Integrand size = 24, antiderivative size = 143 \[ \int (e x)^m (a+b x)^3 (a d-b d x)^2 \, dx=\frac {a^5 d^2 (e x)^{1+m}}{e (1+m)}+\frac {a^4 b d^2 (e x)^{2+m}}{e^2 (2+m)}-\frac {2 a^3 b^2 d^2 (e x)^{3+m}}{e^3 (3+m)}-\frac {2 a^2 b^3 d^2 (e x)^{4+m}}{e^4 (4+m)}+\frac {a b^4 d^2 (e x)^{5+m}}{e^5 (5+m)}+\frac {b^5 d^2 (e x)^{6+m}}{e^6 (6+m)} \]
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Time = 0.06 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {90} \[ \int (e x)^m (a+b x)^3 (a d-b d x)^2 \, dx=\frac {a^5 d^2 (e x)^{m+1}}{e (m+1)}+\frac {a^4 b d^2 (e x)^{m+2}}{e^2 (m+2)}-\frac {2 a^3 b^2 d^2 (e x)^{m+3}}{e^3 (m+3)}-\frac {2 a^2 b^3 d^2 (e x)^{m+4}}{e^4 (m+4)}+\frac {a b^4 d^2 (e x)^{m+5}}{e^5 (m+5)}+\frac {b^5 d^2 (e x)^{m+6}}{e^6 (m+6)} \]
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Rule 90
Rubi steps \begin{align*} \text {integral}& = \int \left (a^5 d^2 (e x)^m+\frac {a^4 b d^2 (e x)^{1+m}}{e}-\frac {2 a^3 b^2 d^2 (e x)^{2+m}}{e^2}-\frac {2 a^2 b^3 d^2 (e x)^{3+m}}{e^3}+\frac {a b^4 d^2 (e x)^{4+m}}{e^4}+\frac {b^5 d^2 (e x)^{5+m}}{e^5}\right ) \, dx \\ & = \frac {a^5 d^2 (e x)^{1+m}}{e (1+m)}+\frac {a^4 b d^2 (e x)^{2+m}}{e^2 (2+m)}-\frac {2 a^3 b^2 d^2 (e x)^{3+m}}{e^3 (3+m)}-\frac {2 a^2 b^3 d^2 (e x)^{4+m}}{e^4 (4+m)}+\frac {a b^4 d^2 (e x)^{5+m}}{e^5 (5+m)}+\frac {b^5 d^2 (e x)^{6+m}}{e^6 (6+m)} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.62 \[ \int (e x)^m (a+b x)^3 (a d-b d x)^2 \, dx=d^2 x (e x)^m \left (\frac {a^5}{1+m}+\frac {a^4 b x}{2+m}-\frac {2 a^3 b^2 x^2}{3+m}-\frac {2 a^2 b^3 x^3}{4+m}+\frac {a b^4 x^4}{5+m}+\frac {b^5 x^5}{6+m}\right ) \]
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Time = 0.42 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.99
method | result | size |
norman | \(\frac {a^{5} d^{2} x \,{\mathrm e}^{m \ln \left (e x \right )}}{1+m}+\frac {b^{5} d^{2} x^{6} {\mathrm e}^{m \ln \left (e x \right )}}{6+m}+\frac {a \,b^{4} d^{2} x^{5} {\mathrm e}^{m \ln \left (e x \right )}}{5+m}+\frac {a^{4} b \,d^{2} x^{2} {\mathrm e}^{m \ln \left (e x \right )}}{2+m}-\frac {2 a^{2} b^{3} d^{2} x^{4} {\mathrm e}^{m \ln \left (e x \right )}}{4+m}-\frac {2 a^{3} b^{2} d^{2} x^{3} {\mathrm e}^{m \ln \left (e x \right )}}{3+m}\) | \(142\) |
gosper | \(\frac {d^{2} \left (e x \right )^{m} \left (b^{5} m^{5} x^{5}+a \,b^{4} m^{5} x^{4}+15 b^{5} m^{4} x^{5}-2 a^{2} b^{3} m^{5} x^{3}+16 a \,b^{4} m^{4} x^{4}+85 b^{5} m^{3} x^{5}-2 a^{3} b^{2} m^{5} x^{2}-34 a^{2} b^{3} m^{4} x^{3}+95 a \,b^{4} m^{3} x^{4}+225 b^{5} m^{2} x^{5}+a^{4} b \,m^{5} x -36 a^{3} b^{2} m^{4} x^{2}-214 a^{2} b^{3} m^{3} x^{3}+260 a \,b^{4} m^{2} x^{4}+274 m \,x^{5} b^{5}+a^{5} m^{5}+19 a^{4} b \,m^{4} x -242 a^{3} b^{2} m^{3} x^{2}-614 a^{2} b^{3} m^{2} x^{3}+324 a \,b^{4} m \,x^{4}+120 b^{5} x^{5}+20 a^{5} m^{4}+137 a^{4} b \,m^{3} x -744 a^{3} b^{2} m^{2} x^{2}-792 a^{2} b^{3} m \,x^{3}+144 a \,b^{4} x^{4}+155 a^{5} m^{3}+461 a^{4} b \,m^{2} x -1016 a^{3} b^{2} m \,x^{2}-360 a^{2} b^{3} x^{3}+580 a^{5} m^{2}+702 a^{4} b m x -480 a^{3} b^{2} x^{2}+1044 a^{5} m +360 a^{4} b x +720 a^{5}\right ) x}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(422\) |
