Integrand size = 35, antiderivative size = 94 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^5} \, dx=-\frac {c^2 d^2 \left (2 c d^2-3 a e^2\right ) x}{e^3}+\frac {c^3 d^3 x^2}{2 e^2}+\frac {\left (c d^2-a e^2\right )^3}{e^4 (d+e x)}+\frac {3 c d \left (c d^2-a e^2\right )^2 \log (d+e x)}{e^4} \]
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Time = 0.06 (sec) , antiderivative size = 94, 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} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^5} \, dx=-\frac {c^2 d^2 x \left (2 c d^2-3 a e^2\right )}{e^3}+\frac {\left (c d^2-a e^2\right )^3}{e^4 (d+e x)}+\frac {3 c d \left (c d^2-a e^2\right )^2 \log (d+e x)}{e^4}+\frac {c^3 d^3 x^2}{2 e^2} \]
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Rule 45
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int \frac {(a e+c d x)^3}{(d+e x)^2} \, dx \\ & = \int \left (-\frac {c^2 d^2 \left (2 c d^2-3 a e^2\right )}{e^3}+\frac {c^3 d^3 x}{e^2}+\frac {\left (-c d^2+a e^2\right )^3}{e^3 (d+e x)^2}+\frac {3 c d \left (c d^2-a e^2\right )^2}{e^3 (d+e x)}\right ) \, dx \\ & = -\frac {c^2 d^2 \left (2 c d^2-3 a e^2\right ) x}{e^3}+\frac {c^3 d^3 x^2}{2 e^2}+\frac {\left (c d^2-a e^2\right )^3}{e^4 (d+e x)}+\frac {3 c d \left (c d^2-a e^2\right )^2 \log (d+e x)}{e^4} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.37 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^5} \, dx=\frac {6 a^2 c d^2 e^4-2 a^3 e^6+6 a c^2 d^2 e^2 \left (-d^2+d e x+e^2 x^2\right )+c^3 d^3 \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+6 c d \left (c d^2-a e^2\right )^2 (d+e x) \log (d+e x)}{2 e^4 (d+e x)} \]
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Time = 2.50 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.33
method | result | size |
default | \(\frac {c^{2} d^{2} \left (\frac {1}{2} c d e \,x^{2}+3 a \,e^{2} x -2 c \,d^{2} x \right )}{e^{3}}-\frac {e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}}{e^{4} \left (e x +d \right )}+\frac {3 c d \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \ln \left (e x +d \right )}{e^{4}}\) | \(125\) |
risch | \(\frac {c^{3} d^{3} x^{2}}{2 e^{2}}+\frac {3 d^{2} c^{2} a x}{e}-\frac {2 d^{4} c^{3} x}{e^{3}}-\frac {e^{2} a^{3}}{e x +d}+\frac {3 d^{2} a^{2} c}{e x +d}-\frac {3 d^{4} c^{2} a}{e^{2} \left (e x +d \right )}+\frac {c^{3} d^{6}}{e^{4} \left (e x +d \right )}+3 c d \ln \left (e x +d \right ) a^{2}-\frac {6 c^{2} d^{3} \ln \left (e x +d \right ) a}{e^{2}}+\frac {3 c^{3} d^{5} \ln \left (e x +d \right )}{e^{4}}\) | \(156\) |
parallelrisch | \(\frac {x^{3} c^{3} d^{3} e^{3}+6 \ln \left (e x +d \right ) x \,a^{2} c d \,e^{5}-12 \ln \left (e x +d \right ) x a \,c^{2} d^{3} e^{3}+6 \ln \left (e x +d \right ) x \,c^{3} d^{5} e +6 x^{2} a \,c^{2} d^{2} e^{4}-3 x^{2} c^{3} d^{4} e^{2}+6 \ln \left (e x +d \right ) a^{2} c \,d^{2} e^{4}-12 \ln \left (e x +d \right ) a \,c^{2} d^{4} e^{2}+6 \ln \left (e x +d \right ) c^{3} d^{6}-2 e^{6} a^{3}+6 d^{2} e^{4} a^{2} c -12 d^{4} e^{2} c^{2} a +6 c^{3} d^{6}}{2 e^{4} \left (e x +d \right )}\) | \(198\) |
norman | \(\frac {-\frac {d^{3} \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +15 d^{4} e^{2} c^{2} a -6 c^{3} d^{6}\right )}{e^{4}}-\frac {\left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +33 d^{4} e^{2} c^{2} a -11 c^{3} d^{6}\right ) x^{3}}{e}-\frac {3 d \left (2 e^{6} a^{3}-6 d^{2} e^{4} a^{2} c +46 d^{4} e^{2} c^{2} a -17 c^{3} d^{6}\right ) x^{2}}{2 e^{2}}-\frac {d^{2} \left (3 e^{6} a^{3}-9 d^{2} e^{4} a^{2} c +54 d^{4} e^{2} c^{2} a -21 c^{3} d^{6}\right ) x}{e^{3}}+\frac {e^{2} c^{3} d^{3} x^{6}}{2}+3 x^{5} e^{3} a \,c^{2} d^{2}}{\left (e x +d \right )^{4}}+\frac {3 c d \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \ln \left (e x +d \right )}{e^{4}}\) | \(274\) |
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Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (92) = 184\).
