Integrand size = 22, antiderivative size = 93 \[ \int (e x)^m (a+b x) (a c-b c x)^2 \, dx=\frac {a^3 c^2 (e x)^{1+m}}{e (1+m)}-\frac {a^2 b c^2 (e x)^{2+m}}{e^2 (2+m)}-\frac {a b^2 c^2 (e x)^{3+m}}{e^3 (3+m)}+\frac {b^3 c^2 (e x)^{4+m}}{e^4 (4+m)} \]
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Time = 0.04 (sec) , antiderivative size = 93, 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} \[ \int (e x)^m (a+b x) (a c-b c x)^2 \, dx=\frac {a^3 c^2 (e x)^{m+1}}{e (m+1)}-\frac {a^2 b c^2 (e x)^{m+2}}{e^2 (m+2)}-\frac {a b^2 c^2 (e x)^{m+3}}{e^3 (m+3)}+\frac {b^3 c^2 (e x)^{m+4}}{e^4 (m+4)} \]
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Rule 76
Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 c^2 (e x)^m-\frac {a^2 b c^2 (e x)^{1+m}}{e}-\frac {a b^2 c^2 (e x)^{2+m}}{e^2}+\frac {b^3 c^2 (e x)^{3+m}}{e^3}\right ) \, dx \\ & = \frac {a^3 c^2 (e x)^{1+m}}{e (1+m)}-\frac {a^2 b c^2 (e x)^{2+m}}{e^2 (2+m)}-\frac {a b^2 c^2 (e x)^{3+m}}{e^3 (3+m)}+\frac {b^3 c^2 (e x)^{4+m}}{e^4 (4+m)} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.95 \[ \int (e x)^m (a+b x) (a c-b c x)^2 \, dx=\frac {c^2 x (e x)^m \left ((-a+b x)^3+\frac {a (5+2 m) \left (a^2 \left (6+5 m+m^2\right )-2 a b \left (3+4 m+m^2\right ) x+b^2 \left (2+3 m+m^2\right ) x^2\right )}{(1+m) (2+m) (3+m)}\right )}{4+m} \]
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Time = 0.40 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.99
method | result | size |
norman | \(\frac {a^{3} c^{2} x \,{\mathrm e}^{m \ln \left (e x \right )}}{1+m}+\frac {b^{3} c^{2} x^{4} {\mathrm e}^{m \ln \left (e x \right )}}{4+m}-\frac {a \,b^{2} c^{2} x^{3} {\mathrm e}^{m \ln \left (e x \right )}}{3+m}-\frac {a^{2} b \,c^{2} x^{2} {\mathrm e}^{m \ln \left (e x \right )}}{2+m}\) | \(92\) |
gosper | \(\frac {c^{2} \left (e x \right )^{m} \left (b^{3} m^{3} x^{3}-a \,b^{2} m^{3} x^{2}+6 b^{3} m^{2} x^{3}-a^{2} b \,m^{3} x -7 a \,b^{2} m^{2} x^{2}+11 m \,x^{3} b^{3}+a^{3} m^{3}-8 a^{2} b \,m^{2} x -14 a \,b^{2} m \,x^{2}+6 b^{3} x^{3}+9 a^{3} m^{2}-19 a^{2} b m x -8 a \,b^{2} x^{2}+26 a^{3} m -12 a^{2} b x +24 a^{3}\right ) x}{\left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(174\) |
risch | \(\frac {c^{2} \left (e x \right )^{m} \left (b^{3} m^{3} x^{3}-a \,b^{2} m^{3} x^{2}+6 b^{3} m^{2} x^{3}-a^{2} b \,m^{3} x -7 a \,b^{2} m^{2} x^{2}+11 m \,x^{3} b^{3}+a^{3} m^{3}-8 a^{2} b \,m^{2} x -14 a \,b^{2} m \,x^{2}+6 b^{3} x^{3}+9 a^{3} m^{2}-19 a^{2} b m x -8 a \,b^{2} x^{2}+26 a^{3} m -12 a^{2} b x +24 a^{3}\right ) x}{\left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(174\) |
parallelrisch | \(\frac {x^{4} \left (e x \right )^{m} b^{3} c^{2} m^{3}+6 x^{4} \left (e x \right )^{m} b^{3} c^{2} m^{2}-x^{3} \left (e x \right )^{m} a \,b^{2} c^{2} m^{3}+11 x^{4} \left (e x \right )^{m} b^{3} c^{2} m -7 x^{3} \left (e x \right )^{m} a \,b^{2} c^{2} m^{2}-x^{2} \left (e x \right )^{m} a^{2} b \,c^{2} m^{3}+6 x^{4} \left (e x \right )^{m} b^{3} c^{2}-14 x^{3} \left (e x \right )^{m} a \,b^{2} c^{2} m -8 x^{2} \left (e x \right )^{m} a^{2} b \,c^{2} m^{2}+x \left (e x \right )^{m} a^{3} c^{2} m^{3}-8 x^{3} \left (e x \right )^{m} a \,b^{2} c^{2}-19 x^{2} \left (e x \right )^{m} a^{2} b \,c^{2} m +9 x \left (e x \right )^{m} a^{3} c^{2} m^{2}-12 x^{2} \left (e x \right )^{m} a^{2} b \,c^{2}+26 x \left (e x \right )^{m} a^{3} c^{2} m +24 x \left (e x \right )^{m} a^{3} c^{2}}{\left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(305\) |
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Leaf count of result is larger than twice the leaf count of optimal. 207 vs. \(2 (93) = 186\).