risch | \(\frac {d^{2} \left (e x \right )^{m} \left (b^{5} m^{5} x^{5}+a \,b^{4} m^{5} x^{4}+15 b^{5} m^{4} x^{5}-2 a^{2} b^{3} m^{5} x^{3}+16 a \,b^{4} m^{4} x^{4}+85 b^{5} m^{3} x^{5}-2 a^{3} b^{2} m^{5} x^{2}-34 a^{2} b^{3} m^{4} x^{3}+95 a \,b^{4} m^{3} x^{4}+225 b^{5} m^{2} x^{5}+a^{4} b \,m^{5} x -36 a^{3} b^{2} m^{4} x^{2}-214 a^{2} b^{3} m^{3} x^{3}+260 a \,b^{4} m^{2} x^{4}+274 m \,x^{5} b^{5}+a^{5} m^{5}+19 a^{4} b \,m^{4} x -242 a^{3} b^{2} m^{3} x^{2}-614 a^{2} b^{3} m^{2} x^{3}+324 a \,b^{4} m \,x^{4}+120 b^{5} x^{5}+20 a^{5} m^{4}+137 a^{4} b \,m^{3} x -744 a^{3} b^{2} m^{2} x^{2}-792 a^{2} b^{3} m \,x^{3}+144 a \,b^{4} x^{4}+155 a^{5} m^{3}+461 a^{4} b \,m^{2} x -1016 a^{3} b^{2} m \,x^{2}-360 a^{2} b^{3} x^{3}+580 a^{5} m^{2}+702 a^{4} b m x -480 a^{3} b^{2} x^{2}+1044 a^{5} m +360 a^{4} b x +720 a^{5}\right ) x}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(422\) |
parallelrisch | \(\frac {702 x^{2} \left (e x \right )^{m} a^{4} b \,d^{2} m +137 x^{2} \left (e x \right )^{m} a^{4} b \,d^{2} m^{3}-1016 x^{3} \left (e x \right )^{m} a^{3} b^{2} d^{2} m +461 x^{2} \left (e x \right )^{m} a^{4} b \,d^{2} m^{2}-34 x^{4} \left (e x \right )^{m} a^{2} b^{3} d^{2} m^{4}-2 x^{3} \left (e x \right )^{m} a^{3} b^{2} d^{2} m^{5}+260 x^{5} \left (e x \right )^{m} a \,b^{4} d^{2} m^{2}-214 x^{4} \left (e x \right )^{m} a^{2} b^{3} d^{2} m^{3}-36 x^{3} \left (e x \right )^{m} a^{3} b^{2} d^{2} m^{4}+x^{2} \left (e x \right )^{m} a^{4} b \,d^{2} m^{5}+324 x^{5} \left (e x \right )^{m} a \,b^{4} d^{2} m -614 x^{4} \left (e x \right )^{m} a^{2} b^{3} d^{2} m^{2}-242 x^{3} \left (e x \right )^{m} a^{3} b^{2} d^{2} m^{3}+19 x^{2} \left (e x \right )^{m} a^{4} b \,d^{2} m^{4}-792 x^{4} \left (e x \right )^{m} a^{2} b^{3} d^{2} m -744 x^{3} \left (e x \right )^{m} a^{3} b^{2} d^{2} m^{2}+x^{5} \left (e x \right )^{m} a \,b^{4} d^{2} m^{5}+16 x^{5} \left (e x \right )^{m} a \,b^{4} d^{2} m^{4}-2 x^{4} \left (e x \right )^{m} a^{2} b^{3} d^{2} m^{5}+95 x^{5} \left (e x \right )^{m} a \,b^{4} d^{2} m^{3}+120 x^{6} \left (e x \right )^{m} b^{5} d^{2}+720 x \left (e x \right )^{m} a^{5} d^{2}+x^{6} \left (e x \right )^{m} b^{5} d^{2} m^{5}+15 x^{6} \left (e x \right )^{m} b^{5} d^{2} m^{4}+85 x^{6} \left (e x \right )^{m} b^{5} d^{2} m^{3}+225 x^{6} \left (e x \right )^{m} b^{5} d^{2} m^{2}+274 x^{6} \left (e x \right )^{m} b^{5} d^{2} m +x \left (e x \right )^{m} a^{5} d^{2} m^{5}+144 x^{5} \left (e x \right )^{m} a \,b^{4} d^{2}+20 x \left (e x \right )^{m} a^{5} d^{2} m^{4}-360 x^{4} \left (e x \right )^{m} a^{2} b^{3} d^{2}+155 x \left (e x \right )^{m} a^{5} d^{2} m^{3}-480 x^{3} \left (e x \right )^{m} a^{3} b^{2} d^{2}+580 x \left (e x \right )^{m} a^{5} d^{2} m^{2}+360 x^{2} \left (e x \right )^{m} a^{4} b \,d^{2}+1044 x \left (e x \right )^{m} a^{5} d^{2} m}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(719\) |
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Leaf count of result is larger than twice the leaf count of optimal. 475 vs. \(2 (143) = 286\).