Time = 0.27 (sec) , antiderivative size = 193, normalized size of antiderivative = 2.05 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^5} \, dx=\frac {c^{3} d^{3} e^{3} x^{3} + 2 \, c^{3} d^{6} - 6 \, a c^{2} d^{4} e^{2} + 6 \, a^{2} c d^{2} e^{4} - 2 \, a^{3} e^{6} - 3 \, {\left (c^{3} d^{4} e^{2} - 2 \, a c^{2} d^{2} e^{4}\right )} x^{2} - 2 \, {\left (2 \, c^{3} d^{5} e - 3 \, a c^{2} d^{3} e^{3}\right )} x + 6 \, {\left (c^{3} d^{6} - 2 \, a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} + {\left (c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (e^{5} x + d e^{4}\right )}} \]
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Time = 0.33 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.24 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^5} \, dx=\frac {c^{3} d^{3} x^{2}}{2 e^{2}} + \frac {3 c d \left (a e^{2} - c d^{2}\right )^{2} \log {\left (d + e x \right )}}{e^{4}} + x \left (\frac {3 a c^{2} d^{2}}{e} - \frac {2 c^{3} d^{4}}{e^{3}}\right ) + \frac {- a^{3} e^{6} + 3 a^{2} c d^{2} e^{4} - 3 a c^{2} d^{4} e^{2} + c^{3} d^{6}}{d e^{4} + e^{5} x} \]
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Time = 0.20 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.45 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^5} \, dx=\frac {c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}}{e^{5} x + d e^{4}} + \frac {c^{3} d^{3} e x^{2} - 2 \, {\left (2 \, c^{3} d^{4} - 3 \, a c^{2} d^{2} e^{2}\right )} x}{2 \, e^{3}} + \frac {3 \, {\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} \log \left (e x + d\right )}{e^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.94 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^5} \, dx=\frac {{\left (c^{3} d^{3} - \frac {6 \, {\left (c^{3} d^{4} e - a c^{2} d^{2} e^{3}\right )}}{{\left (e x + d\right )} e}\right )} {\left (e x + d\right )}^{2}}{2 \, e^{4}} - \frac {3 \, {\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} \log \left (\frac {{\left | e x + d \right |}}{{\left (e x + d\right )}^{2} {\left | e \right |}}\right )}{e^{4}} + \frac {\frac {c^{3} d^{6} e^{2}}{e x + d} - \frac {3 \, a c^{2} d^{4} e^{4}}{e x + d} + \frac {3 \, a^{2} c d^{2} e^{6}}{e x + d} - \frac {a^{3} e^{8}}{e x + d}}{e^{6}} \]
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Time = 0.08 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.50 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^5} \, dx=\frac {\ln \left (d+e\,x\right )\,\left (3\,a^2\,c\,d\,e^4-6\,a\,c^2\,d^3\,e^2+3\,c^3\,d^5\right )}{e^4}-\frac {a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6}{e\,\left (x\,e^4+d\,e^3\right )}-x\,\left (\frac {2\,c^3\,d^4}{e^3}-\frac {3\,a\,c^2\,d^2}{e}\right )+\frac {c^3\,d^3\,x^2}{2\,e^2} \]
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