Integrand size = 35, antiderivative size = 90 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {\left (c d^2-a e^2\right )^2 (d+e x)^{3+m}}{e^3 (3+m)}-\frac {2 c d \left (c d^2-a e^2\right ) (d+e x)^{4+m}}{e^3 (4+m)}+\frac {c^2 d^2 (d+e x)^{5+m}}{e^3 (5+m)} \]
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Time = 0.04 (sec) , antiderivative size = 90, 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} \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {\left (c d^2-a e^2\right )^2 (d+e x)^{m+3}}{e^3 (m+3)}-\frac {2 c d \left (c d^2-a e^2\right ) (d+e x)^{m+4}}{e^3 (m+4)}+\frac {c^2 d^2 (d+e x)^{m+5}}{e^3 (m+5)} \]
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Rule 45
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int (a e+c d x)^2 (d+e x)^{2+m} \, dx \\ & = \int \left (\frac {\left (-c d^2+a e^2\right )^2 (d+e x)^{2+m}}{e^2}-\frac {2 c d \left (c d^2-a e^2\right ) (d+e x)^{3+m}}{e^2}+\frac {c^2 d^2 (d+e x)^{4+m}}{e^2}\right ) \, dx \\ & = \frac {\left (c d^2-a e^2\right )^2 (d+e x)^{3+m}}{e^3 (3+m)}-\frac {2 c d \left (c d^2-a e^2\right ) (d+e x)^{4+m}}{e^3 (4+m)}+\frac {c^2 d^2 (d+e x)^{5+m}}{e^3 (5+m)} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.88 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {(d+e x)^{3+m} \left (\frac {\left (c d^2-a e^2\right )^2}{3+m}-\frac {2 c d \left (c d^2-a e^2\right ) (d+e x)}{4+m}+\frac {c^2 d^2 (d+e x)^2}{5+m}\right )}{e^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(182\) vs. \(2(90)=180\).
Time = 2.98 (sec) , antiderivative size = 183, normalized size of antiderivative = 2.03
method | result | size |
gosper | \(\frac {\left (e x +d \right )^{3+m} \left (c^{2} d^{2} e^{2} m^{2} x^{2}+2 a c d \,e^{3} m^{2} x +7 c^{2} d^{2} e^{2} m \,x^{2}+a^{2} e^{4} m^{2}+16 a c d \,e^{3} m x -2 c^{2} d^{3} e m x +12 x^{2} c^{2} d^{2} e^{2}+9 a^{2} e^{4} m -2 a c \,d^{2} e^{2} m +30 x a c d \,e^{3}-6 x \,c^{2} d^{3} e +20 a^{2} e^{4}-10 a c \,d^{2} e^{2}+2 c^{2} d^{4}\right )}{e^{3} \left (m^{3}+12 m^{2}+47 m +60\right )}\) | \(183\) |
norman | \(\frac {\left (a^{2} e^{4} m^{2}+6 a c \,d^{2} e^{2} m^{2}+3 c^{2} d^{4} m^{2}+9 a^{2} e^{4} m +46 a c \,d^{2} e^{2} m +15 c^{2} d^{4} m +20 a^{2} e^{4}+80 a c \,d^{2} e^{2}+20 c^{2} d^{4}\right ) x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{m^{3}+12 m^{2}+47 m +60}+\frac {d^{3} \left (a^{2} e^{4} m^{2}+9 a^{2} e^{4} m -2 a c \,d^{2} e^{2} m +20 a^{2} e^{4}-10 a c \,d^{2} e^{2}+2 c^{2} d^{4}\right ) {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{3} \left (m^{3}+12 m^{2}+47 m +60\right )}+\frac {c^{2} d^{2} e^{2} x^{5} {\mathrm e}^{m \ln \left (e x +d \right )}}{5+m}+\frac {d^{2} \left (3 a^{2} e^{4} m^{2}+2 a c \,d^{2} e^{2} m^{2}+27 a^{2} e^{4} m +10 a c \,d^{2} e^{2} m -2 c^{2} d^{4} m +60 a^{2} e^{4}\right ) x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e^{2} \left (m^{3}+12 m^{2}+47 m +60\right )}+\frac {d \left (3 a^{2} e^{4} m^{2}+6 a c \,d^{2} e^{2} m^{2}+c^{2} d^{4} m^{2}+27 a^{2} e^{4} m +42 a c \,d^{2} e^{2} m +c^{2} d^{4} m +60 a^{2} e^{4}+60 a c \,d^{2} e^{2}\right ) x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (m^{3}+12 m^{2}+47 m +60\right )}+\frac {c d e \left (2 a \,e^{2} m +3 c \,d^{2} m +10 e^{2} a +10 c \,d^{2}\right ) x^{4} {\mathrm e}^{m \ln \left (e x +d \right )}}{m^{2}+9 m +20}\) | \(495\) |
risch | \(\frac {\left (c^{2} d^{2} e^{5} m^{2} x^{5}+2 a c d \,e^{6} m^{2} x^{4}+3 c^{2} d^{3} e^{4} m^{2} x^{4}+7 c^{2} d^{2} e^{5} m \,x^{5}+a^{2} e^{7} m^{2} x^{3}+6 a c \,d^{2} e^{5} m^{2} x^{3}+16 a c d \,e^{6} m \,x^{4}+3 c^{2} d^{4} e^{3} m^{2} x^{3}+19 c^{2} d^{3} e^{4} m \,x^{4}+12 c^{2} d^{2} e^{5} x^{5}+3 a^{2} d \,e^{6} m^{2} x^{2}+9 a^{2} e^{7} m \,x^{3}+6 a c \,d^{3} e^{4} m^{2} x^{2}+46 a c \,d^{2} e^{5} m \,x^{3}+30 a c d \,e^{6} x^{4}+c^{2} d^{5} e^{2} m^{2} x^{2}+15 c^{2} d^{4} e^{3} m \,x^{3}+30 c^{2} d^{3} e^{4} x^{4}+3 a^{2} d^{2} e^{5} m^{2} x +27 a^{2} d \,e^{6} m \,x^{2}+20 a^{2} e^{7} x^{3}+2 a c \,d^{4} e^{3} m^{2} x +42 a c \,d^{3} e^{4} m \,x^{2}+80 a c \,d^{2} e^{5} x^{3}+c^{2} d^{5} e^{2} m \,x^{2}+20 c^{2} d^{4} e^{3} x^{3}+a^{2} d^{3} e^{4} m^{2}+27 a^{2} d^{2} e^{5} m x +60 a^{2} d \,e^{6} x^{2}+10 a c \,d^{4} e^{3} m x +60 a c \,d^{3} e^{4} x^{2}-2 c^{2} d^{6} e m x +9 a^{2} d^{3} e^{4} m +60 a^{2} d^{2} e^{5} x -2 a c \,d^{5} e^{2} m +20 a^{2} d^{3} e^{4}-10 a c \,d^{5} e^{2}+2 c^{2} d^{7}\right ) \left (e x +d \right )^{m}}{\left (4+m \right ) \left (5+m \right ) \left (3+m \right ) e^{3}}\) | \(536\) |
parallelrisch | \(\frac {20 \left (e x +d \right )^{m} a^{2} d^{4} e^{4}+2 \left (e x +d \right )^{m} c^{2} d^{8}+9 x^{3} \left (e x +d \right )^{m} a^{2} d \,e^{7} m +15 x^{3} \left (e x +d \right )^{m} c^{2} d^{5} e^{3} m +3 x^{2} \left (e x +d \right )^{m} a^{2} d^{2} e^{6} m^{2}+x^{2} \left (e x +d \right )^{m} c^{2} d^{6} e^{2} m^{2}+80 x^{3} \left (e x +d \right )^{m} a c \,d^{3} e^{5}+27 x^{2} \left (e x +d \right )^{m} a^{2} d^{2} e^{6} m +x^{2} \left (e x +d \right )^{m} c^{2} d^{6} e^{2} m +3 x \left (e x +d \right )^{m} a^{2} d^{3} e^{5} m^{2}+60 x^{2} \left (e x +d \right )^{m} a c \,d^{4} e^{4}+27 x \left (e x +d \right )^{m} a^{2} d^{3} e^{5} m -2 x \left (e x +d \right )^{m} c^{2} d^{7} e m -2 \left (e x +d \right )^{m} a c \,d^{6} e^{2} m +30 x^{4} \left (e x +d \right )^{m} c^{2} d^{4} e^{4}+20 x^{3} \left (e x +d \right )^{m} a^{2} d \,e^{7}+20 x^{3} \left (e x +d \right )^{m} c^{2} d^{5} e^{3}+60 x^{2} \left (e x +d \right )^{m} a^{2} d^{2} e^{6}+\left (e x +d \right )^{m} a^{2} d^{4} e^{4} m^{2}+60 x \left (e x +d \right )^{m} a^{2} d^{3} e^{5}+9 \left (e x +d \right )^{m} a^{2} d^{4} e^{4} m -10 \left (e x +d \right )^{m} a c \,d^{6} e^{2}+6 x^{2} \left (e x +d \right )^{m} a c \,d^{4} e^{4} m^{2}+42 x^{2} \left (e x +d \right )^{m} a c \,d^{4} e^{4} m +2 x \left (e x +d \right )^{m} a c \,d^{5} e^{3} m^{2}+10 x \left (e x +d \right )^{m} a c \,d^{5} e^{3} m +2 x^{4} \left (e x +d \right )^{m} a c \,d^{2} e^{6} m^{2}+16 x^{4} \left (e x +d \right )^{m} a c \,d^{2} e^{6} m +6 x^{3} \left (e x +d \right )^{m} a c \,d^{3} e^{5} m^{2}+46 x^{3} \left (e x +d \right )^{m} a c \,d^{3} e^{5} m +12 x^{5} \left (e x +d \right )^{m} c^{2} d^{3} e^{5}+x^{5} \left (e x +d \right )^{m} c^{2} d^{3} e^{5} m^{2}+7 x^{5} \left (e x +d \right )^{m} c^{2} d^{3} e^{5} m +3 x^{4} \left (e x +d \right )^{m} c^{2} d^{4} e^{4} m^{2}+19 x^{4} \left (e x +d \right )^{m} c^{2} d^{4} e^{4} m +x^{3} \left (e x +d \right )^{m} a^{2} d \,e^{7} m^{2}+3 x^{3} \left (e x +d \right )^{m} c^{2} d^{5} e^{3} m^{2}+30 x^{4} \left (e x +d \right )^{m} a c \,d^{2} e^{6}}{\left (m^{2}+9 m +20\right ) d \left (3+m \right ) e^{3}}\) | \(813\) |
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Leaf count of result is larger than twice the leaf count of optimal. 479 vs. \(2 (90) = 180\).
Time = 0.38 (sec) , antiderivative size = 479, normalized size of antiderivative = 5.32 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {{\left (a^{2} d^{3} e^{4} m^{2} + 2 \, c^{2} d^{7} - 10 \, a c d^{5} e^{2} + 20 \, a^{2} d^{3} e^{4} + {\left (c^{2} d^{2} e^{5} m^{2} + 7 \, c^{2} d^{2} e^{5} m + 12 \, c^{2} d^{2} e^{5}\right )} x^{5} + {\left (30 \, c^{2} d^{3} e^{4} + 30 \, a c d e^{6} + {\left (3 \, c^{2} d^{3} e^{4} + 2 \, a c d e^{6}\right )} m^{2} + {\left (19 \, c^{2} d^{3} e^{4} + 16 \, a c d e^{6}\right )} m\right )} x^{4} + {\left (20 \, c^{2} d^{4} e^{3} + 80 \, a c d^{2} e^{5} + 20 \, a^{2} e^{7} + {\left (3 \, c^{2} d^{4} e^{3} + 6 \, a c d^{2} e^{5} + a^{2} e^{7}\right )} m^{2} + {\left (15 \, c^{2} d^{4} e^{3} + 46 \, a c d^{2} e^{5} + 9 \, a^{2} e^{7}\right )} m\right )} x^{3} + {\left (60 \, a c d^{3} e^{4} + 60 \, a^{2} d e^{6} + {\left (c^{2} d^{5} e^{2} + 6 \, a c d^{3} e^{4} + 3 \, a^{2} d e^{6}\right )} m^{2} + {\left (c^{2} d^{5} e^{2} + 42 \, a c d^{3} e^{4} + 27 \, a^{2} d e^{6}\right )} m\right )} x^{2} - {\left (2 \, a c d^{5} e^{2} - 9 \, a^{2} d^{3} e^{4}\right )} m + {\left (60 \, a^{2} d^{2} e^{5} + {\left (2 \, a c d^{4} e^{3} + 3 \, a^{2} d^{2} e^{5}\right )} m^{2} - {\left (2 \, c^{2} d^{6} e - 10 \, a c d^{4} e^{3} - 27 \, a^{2} d^{2} e^{5}\right )} m\right )} x\right )} {\left (e x + d\right )}^{m}}{e^{3} m^{3} + 12 \, e^{3} m^{2} + 47 \, e^{3} m + 60 \, e^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2494 vs. \(2 (78) = 156\).