Time = 0.23 (sec) , antiderivative size = 475, normalized size of antiderivative = 3.32 \[ \int (e x)^m (a+b x)^3 (a d-b d x)^2 \, dx=\frac {{\left ({\left (b^{5} d^{2} m^{5} + 15 \, b^{5} d^{2} m^{4} + 85 \, b^{5} d^{2} m^{3} + 225 \, b^{5} d^{2} m^{2} + 274 \, b^{5} d^{2} m + 120 \, b^{5} d^{2}\right )} x^{6} + {\left (a b^{4} d^{2} m^{5} + 16 \, a b^{4} d^{2} m^{4} + 95 \, a b^{4} d^{2} m^{3} + 260 \, a b^{4} d^{2} m^{2} + 324 \, a b^{4} d^{2} m + 144 \, a b^{4} d^{2}\right )} x^{5} - 2 \, {\left (a^{2} b^{3} d^{2} m^{5} + 17 \, a^{2} b^{3} d^{2} m^{4} + 107 \, a^{2} b^{3} d^{2} m^{3} + 307 \, a^{2} b^{3} d^{2} m^{2} + 396 \, a^{2} b^{3} d^{2} m + 180 \, a^{2} b^{3} d^{2}\right )} x^{4} - 2 \, {\left (a^{3} b^{2} d^{2} m^{5} + 18 \, a^{3} b^{2} d^{2} m^{4} + 121 \, a^{3} b^{2} d^{2} m^{3} + 372 \, a^{3} b^{2} d^{2} m^{2} + 508 \, a^{3} b^{2} d^{2} m + 240 \, a^{3} b^{2} d^{2}\right )} x^{3} + {\left (a^{4} b d^{2} m^{5} + 19 \, a^{4} b d^{2} m^{4} + 137 \, a^{4} b d^{2} m^{3} + 461 \, a^{4} b d^{2} m^{2} + 702 \, a^{4} b d^{2} m + 360 \, a^{4} b d^{2}\right )} x^{2} + {\left (a^{5} d^{2} m^{5} + 20 \, a^{5} d^{2} m^{4} + 155 \, a^{5} d^{2} m^{3} + 580 \, a^{5} d^{2} m^{2} + 1044 \, a^{5} d^{2} m + 720 \, a^{5} d^{2}\right )} x\right )} \left (e x\right )^{m}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2259 vs. \(2 (133) = 266\).
Time = 0.49 (sec) , antiderivative size = 2259, normalized size of antiderivative = 15.80 \[ \int (e x)^m (a+b x)^3 (a d-b d x)^2 \, dx=\text {Too large to display} \]
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Time = 0.27 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.97 \[ \int (e x)^m (a+b x)^3 (a d-b d x)^2 \, dx=\frac {b^{5} d^{2} e^{m} x^{6} x^{m}}{m + 6} + \frac {a b^{4} d^{2} e^{m} x^{5} x^{m}}{m + 5} - \frac {2 \, a^{2} b^{3} d^{2} e^{m} x^{4} x^{m}}{m + 4} - \frac {2 \, a^{3} b^{2} d^{2} e^{m} x^{3} x^{m}}{m + 3} + \frac {a^{4} b d^{2} e^{m} x^{2} x^{m}}{m + 2} + \frac {\left (e x\right )^{m + 1} a^{5} d^{2}}{e {\left (m + 1\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 718 vs. \(2 (143) = 286\).
