Integrand size = 21, antiderivative size = 95 \[ \int (d+e x)^m \left (c d x+c e x^2\right )^3 \, dx=-\frac {c^3 d^3 (d+e x)^{4+m}}{e^4 (4+m)}+\frac {3 c^3 d^2 (d+e x)^{5+m}}{e^4 (5+m)}-\frac {3 c^3 d (d+e x)^{6+m}}{e^4 (6+m)}+\frac {c^3 (d+e x)^{7+m}}{e^4 (7+m)} \]
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Time = 0.05 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {640, 12, 45} \[ \int (d+e x)^m \left (c d x+c e x^2\right )^3 \, dx=-\frac {c^3 d^3 (d+e x)^{m+4}}{e^4 (m+4)}+\frac {3 c^3 d^2 (d+e x)^{m+5}}{e^4 (m+5)}-\frac {3 c^3 d (d+e x)^{m+6}}{e^4 (m+6)}+\frac {c^3 (d+e x)^{m+7}}{e^4 (m+7)} \]
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Rule 12
Rule 45
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int c^3 x^3 (d+e x)^{3+m} \, dx \\ & = c^3 \int x^3 (d+e x)^{3+m} \, dx \\ & = c^3 \int \left (-\frac {d^3 (d+e x)^{3+m}}{e^3}+\frac {3 d^2 (d+e x)^{4+m}}{e^3}-\frac {3 d (d+e x)^{5+m}}{e^3}+\frac {(d+e x)^{6+m}}{e^3}\right ) \, dx \\ & = -\frac {c^3 d^3 (d+e x)^{4+m}}{e^4 (4+m)}+\frac {3 c^3 d^2 (d+e x)^{5+m}}{e^4 (5+m)}-\frac {3 c^3 d (d+e x)^{6+m}}{e^4 (6+m)}+\frac {c^3 (d+e x)^{7+m}}{e^4 (7+m)} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.74 \[ \int (d+e x)^m \left (c d x+c e x^2\right )^3 \, dx=\frac {c^3 (d+e x)^{4+m} \left (-\frac {d^3}{4+m}+\frac {3 d^2 (d+e x)}{5+m}-\frac {3 d (d+e x)^2}{6+m}+\frac {(d+e x)^3}{7+m}\right )}{e^4} \]
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Time = 2.04 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.36
method | result | size |
gosper | \(-\frac {c^{3} \left (e x +d \right )^{4+m} \left (-e^{3} m^{3} x^{3}-15 e^{3} m^{2} x^{3}+3 d \,e^{2} m^{2} x^{2}-74 e^{3} m \,x^{3}+27 d \,e^{2} m \,x^{2}-120 e^{3} x^{3}-6 d^{2} e m x +60 d \,e^{2} x^{2}-24 d^{2} e x +6 d^{3}\right )}{e^{4} \left (m^{4}+22 m^{3}+179 m^{2}+638 m +840\right )}\) | \(129\) |
risch | \(-\frac {c^{3} \left (-e^{7} m^{3} x^{7}-4 d \,e^{6} m^{3} x^{6}-15 e^{7} m^{2} x^{7}-6 d^{2} e^{5} m^{3} x^{5}-57 d \,e^{6} m^{2} x^{6}-74 e^{7} m \,x^{7}-4 d^{3} e^{4} m^{3} x^{4}-78 d^{2} e^{5} m^{2} x^{5}-269 d \,e^{6} m \,x^{6}-120 e^{7} x^{7}-d^{4} e^{3} m^{3} x^{3}-42 d^{3} e^{4} m^{2} x^{4}-342 d^{2} e^{5} m \,x^{5}-420 d \,e^{6} x^{6}-3 d^{4} e^{3} m^{2} x^{3}-158 d^{3} e^{4} m \,x^{4}-504 d^{2} e^{5} x^{5}+3 d^{5} e^{2} m^{2} x^{2}-2 d^{4} e^{3} m \,x^{3}-210 d^{3} e^{4} x^{4}+3 d^{5} e^{2} m \,x^{2}-6 x \,d^{6} m e +6 d^{7}\right ) \left (e x +d \right )^{m}}{\left (6+m \right ) \left (7+m \right ) \left (5+m \right ) \left (4+m \right ) e^{4}}\) | \(300\) |
