Integrand size = 33, antiderivative size = 52 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=-\frac {\left (c d^2-a e^2\right ) (d+e x)^{2+m}}{e^2 (2+m)}+\frac {c d (d+e x)^{3+m}}{e^2 (3+m)} \]
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Time = 0.02 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {640, 45} \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {c d (d+e x)^{m+3}}{e^2 (m+3)}-\frac {\left (c d^2-a e^2\right ) (d+e x)^{m+2}}{e^2 (m+2)} \]
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Rule 45
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int (a e+c d x) (d+e x)^{1+m} \, dx \\ & = \int \left (\frac {\left (-c d^2+a e^2\right ) (d+e x)^{1+m}}{e}+\frac {c d (d+e x)^{2+m}}{e}\right ) \, dx \\ & = -\frac {\left (c d^2-a e^2\right ) (d+e x)^{2+m}}{e^2 (2+m)}+\frac {c d (d+e x)^{3+m}}{e^2 (3+m)} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.87 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {(d+e x)^{2+m} \left (a e^2 (3+m)+c d (-d+e (2+m) x)\right )}{e^2 (2+m) (3+m)} \]
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Time = 4.10 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.06
method | result | size |
gosper | \(\frac {\left (e x +d \right )^{2+m} \left (c d e m x +a \,e^{2} m +2 x c d e +3 e^{2} a -c \,d^{2}\right )}{e^{2} \left (m^{2}+5 m +6\right )}\) | \(55\) |
risch | \(\frac {\left (c d \,e^{3} m \,x^{3}+a \,e^{4} m \,x^{2}+2 c \,d^{2} e^{2} m \,x^{2}+2 c d \,e^{3} x^{3}+2 a d \,e^{3} m x +3 a \,e^{4} x^{2}+c \,d^{3} e m x +3 c \,d^{2} e^{2} x^{2}+a \,d^{2} e^{2} m +6 a d \,e^{3} x +3 a \,d^{2} e^{2}-c \,d^{4}\right ) \left (e x +d \right )^{m}}{\left (2+m \right ) \left (3+m \right ) e^{2}}\) | \(135\) |
norman | \(\frac {\left (a \,e^{2} m +2 c \,d^{2} m +3 e^{2} a +3 c \,d^{2}\right ) x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{m^{2}+5 m +6}+\frac {d^{2} \left (a \,e^{2} m +3 e^{2} a -c \,d^{2}\right ) {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{2} \left (m^{2}+5 m +6\right )}+\frac {c d e \,x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{3+m}+\frac {d \left (2 a \,e^{2} m +c \,d^{2} m +6 e^{2} a \right ) x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (m^{2}+5 m +6\right )}\) | \(162\) |
parallelrisch | \(\frac {x^{3} \left (e x +d \right )^{m} c d \,e^{3} m +2 x^{3} \left (e x +d \right )^{m} c d \,e^{3}+x^{2} \left (e x +d \right )^{m} a \,e^{4} m +2 x^{2} \left (e x +d \right )^{m} c \,d^{2} e^{2} m +3 x^{2} \left (e x +d \right )^{m} a \,e^{4}+3 x^{2} \left (e x +d \right )^{m} c \,d^{2} e^{2}+2 x \left (e x +d \right )^{m} a d \,e^{3} m +x \left (e x +d \right )^{m} c \,d^{3} e m +6 x \left (e x +d \right )^{m} a d \,e^{3}+\left (e x +d \right )^{m} a \,d^{2} e^{2} m +3 \left (e x +d \right )^{m} a \,d^{2} e^{2}-\left (e x +d \right )^{m} c \,d^{4}}{e^{2} \left (m^{2}+5 m +6\right )}\) | \(212\) |
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Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (52) = 104\).