Time = 0.23 (sec) , antiderivative size = 207, normalized size of antiderivative = 2.23 \[ \int (e x)^m (a+b x) (a c-b c x)^2 \, dx=\frac {{\left ({\left (b^{3} c^{2} m^{3} + 6 \, b^{3} c^{2} m^{2} + 11 \, b^{3} c^{2} m + 6 \, b^{3} c^{2}\right )} x^{4} - {\left (a b^{2} c^{2} m^{3} + 7 \, a b^{2} c^{2} m^{2} + 14 \, a b^{2} c^{2} m + 8 \, a b^{2} c^{2}\right )} x^{3} - {\left (a^{2} b c^{2} m^{3} + 8 \, a^{2} b c^{2} m^{2} + 19 \, a^{2} b c^{2} m + 12 \, a^{2} b c^{2}\right )} x^{2} + {\left (a^{3} c^{2} m^{3} + 9 \, a^{3} c^{2} m^{2} + 26 \, a^{3} c^{2} m + 24 \, a^{3} c^{2}\right )} x\right )} \left (e x\right )^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \]
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Leaf count of result is larger than twice the leaf count of optimal. 794 vs. \(2 (82) = 164\).
Time = 0.33 (sec) , antiderivative size = 794, normalized size of antiderivative = 8.54 \[ \int (e x)^m (a+b x) (a c-b c x)^2 \, dx=\begin {cases} \frac {- \frac {a^{3} c^{2}}{3 x^{3}} + \frac {a^{2} b c^{2}}{2 x^{2}} + \frac {a b^{2} c^{2}}{x} + b^{3} c^{2} \log {\left (x \right )}}{e^{4}} & \text {for}\: m = -4 \\\frac {- \frac {a^{3} c^{2}}{2 x^{2}} + \frac {a^{2} b c^{2}}{x} - a b^{2} c^{2} \log {\left (x \right )} + b^{3} c^{2} x}{e^{3}} & \text {for}\: m = -3 \\\frac {- \frac {a^{3} c^{2}}{x} - a^{2} b c^{2} \log {\left (x \right )} - a b^{2} c^{2} x + \frac {b^{3} c^{2} x^{2}}{2}}{e^{2}} & \text {for}\: m = -2 \\\frac {a^{3} c^{2} \log {\left (x \right )} - a^{2} b c^{2} x - \frac {a b^{2} c^{2} x^{2}}{2} + \frac {b^{3} c^{2} x^{3}}{3}}{e} & \text {for}\: m = -1 \\\frac {a^{3} c^{2} m^{3} x \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {9 a^{3} c^{2} m^{2} x \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {26 a^{3} c^{2} m x \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {24 a^{3} c^{2} x \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {a^{2} b c^{2} m^{3} x^{2} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {8 a^{2} b c^{2} m^{2} x^{2} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {19 a^{2} b c^{2} m x^{2} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {12 a^{2} b c^{2} x^{2} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {a b^{2} c^{2} m^{3} x^{3} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {7 a b^{2} c^{2} m^{2} x^{3} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {14 a b^{2} c^{2} m x^{3} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {8 a b^{2} c^{2} x^{3} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {b^{3} c^{2} m^{3} x^{4} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {6 b^{3} c^{2} m^{2} x^{4} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {11 b^{3} c^{2} m x^{4} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {6 b^{3} c^{2} x^{4} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.97 \[ \int (e x)^m (a+b x) (a c-b c x)^2 \, dx=\frac {b^{3} c^{2} e^{m} x^{4} x^{m}}{m + 4} - \frac {a b^{2} c^{2} e^{m} x^{3} x^{m}}{m + 3} - \frac {a^{2} b c^{2} e^{m} x^{2} x^{m}}{m + 2} + \frac {\left (e x\right )^{m + 1} a^{3} c^{2}}{e {\left (m + 1\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (93) = 186\).
Time = 0.30 (sec) , antiderivative size = 304, normalized size of antiderivative = 3.27 \[ \int (e x)^m (a+b x) (a c-b c x)^2 \, dx=\frac {\left (e x\right )^{m} b^{3} c^{2} m^{3} x^{4} - \left (e x\right )^{m} a b^{2} c^{2} m^{3} x^{3} + 6 \, \left (e x\right )^{m} b^{3} c^{2} m^{2} x^{4} - \left (e x\right )^{m} a^{2} b c^{2} m^{3} x^{2} - 7 \, \left (e x\right )^{m} a b^{2} c^{2} m^{2} x^{3} + 11 \, \left (e x\right )^{m} b^{3} c^{2} m x^{4} + \left (e x\right )^{m} a^{3} c^{2} m^{3} x - 8 \, \left (e x\right )^{m} a^{2} b c^{2} m^{2} x^{2} - 14 \, \left (e x\right )^{m} a b^{2} c^{2} m x^{3} + 6 \, \left (e x\right )^{m} b^{3} c^{2} x^{4} + 9 \, \left (e x\right )^{m} a^{3} c^{2} m^{2} x - 19 \, \left (e x\right )^{m} a^{2} b c^{2} m x^{2} - 8 \, \left (e x\right )^{m} a b^{2} c^{2} x^{3} + 26 \, \left (e x\right )^{m} a^{3} c^{2} m x - 12 \, \left (e x\right )^{m} a^{2} b c^{2} x^{2} + 24 \, \left (e x\right )^{m} a^{3} c^{2} x}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \]
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Time = 0.42 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.95 \[ \int (e x)^m (a+b x) (a c-b c x)^2 \, dx={\left (e\,x\right )}^m\,\left (\frac {a^3\,c^2\,x\,\left (m^3+9\,m^2+26\,m+24\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {b^3\,c^2\,x^4\,\left (m^3+6\,m^2+11\,m+6\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}-\frac {a\,b^2\,c^2\,x^3\,\left (m^3+7\,m^2+14\,m+8\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}-\frac {a^2\,b\,c^2\,x^2\,\left (m^3+8\,m^2+19\,m+12\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}\right ) \]
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