norman | \(\frac {c^{3} e^{3} x^{7} {\mathrm e}^{m \ln \left (e x +d \right )}}{7+m}+\frac {c^{3} d \,e^{2} \left (4 m +21\right ) x^{6} {\mathrm e}^{m \ln \left (e x +d \right )}}{m^{2}+13 m +42}+\frac {\left (2+m \right ) m \left (1+m \right ) c^{3} d^{4} x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (m^{4}+22 m^{3}+179 m^{2}+638 m +840\right )}-\frac {6 c^{3} d^{7} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{4} \left (m^{4}+22 m^{3}+179 m^{2}+638 m +840\right )}+\frac {2 c^{3} d^{3} \left (2 m^{3}+21 m^{2}+79 m +105\right ) x^{4} {\mathrm e}^{m \ln \left (e x +d \right )}}{m^{4}+22 m^{3}+179 m^{2}+638 m +840}+\frac {6 m \,c^{3} d^{6} x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e^{3} \left (m^{4}+22 m^{3}+179 m^{2}+638 m +840\right )}+\frac {6 \left (m^{2}+9 m +21\right ) c^{3} d^{2} e \,x^{5} {\mathrm e}^{m \ln \left (e x +d \right )}}{m^{3}+18 m^{2}+107 m +210}-\frac {3 m \left (1+m \right ) c^{3} d^{5} x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{2} \left (m^{4}+22 m^{3}+179 m^{2}+638 m +840\right )}\) | \(338\) |
parallelrisch | \(\frac {-6 c^{3} d^{8} \left (e x +d \right )^{m}+78 x^{5} \left (e x +d \right )^{m} c^{3} d^{3} e^{5} m^{2}+4 x^{4} \left (e x +d \right )^{m} c^{3} d^{4} e^{4} m^{3}+342 x^{5} \left (e x +d \right )^{m} c^{3} d^{3} e^{5} m +42 x^{4} \left (e x +d \right )^{m} c^{3} d^{4} e^{4} m^{2}+x^{3} \left (e x +d \right )^{m} c^{3} d^{5} e^{3} m^{3}+158 x^{4} \left (e x +d \right )^{m} c^{3} d^{4} e^{4} m +3 x^{3} \left (e x +d \right )^{m} c^{3} d^{5} e^{3} m^{2}+2 x^{3} \left (e x +d \right )^{m} c^{3} d^{5} e^{3} m -3 x^{2} \left (e x +d \right )^{m} c^{3} d^{6} e^{2} m^{2}-3 x^{2} \left (e x +d \right )^{m} c^{3} d^{6} e^{2} m +x^{7} \left (e x +d \right )^{m} c^{3} d \,e^{7} m^{3}+15 x^{7} \left (e x +d \right )^{m} c^{3} d \,e^{7} m^{2}+4 x^{6} \left (e x +d \right )^{m} c^{3} d^{2} e^{6} m^{3}+74 x^{7} \left (e x +d \right )^{m} c^{3} d \,e^{7} m +57 x^{6} \left (e x +d \right )^{m} c^{3} d^{2} e^{6} m^{2}+6 x^{5} \left (e x +d \right )^{m} c^{3} d^{3} e^{5} m^{3}+269 x^{6} \left (e x +d \right )^{m} c^{3} d^{2} e^{6} m +6 x \left (e x +d \right )^{m} c^{3} d^{7} e m +120 x^{7} \left (e x +d \right )^{m} c^{3} d \,e^{7}+420 x^{6} \left (e x +d \right )^{m} c^{3} d^{2} e^{6}+504 x^{5} \left (e x +d \right )^{m} c^{3} d^{3} e^{5}+210 x^{4} \left (e x +d \right )^{m} c^{3} d^{4} e^{4}}{\left (m^{3}+18 m^{2}+107 m +210\right ) d \left (4+m \right ) e^{4}}\) | \(532\) |
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Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (95) = 190\).
