Integrand size = 22, antiderivative size = 94 \[ \int (e x)^m (a+b x) (a c-b c x)^3 \, dx=\frac {a^4 c^3 (e x)^{1+m}}{e (1+m)}-\frac {2 a^3 b c^3 (e x)^{2+m}}{e^2 (2+m)}+\frac {2 a b^3 c^3 (e x)^{4+m}}{e^4 (4+m)}-\frac {b^4 c^3 (e x)^{5+m}}{e^5 (5+m)} \]
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Time = 0.04 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {76} \[ \int (e x)^m (a+b x) (a c-b c x)^3 \, dx=\frac {a^4 c^3 (e x)^{m+1}}{e (m+1)}-\frac {2 a^3 b c^3 (e x)^{m+2}}{e^2 (m+2)}+\frac {2 a b^3 c^3 (e x)^{m+4}}{e^4 (m+4)}-\frac {b^4 c^3 (e x)^{m+5}}{e^5 (m+5)} \]
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Rule 76
Rubi steps \begin{align*} \text {integral}& = \int \left (a^4 c^3 (e x)^m-\frac {2 a^3 b c^3 (e x)^{1+m}}{e}+\frac {2 a b^3 c^3 (e x)^{3+m}}{e^3}-\frac {b^4 c^3 (e x)^{4+m}}{e^4}\right ) \, dx \\ & = \frac {a^4 c^3 (e x)^{1+m}}{e (1+m)}-\frac {2 a^3 b c^3 (e x)^{2+m}}{e^2 (2+m)}+\frac {2 a b^3 c^3 (e x)^{4+m}}{e^4 (4+m)}-\frac {b^4 c^3 (e x)^{5+m}}{e^5 (5+m)} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.19 \[ \int (e x)^m (a+b x) (a c-b c x)^3 \, dx=-\frac {c^3 x (e x)^m \left (-a^4 \left (40+38 m+11 m^2+m^3\right )+2 a^3 b \left (20+29 m+10 m^2+m^3\right ) x-2 a b^3 \left (10+17 m+8 m^2+m^3\right ) x^3+b^4 \left (8+14 m+7 m^2+m^3\right ) x^4\right )}{(1+m) (2+m) (4+m) (5+m)} \]
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Time = 0.41 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.99
method | result | size |
norman | \(\frac {a^{4} c^{3} x \,{\mathrm e}^{m \ln \left (e x \right )}}{1+m}-\frac {b^{4} c^{3} x^{5} {\mathrm e}^{m \ln \left (e x \right )}}{5+m}+\frac {2 a \,b^{3} c^{3} x^{4} {\mathrm e}^{m \ln \left (e x \right )}}{4+m}-\frac {2 a^{3} c^{3} b \,x^{2} {\mathrm e}^{m \ln \left (e x \right )}}{2+m}\) | \(93\) |
gosper | \(\frac {c^{3} \left (e x \right )^{m} \left (-b^{4} m^{3} x^{4}+2 a \,b^{3} m^{3} x^{3}-7 b^{4} m^{2} x^{4}+16 a \,b^{3} m^{2} x^{3}-14 m \,x^{4} b^{4}-2 a^{3} b \,m^{3} x +34 a \,b^{3} m \,x^{3}-8 b^{4} x^{4}+a^{4} m^{3}-20 a^{3} b \,m^{2} x +20 a \,b^{3} x^{3}+11 a^{4} m^{2}-58 a^{3} b m x +38 a^{4} m -40 a^{3} b x +40 a^{4}\right ) x}{\left (5+m \right ) \left (4+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(175\) |
risch | \(\frac {c^{3} \left (e x \right )^{m} \left (-b^{4} m^{3} x^{4}+2 a \,b^{3} m^{3} x^{3}-7 b^{4} m^{2} x^{4}+16 a \,b^{3} m^{2} x^{3}-14 m \,x^{4} b^{4}-2 a^{3} b \,m^{3} x +34 a \,b^{3} m \,x^{3}-8 b^{4} x^{4}+a^{4} m^{3}-20 a^{3} b \,m^{2} x +20 a \,b^{3} x^{3}+11 a^{4} m^{2}-58 a^{3} b m x +38 a^{4} m -40 a^{3} b x +40 a^{4}\right ) x}{\left (5+m \right ) \left (4+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(175\) |