Time = 0.29 (sec) , antiderivative size = 718, normalized size of antiderivative = 5.02 \[ \int (e x)^m (a+b x)^3 (a d-b d x)^2 \, dx=\frac {\left (e x\right )^{m} b^{5} d^{2} m^{5} x^{6} + \left (e x\right )^{m} a b^{4} d^{2} m^{5} x^{5} + 15 \, \left (e x\right )^{m} b^{5} d^{2} m^{4} x^{6} - 2 \, \left (e x\right )^{m} a^{2} b^{3} d^{2} m^{5} x^{4} + 16 \, \left (e x\right )^{m} a b^{4} d^{2} m^{4} x^{5} + 85 \, \left (e x\right )^{m} b^{5} d^{2} m^{3} x^{6} - 2 \, \left (e x\right )^{m} a^{3} b^{2} d^{2} m^{5} x^{3} - 34 \, \left (e x\right )^{m} a^{2} b^{3} d^{2} m^{4} x^{4} + 95 \, \left (e x\right )^{m} a b^{4} d^{2} m^{3} x^{5} + 225 \, \left (e x\right )^{m} b^{5} d^{2} m^{2} x^{6} + \left (e x\right )^{m} a^{4} b d^{2} m^{5} x^{2} - 36 \, \left (e x\right )^{m} a^{3} b^{2} d^{2} m^{4} x^{3} - 214 \, \left (e x\right )^{m} a^{2} b^{3} d^{2} m^{3} x^{4} + 260 \, \left (e x\right )^{m} a b^{4} d^{2} m^{2} x^{5} + 274 \, \left (e x\right )^{m} b^{5} d^{2} m x^{6} + \left (e x\right )^{m} a^{5} d^{2} m^{5} x + 19 \, \left (e x\right )^{m} a^{4} b d^{2} m^{4} x^{2} - 242 \, \left (e x\right )^{m} a^{3} b^{2} d^{2} m^{3} x^{3} - 614 \, \left (e x\right )^{m} a^{2} b^{3} d^{2} m^{2} x^{4} + 324 \, \left (e x\right )^{m} a b^{4} d^{2} m x^{5} + 120 \, \left (e x\right )^{m} b^{5} d^{2} x^{6} + 20 \, \left (e x\right )^{m} a^{5} d^{2} m^{4} x + 137 \, \left (e x\right )^{m} a^{4} b d^{2} m^{3} x^{2} - 744 \, \left (e x\right )^{m} a^{3} b^{2} d^{2} m^{2} x^{3} - 792 \, \left (e x\right )^{m} a^{2} b^{3} d^{2} m x^{4} + 144 \, \left (e x\right )^{m} a b^{4} d^{2} x^{5} + 155 \, \left (e x\right )^{m} a^{5} d^{2} m^{3} x + 461 \, \left (e x\right )^{m} a^{4} b d^{2} m^{2} x^{2} - 1016 \, \left (e x\right )^{m} a^{3} b^{2} d^{2} m x^{3} - 360 \, \left (e x\right )^{m} a^{2} b^{3} d^{2} x^{4} + 580 \, \left (e x\right )^{m} a^{5} d^{2} m^{2} x + 702 \, \left (e x\right )^{m} a^{4} b d^{2} m x^{2} - 480 \, \left (e x\right )^{m} a^{3} b^{2} d^{2} x^{3} + 1044 \, \left (e x\right )^{m} a^{5} d^{2} m x + 360 \, \left (e x\right )^{m} a^{4} b d^{2} x^{2} + 720 \, \left (e x\right )^{m} a^{5} d^{2} x}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \]
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Time = 0.80 (sec) , antiderivative size = 393, normalized size of antiderivative = 2.75 \[ \int (e x)^m (a+b x)^3 (a d-b d x)^2 \, dx={\left (e\,x\right )}^m\,\left (\frac {b^5\,d^2\,x^6\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {a^5\,d^2\,x\,\left (m^5+20\,m^4+155\,m^3+580\,m^2+1044\,m+720\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {a\,b^4\,d^2\,x^5\,\left (m^5+16\,m^4+95\,m^3+260\,m^2+324\,m+144\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {a^4\,b\,d^2\,x^2\,\left (m^5+19\,m^4+137\,m^3+461\,m^2+702\,m+360\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}-\frac {2\,a^2\,b^3\,d^2\,x^4\,\left (m^5+17\,m^4+107\,m^3+307\,m^2+396\,m+180\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}-\frac {2\,a^3\,b^2\,d^2\,x^3\,\left (m^5+18\,m^4+121\,m^3+372\,m^2+508\,m+240\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}\right ) \]
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