Time = 0.50 (sec) , antiderivative size = 343, normalized size of antiderivative = 3.61 \[ \int (d+e x)^m \left (c d x+c e x^2\right )^3 \, dx=\frac {{\left (6 \, c^{3} d^{6} e m x - 6 \, c^{3} d^{7} + {\left (c^{3} e^{7} m^{3} + 15 \, c^{3} e^{7} m^{2} + 74 \, c^{3} e^{7} m + 120 \, c^{3} e^{7}\right )} x^{7} + {\left (4 \, c^{3} d e^{6} m^{3} + 57 \, c^{3} d e^{6} m^{2} + 269 \, c^{3} d e^{6} m + 420 \, c^{3} d e^{6}\right )} x^{6} + 6 \, {\left (c^{3} d^{2} e^{5} m^{3} + 13 \, c^{3} d^{2} e^{5} m^{2} + 57 \, c^{3} d^{2} e^{5} m + 84 \, c^{3} d^{2} e^{5}\right )} x^{5} + 2 \, {\left (2 \, c^{3} d^{3} e^{4} m^{3} + 21 \, c^{3} d^{3} e^{4} m^{2} + 79 \, c^{3} d^{3} e^{4} m + 105 \, c^{3} d^{3} e^{4}\right )} x^{4} + {\left (c^{3} d^{4} e^{3} m^{3} + 3 \, c^{3} d^{4} e^{3} m^{2} + 2 \, c^{3} d^{4} e^{3} m\right )} x^{3} - 3 \, {\left (c^{3} d^{5} e^{2} m^{2} + c^{3} d^{5} e^{2} m\right )} x^{2}\right )} {\left (e x + d\right )}^{m}}{e^{4} m^{4} + 22 \, e^{4} m^{3} + 179 \, e^{4} m^{2} + 638 \, e^{4} m + 840 \, e^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2218 vs. \(2 (85) = 170\).
Time = 1.55 (sec) , antiderivative size = 2218, normalized size of antiderivative = 23.35 \[ \int (d+e x)^m \left (c d x+c e x^2\right )^3 \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 674 vs. \(2 (95) = 190\).
Time = 0.21 (sec) , antiderivative size = 674, normalized size of antiderivative = 7.09 \[ \int (d+e x)^m \left (c d x+c e x^2\right )^3 \, dx=\frac {{\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^{3} d^{3}}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} + \frac {3 \, {\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^{3} d^{2}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{4}} + \frac {3 \, {\left ({\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{6} x^{6} + {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} d e^{5} x^{5} - 5 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d^{2} e^{4} x^{4} + 20 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{3} e^{3} x^{3} - 60 \, {\left (m^{2} + m\right )} d^{4} e^{2} x^{2} + 120 \, d^{5} e m x - 120 \, d^{6}\right )} {\left (e x + d\right )}^{m} c^{3} d}{{\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} e^{4}} + \frac {{\left ({\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} e^{7} x^{7} + {\left (m^{6} + 15 \, m^{5} + 85 \, m^{4} + 225 \, m^{3} + 274 \, m^{2} + 120 \, m\right )} d e^{6} x^{6} - 6 \, {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} d^{2} e^{5} x^{5} + 30 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d^{3} e^{4} x^{4} - 120 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{4} e^{3} x^{3} + 360 \, {\left (m^{2} + m\right )} d^{5} e^{2} x^{2} - 720 \, d^{6} e m x + 720 \, d^{7}\right )} {\left (e x + d\right )}^{m} c^{3}}{{\left (m^{7} + 28 \, m^{6} + 322 \, m^{5} + 1960 \, m^{4} + 6769 \, m^{3} + 13132 \, m^{2} + 13068 \, m + 5040\right )} e^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 530 vs. \(2 (95) = 190\).