parallelrisch | \(-\frac {x^{5} \left (e x \right )^{m} b^{4} c^{3} m^{3}+7 x^{5} \left (e x \right )^{m} b^{4} c^{3} m^{2}-2 x^{4} \left (e x \right )^{m} a \,b^{3} c^{3} m^{3}+14 x^{5} \left (e x \right )^{m} b^{4} c^{3} m -16 x^{4} \left (e x \right )^{m} a \,b^{3} c^{3} m^{2}+8 x^{5} \left (e x \right )^{m} b^{4} c^{3}-34 x^{4} \left (e x \right )^{m} a \,b^{3} c^{3} m +2 x^{2} \left (e x \right )^{m} a^{3} b \,c^{3} m^{3}-20 x^{4} \left (e x \right )^{m} a \,b^{3} c^{3}+20 x^{2} \left (e x \right )^{m} a^{3} b \,c^{3} m^{2}-x \left (e x \right )^{m} a^{4} c^{3} m^{3}+58 x^{2} \left (e x \right )^{m} a^{3} b \,c^{3} m -11 x \left (e x \right )^{m} a^{4} c^{3} m^{2}+40 x^{2} \left (e x \right )^{m} a^{3} b \,c^{3}-38 x \left (e x \right )^{m} a^{4} c^{3} m -40 x \left (e x \right )^{m} a^{4} c^{3}}{\left (5+m \right ) \left (4+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(307\) |
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Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (94) = 188\).
Time = 0.24 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.22 \[ \int (e x)^m (a+b x) (a c-b c x)^3 \, dx=-\frac {{\left ({\left (b^{4} c^{3} m^{3} + 7 \, b^{4} c^{3} m^{2} + 14 \, b^{4} c^{3} m + 8 \, b^{4} c^{3}\right )} x^{5} - 2 \, {\left (a b^{3} c^{3} m^{3} + 8 \, a b^{3} c^{3} m^{2} + 17 \, a b^{3} c^{3} m + 10 \, a b^{3} c^{3}\right )} x^{4} + 2 \, {\left (a^{3} b c^{3} m^{3} + 10 \, a^{3} b c^{3} m^{2} + 29 \, a^{3} b c^{3} m + 20 \, a^{3} b c^{3}\right )} x^{2} - {\left (a^{4} c^{3} m^{3} + 11 \, a^{4} c^{3} m^{2} + 38 \, a^{4} c^{3} m + 40 \, a^{4} c^{3}\right )} x\right )} \left (e x\right )^{m}}{m^{4} + 12 \, m^{3} + 49 \, m^{2} + 78 \, m + 40} \]
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Leaf count of result is larger than twice the leaf count of optimal. 811 vs. \(2 (85) = 170\).
Time = 0.36 (sec) , antiderivative size = 811, normalized size of antiderivative = 8.63 \[ \int (e x)^m (a+b x) (a c-b c x)^3 \, dx=\begin {cases} \frac {- \frac {a^{4} c^{3}}{4 x^{4}} + \frac {2 a^{3} b c^{3}}{3 x^{3}} - \frac {2 a b^{3} c^{3}}{x} - b^{4} c^{3} \log {\left (x \right )}}{e^{5}} & \text {for}\: m = -5 \\\frac {- \frac {a^{4} c^{3}}{3 x^{3}} + \frac {a^{3} b c^{3}}{x^{2}} + 2 a b^{3} c^{3} \log {\left (x \right )} - b^{4} c^{3} x}{e^{4}} & \text {for}\: m = -4 \\\frac {- \frac {a^{4} c^{3}}{x} - 2 a^{3} b c^{3} \log {\left (x \right )} + a b^{3} c^{3} x^{2} - \frac {b^{4} c^{3} x^{3}}{3}}{e^{2}} & \text {for}\: m = -2 \\\frac {a^{4} c^{3} \log {\left (x \right )} - 2 a^{3} b c^{3} x + \frac {2 a b^{3} c^{3} x^{3}}{3} - \frac {b^{4} c^{3} x^{4}}{4}}{e} & \text {for}\: m = -1 \\\frac {a^{4} c^{3} m^{3} x \left (e x\right )^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {11 a^{4} c^{3} m^{2} x \left (e x\right )^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {38 a^{4} c^{3} m x \left (e x\right )^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {40 a^{4} c^{3} x \left (e x\right )^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {2 a^{3} b c^{3} m^{3} x^{2} \left (e x\right )^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {20 a^{3} b c^{3} m^{2} x^{2} \left (e x\right )^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {58 