Time = 1.11 (sec) , antiderivative size = 2494, normalized size of antiderivative = 27.71 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 691 vs. \(2 (90) = 180\).
Time = 0.22 (sec) , antiderivative size = 691, normalized size of antiderivative = 7.68 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {2 \, {\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} a c d^{3}}{{\left (m^{2} + 3 \, m + 2\right )} e} + \frac {2 \, {\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} a^{2} d e}{m^{2} + 3 \, m + 2} + \frac {{\left (e x + d\right )}^{m + 1} a^{2} d^{2} e}{m + 1} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} c^{2} d^{4}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {4 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} a c d^{2}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} a^{2} e}{m^{3} + 6 \, m^{2} + 11 \, m + 6} + \frac {2 \, {\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} c^{2} d^{3}}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{3}} + \frac {2 \, {\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} a c d}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e} + \frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{5} x^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d e^{4} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} e^{3} x^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} e^{2} x^{2} - 24 \, d^{4} e m x + 24 \, d^{5}\right )} {\left (e x + d\right )}^{m} c^{2} d^{2}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 805 vs. \(2 (90) = 180\).
Time = 0.28 (sec) , antiderivative size = 805, normalized size of antiderivative = 8.94 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {{\left (e x + d\right )}^{m} c^{2} d^{2} e^{5} m^{2} x^{5} + 3 \, {\left (e x + d\right )}^{m} c^{2} d^{3} e^{4} m^{2} x^{4} + 2 \, {\left (e x + d\right )}^{m} a c d e^{6} m^{2} x^{4} + 7 \, {\left (e x + d\right )}^{m} c^{2} d^{2} e^{5} m x^{5} + 3 \, {\left (e x + d\right )}^{m} c^{2} d^{4} e^{3} m^{2} x^{3} + 6 \, {\left (e x + d\right )}^{m} a c d^{2} e^{5} m^{2} x^{3} + {\left (e x + d\right )}^{m} a^{2} e^{7} m^{2} x^{3} + 19 \, {\left (e x + d\right )}^{m} c^{2} d^{3} e^{4} m x^{4} + 16 \, {\left (e x + d\right )}^{m} a c d e^{6} m x^{4} + 12 \, {\left (e x + d\right )}^{m} c^{2} d^{2} e^{5} x^{5} + {\left (e x + d\right )}^{m} c^{2} d^{5} e^{2} m^{2} x^{2} + 6 \, {\left (e x + d\right )}^{m} a c d^{3} e^{4} m^{2} x^{2} + 3 \, {\left (e x + d\right )}^{m} a^{2} d e^{6} m^{2} x^{2} + 15 \, {\left (e x + d\right )}^{m} c^{2} d^{4} e^{3} m x^{3} + 46 \, {\left (e x + d\right )}^{m} a c d^{2} e^{5} m x^{3} + 9 \, {\left (e