Time = 0.30 (sec) , antiderivative size = 530, normalized size of antiderivative = 5.58 \[ \int (d+e x)^m \left (c d x+c e x^2\right )^3 \, dx=\frac {{\left (e x + d\right )}^{m} c^{3} e^{7} m^{3} x^{7} + 4 \, {\left (e x + d\right )}^{m} c^{3} d e^{6} m^{3} x^{6} + 15 \, {\left (e x + d\right )}^{m} c^{3} e^{7} m^{2} x^{7} + 6 \, {\left (e x + d\right )}^{m} c^{3} d^{2} e^{5} m^{3} x^{5} + 57 \, {\left (e x + d\right )}^{m} c^{3} d e^{6} m^{2} x^{6} + 74 \, {\left (e x + d\right )}^{m} c^{3} e^{7} m x^{7} + 4 \, {\left (e x + d\right )}^{m} c^{3} d^{3} e^{4} m^{3} x^{4} + 78 \, {\left (e x + d\right )}^{m} c^{3} d^{2} e^{5} m^{2} x^{5} + 269 \, {\left (e x + d\right )}^{m} c^{3} d e^{6} m x^{6} + 120 \, {\left (e x + d\right )}^{m} c^{3} e^{7} x^{7} + {\left (e x + d\right )}^{m} c^{3} d^{4} e^{3} m^{3} x^{3} + 42 \, {\left (e x + d\right )}^{m} c^{3} d^{3} e^{4} m^{2} x^{4} + 342 \, {\left (e x + d\right )}^{m} c^{3} d^{2} e^{5} m x^{5} + 420 \, {\left (e x + d\right )}^{m} c^{3} d e^{6} x^{6} + 3 \, {\left (e x + d\right )}^{m} c^{3} d^{4} e^{3} m^{2} x^{3} + 158 \, {\left (e x + d\right )}^{m} c^{3} d^{3} e^{4} m x^{4} + 504 \, {\left (e x + d\right )}^{m} c^{3} d^{2} e^{5} x^{5} - 3 \, {\left (e x + d\right )}^{m} c^{3} d^{5} e^{2} m^{2} x^{2} + 2 \, {\left (e x + d\right )}^{m} c^{3} d^{4} e^{3} m x^{3} + 210 \, {\left (e x + d\right )}^{m} c^{3} d^{3} e^{4} x^{4} - 3 \, {\left (e x + d\right )}^{m} c^{3} d^{5} e^{2} m x^{2} + 6 \, {\left (e x + d\right )}^{m} c^{3} d^{6} e m x - 6 \, {\left (e x + d\right )}^{m} c^{3} d^{7}}{e^{4} m^{4} + 22 \, e^{4} m^{3} + 179 \, e^{4} m^{2} + 638 \, e^{4} m + 840 \, e^{4}} \]
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Time = 9.85 (sec) , antiderivative size = 333, normalized size of antiderivative = 3.51 \[ \int (d+e x)^m \left (c d x+c e x^2\right )^3 \, dx={\left (d+e\,x\right )}^m\,\left (\frac {c^3\,e^3\,x^7\,\left (m^3+15\,m^2+74\,m+120\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}-\frac {6\,c^3\,d^7}{e^4\,\left (m^4+22\,m^3+179\,m^2+638\,m+840\right )}+\frac {2\,c^3\,d^3\,x^4\,\left (2\,m^3+21\,m^2+79\,m+105\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}+\frac {6\,c^3\,d^6\,m\,x}{e^3\,\left (m^4+22\,m^3+179\,m^2+638\,m+840\right )}+\frac {c^3\,d\,e^2\,x^6\,\left (4\,m^3+57\,m^2+269\,m+420\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}+\frac {6\,c^3\,d^2\,e\,x^5\,\left (m^3+13\,m^2+57\,m+84\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}-\frac {3\,c^3\,d^5\,m\,x^2\,\left (m+1\right )}{e^2\,\left (m^4+22\,m^3+179\,m^2+638\,m+840\right )}+\frac {c^3\,d^4\,m\,x^3\,\left (m^2+3\,m+2\right )}{e\,\left (m^4+22\,m^3+179\,m^2+638\,m+840\right )}\right ) \]
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