\left (c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} + 5 \, a^{2} c d e^{5}\right )} x}{6 \, {\left (c d^{2} - a e^{2}\right )}^{4} {\left (c d x + a e\right )}^{3}} \]
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Time = 10.04 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.68 \[ \int \frac {(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx=\frac {\frac {11\,a^2\,e^4-7\,a\,c\,d^2\,e^2+2\,c^2\,d^4}{6\,\left (a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6\right )}-\frac {e\,x\,\left (c^2\,d^3-5\,a\,c\,d\,e^2\right )}{2\,\left (a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6\right )}+\frac {c^2\,d^2\,e^2\,x^2}{a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6}}{a^3\,e^3+3\,a^2\,c\,d\,e^2\,x+3\,a\,c^2\,d^2\,e\,x^2+c^3\,d^3\,x^3}-\frac {2\,e^3\,\mathrm {atanh}\left (\frac {a^4\,e^8-2\,a^3\,c\,d^2\,e^6+2\,a\,c^3\,d^6\,e^2-c^4\,d^8}{{\left (a\,e^2-c\,d^2\right )}^4}+\frac {2\,c\,d\,e\,x\,\left (a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6\right )}{{\left (a\,e^2-c\,d^2\right )}^4}\right )}{{\left (a\,e^2-c\,d^2\right )}^4} \]
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