x + d\right )}^{m} a^{2} e^{7} m x^{3} + 30 \, {\left (e x + d\right )}^{m} c^{2} d^{3} e^{4} x^{4} + 30 \, {\left (e x + d\right )}^{m} a c d e^{6} x^{4} + 2 \, {\left (e x + d\right )}^{m} a c d^{4} e^{3} m^{2} x + 3 \, {\left (e x + d\right )}^{m} a^{2} d^{2} e^{5} m^{2} x + {\left (e x + d\right )}^{m} c^{2} d^{5} e^{2} m x^{2} + 42 \, {\left (e x + d\right )}^{m} a c d^{3} e^{4} m x^{2} + 27 \, {\left (e x + d\right )}^{m} a^{2} d e^{6} m x^{2} + 20 \, {\left (e x + d\right )}^{m} c^{2} d^{4} e^{3} x^{3} + 80 \, {\left (e x + d\right )}^{m} a c d^{2} e^{5} x^{3} + 20 \, {\left (e x + d\right )}^{m} a^{2} e^{7} x^{3} + {\left (e x + d\right )}^{m} a^{2} d^{3} e^{4} m^{2} - 2 \, {\left (e x + d\right )}^{m} c^{2} d^{6} e m x + 10 \, {\left (e x + d\right )}^{m} a c d^{4} e^{3} m x + 27 \, {\left (e x + d\right )}^{m} a^{2} d^{2} e^{5} m x + 60 \, {\left (e x + d\right )}^{m} a c d^{3} e^{4} x^{2} + 60 \, {\left (e x + d\right )}^{m} a^{2} d e^{6} x^{2} - 2 \, {\left (e x + d\right )}^{m} a c d^{5} e^{2} m + 9 \, {\left (e x + d\right )}^{m} a^{2} d^{3} e^{4} m + 60 \, {\left (e x + d\right )}^{m} a^{2} d^{2} e^{5} x + 2 \, {\left (e x + d\right )}^{m} c^{2} d^{7} - 10 \, {\left (e x + d\right )}^{m} a c d^{5} e^{2} + 20 \, {\left (e x + d\right )}^{m} a^{2} d^{3} e^{4}}{e^{3} m^{3} + 12 \, e^{3} m^{2} + 47 \, e^{3} m + 60 \, e^{3}} \]
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Time = 10.13 (sec) , antiderivative size = 486, normalized size of antiderivative = 5.40 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx={\left (d+e\,x\right )}^m\,\left (\frac {x^3\,\left (a^2\,e^7\,m^2+9\,a^2\,e^7\,m+20\,a^2\,e^7+6\,a\,c\,d^2\,e^5\,m^2+46\,a\,c\,d^2\,e^5\,m+80\,a\,c\,d^2\,e^5+3\,c^2\,d^4\,e^3\,m^2+15\,c^2\,d^4\,e^3\,m+20\,c^2\,d^4\,e^3\right )}{e^3\,\left (m^3+12\,m^2+47\,m+60\right )}+\frac {d^3\,\left (a^2\,e^4\,m^2+9\,a^2\,e^4\,m+20\,a^2\,e^4-2\,a\,c\,d^2\,e^2\,m-10\,a\,c\,d^2\,e^2+2\,c^2\,d^4\right )}{e^3\,\left (m^3+12\,m^2+47\,m+60\right )}+\frac {d^2\,x\,\left (3\,a^2\,e^4\,m^2+27\,a^2\,e^4\,m+60\,a^2\,e^4+2\,a\,c\,d^2\,e^2\,m^2+10\,a\,c\,d^2\,e^2\,m-2\,c^2\,d^4\,m\right )}{e^2\,\left (m^3+12\,m^2+47\,m+60\right )}+\frac {d\,x^2\,\left (3\,a^2\,e^4\,m^2+27\,a^2\,e^4\,m+60\,a^2\,e^4+6\,a\,c\,d^2\,e^2\,m^2+42\,a\,c\,d^2\,e^2\,m+60\,a\,c\,d^2\,e^2+c^2\,d^4\,m^2+c^2\,d^4\,m\right )}{e\,\left (m^3+12\,m^2+47\,m+60\right )}+\frac {c^2\,d^2\,e^2\,x^5\,\left (m^2+7\,m+12\right )}{m^3+12\,m^2+47\,m+60}+\frac {c\,d\,e\,x^4\,\left (m+3\right )\,\left (10\,a\,e^2+10\,c\,d^2+2\,a\,e^2\,m+3\,c\,d^2\,m\right )}{m^3+12\,m^2+47\,m+60}\right ) \]
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