Time = 0.40 (sec) , antiderivative size = 136, normalized size of antiderivative = 2.62 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {{\left (a d^{2} e^{2} m - c d^{4} + 3 \, a d^{2} e^{2} + {\left (c d e^{3} m + 2 \, c d e^{3}\right )} x^{3} + {\left (3 \, c d^{2} e^{2} + 3 \, a e^{4} + {\left (2 \, c d^{2} e^{2} + a e^{4}\right )} m\right )} x^{2} + {\left (6 \, a d e^{3} + {\left (c d^{3} e + 2 \, a d e^{3}\right )} m\right )} x\right )} {\left (e x + d\right )}^{m}}{e^{2} m^{2} + 5 \, e^{2} m + 6 \, e^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 556 vs. \(2 (42) = 84\).
Time = 0.47 (sec) , antiderivative size = 556, normalized size of antiderivative = 10.69 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\begin {cases} \frac {c d^{2} d^{m} x^{2}}{2} & \text {for}\: e = 0 \\- \frac {a e^{2}}{d e^{2} + e^{3} x} + \frac {c d^{2} \log {\left (\frac {d}{e} + x \right )}}{d e^{2} + e^{3} x} + \frac {c d^{2}}{d e^{2} + e^{3} x} + \frac {c d e x \log {\left (\frac {d}{e} + x \right )}}{d e^{2} + e^{3} x} & \text {for}\: m = -3 \\a \log {\left (\frac {d}{e} + x \right )} - \frac {c d^{2} \log {\left (\frac {d}{e} + x \right )}}{e^{2}} + \frac {c d x}{e} & \text {for}\: m = -2 \\\frac {a d^{2} e^{2} m \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {3 a d^{2} e^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {2 a d e^{3} m x \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {6 a d e^{3} x \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {a e^{4} m x^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {3 a e^{4} x^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} - \frac {c d^{4} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {c d^{3} e m x \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {2 c d^{2} e^{2} m x^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {3 c d^{2} e^{2} x^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {c d e^{3} m x^{3} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {2 c d e^{3} x^{3} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (52) = 104\).
Time = 0.20 (sec) , antiderivative size = 174, normalized size of antiderivative = 3.35 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {{\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} a}{m^{2} + 3 \, m + 2} + \frac {{\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} c d^{2}}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left (e x + d\right )}^{m + 1} a d}{m + 1} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} c d}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (52) = 104\).
Time = 0.26 (sec) , antiderivative size = 219, normalized size of antiderivative = 4.21 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {{\left (e x + d\right )}^{m} c d e^{3} m x^{3} + 2 \, {\left (e x + d\right )}^{m} c d^{2} e^{2} m x^{2} + {\left (e x + d\right )}^{m} a e^{4} m x^{2} + 2 \, {\left (e x + d\right )}^{m} c d e^{3} x^{3} + {\left (e x + d\right )}^{m} c d^{3} e m x + 2 \, {\left (e x + d\right )}^{m} a d e^{3} m x + 3 \, {\left (e x + d\right )}^{m} c d^{2} e^{2} x^{2} + 3 \, {\left (e x + d\right )}^{m} a e^{4} x^{2} + {\left (e x + d\right )}^{m} a d^{2} e^{2} m + 6 \, {\left (e x + d\right )}^{m} a d e^{3} x - {\left (e x + d\right )}^{m} c d^{4} + 3 \, {\left (e x + d\right )}^{m} a d^{2} e^{2}}{e^{2} m^{2} + 5 \, e^{2} m + 6 \, e^{2}} \]
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Time = 9.86 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.71 \[ \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx={\left (d+e\,x\right )}^m\,\left (\frac {x^2\,\left (3\,a\,e^2+3\,c\,d^2+a\,e^2\,m+2\,c\,d^2\,m\right )}{m^2+5\,m+6}+\frac {d^2\,\left (3\,a\,e^2-c\,d^2+a\,e^2\,m\right )}{e^2\,\left (m^2+5\,m+6\right )}+\frac {d\,x\,\left (6\,a\,e^2+2\,a\,e^2\,m+c\,d^2\,m\right )}{e\,\left (m^2+5\,m+6\right )}+\frac {c\,d\,e\,x^3\,\left (m+2\right )}{m^2+5\,m+6}\right ) \]
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