a^{3} b c^{3} m x^{2} \left (e x\right )^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {40 a^{3} b c^{3} x^{2} \left (e x\right )^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {2 a b^{3} c^{3} m^{3} x^{4} \left (e x\right )^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {16 a b^{3} c^{3} m^{2} x^{4} \left (e x\right )^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {34 a b^{3} c^{3} m x^{4} \left (e x\right )^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {20 a b^{3} c^{3} x^{4} \left (e x\right )^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {b^{4} c^{3} m^{3} x^{5} \left (e x\right )^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {7 b^{4} c^{3} m^{2} x^{5} \left (e x\right )^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {14 b^{4} c^{3} m x^{5} \left (e x\right )^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {8 b^{4} c^{3} x^{5} \left (e x\right )^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.97 \[ \int (e x)^m (a+b x) (a c-b c x)^3 \, dx=-\frac {b^{4} c^{3} e^{m} x^{5} x^{m}}{m + 5} + \frac {2 \, a b^{3} c^{3} e^{m} x^{4} x^{m}}{m + 4} - \frac {2 \, a^{3} b c^{3} e^{m} x^{2} x^{m}}{m + 2} + \frac {\left (e x\right )^{m + 1} a^{4} c^{3}}{e {\left (m + 1\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 306 vs. \(2 (94) = 188\).
Time = 0.28 (sec) , antiderivative size = 306, normalized size of antiderivative = 3.26 \[ \int (e x)^m (a+b x) (a c-b c x)^3 \, dx=-\frac {\left (e x\right )^{m} b^{4} c^{3} m^{3} x^{5} - 2 \, \left (e x\right )^{m} a b^{3} c^{3} m^{3} x^{4} + 7 \, \left (e x\right )^{m} b^{4} c^{3} m^{2} x^{5} - 16 \, \left (e x\right )^{m} a b^{3} c^{3} m^{2} x^{4} + 14 \, \left (e x\right )^{m} b^{4} c^{3} m x^{5} + 2 \, \left (e x\right )^{m} a^{3} b c^{3} m^{3} x^{2} - 34 \, \left (e x\right )^{m} a b^{3} c^{3} m x^{4} + 8 \, \left (e x\right )^{m} b^{4} c^{3} x^{5} - \left (e x\right )^{m} a^{4} c^{3} m^{3} x + 20 \, \left (e x\right )^{m} a^{3} b c^{3} m^{2} x^{2} - 20 \, \left (e x\right )^{m} a b^{3} c^{3} x^{4} - 11 \, \left (e x\right )^{m} a^{4} c^{3} m^{2} x + 58 \, \left (e x\right )^{m} a^{3} b c^{3} m x^{2} - 38 \, \left (e x\right )^{m} a^{4} c^{3} m x + 40 \, \left (e x\right )^{m} a^{3} b c^{3} x^{2} - 40 \, \left (e x\right )^{m} a^{4} c^{3} x}{m^{4} + 12 \, m^{3} + 49 \, m^{2} + 78 \, m + 40} \]
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Time = 0.46 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.94 \[ \int (e x)^m (a+b x) (a c-b c x)^3 \, dx={\left (e\,x\right )}^m\,\left (\frac {a^4\,c^3\,x\,\left (m^3+11\,m^2+38\,m+40\right )}{m^4+12\,m^3+49\,m^2+78\,m+40}-\frac {b^4\,c^3\,x^5\,\left (m^3+7\,m^2+14\,m+8\right )}{m^4+12\,m^3+49\,m^2+78\,m+40}+\frac {2\,a\,b^3\,c^3\,x^4\,\left (m^3+8\,m^2+17\,m+10\right )}{m^4+12\,m^3+49\,m^2+78\,m+40}-\frac {2\,a^3\,b\,c^3\,x^2\,\left (m^3+10\,m^2+29\,m+20\right )}{m^4+12\,m^3+49\,m^2+78\,m+40}\right ) \]
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