Integrand size = 19, antiderivative size = 41 \[ \int (d+e x)^m \left (c d x+c e x^2\right ) \, dx=-\frac {c d (d+e x)^{2+m}}{e^2 (2+m)}+\frac {c (d+e x)^{3+m}}{e^2 (3+m)} \]
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Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {640, 12, 45} \[ \int (d+e x)^m \left (c d x+c e x^2\right ) \, dx=\frac {c (d+e x)^{m+3}}{e^2 (m+3)}-\frac {c d (d+e x)^{m+2}}{e^2 (m+2)} \]
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Rule 12
Rule 45
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int c x (d+e x)^{1+m} \, dx \\ & = c \int x (d+e x)^{1+m} \, dx \\ & = c \int \left (-\frac {d (d+e x)^{1+m}}{e}+\frac {(d+e x)^{2+m}}{e}\right ) \, dx \\ & = -\frac {c d (d+e x)^{2+m}}{e^2 (2+m)}+\frac {c (d+e x)^{3+m}}{e^2 (3+m)} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.83 \[ \int (d+e x)^m \left (c d x+c e x^2\right ) \, dx=\frac {c (d+e x)^{2+m} (-d+e (2+m) x)}{e^2 (2+m) (3+m)} \]
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Time = 1.91 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.90
method | result | size |
gosper | \(-\frac {c \left (e x +d \right )^{2+m} \left (-m e x -2 e x +d \right )}{e^{2} \left (m^{2}+5 m +6\right )}\) | \(37\) |
risch | \(-\frac {c \left (-e^{3} m \,x^{3}-2 d \,e^{2} m \,x^{2}-2 e^{3} x^{3}-d^{2} e m x -3 d \,e^{2} x^{2}+d^{3}\right ) \left (e x +d \right )^{m}}{\left (2+m \right ) \left (3+m \right ) e^{2}}\) | \(72\) |
norman | \(\frac {c e \,x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{3+m}+\frac {d c \left (3+2 m \right ) x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{m^{2}+5 m +6}+\frac {m \,d^{2} c x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (m^{2}+5 m +6\right )}-\frac {d^{3} c \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e^{2} \left (m^{2}+5 m +6\right )}\) | \(109\) |
parallelrisch | \(\frac {x^{3} \left (e x +d \right )^{m} c d \,e^{3} m +2 x^{3} \left (e x +d \right )^{m} c d \,e^{3}+2 x^{2} \left (e x +d \right )^{m} c \,d^{2} e^{2} m +3 x^{2} \left (e x +d \right )^{m} c \,d^{2} e^{2}+x \left (e x +d \right )^{m} c \,d^{3} e m -c \,d^{4} \left (e x +d \right )^{m}}{\left (3+m \right ) \left (2+m \right ) d \,e^{2}}\) | \(120\) |
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none
Time = 0.27 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.98 \[ \int (d+e x)^m \left (c d x+c e x^2\right ) \, dx=\frac {{\left (c d^{2} e m x - c d^{3} + {\left (c e^{3} m + 2 \, c e^{3}\right )} x^{3} + {\left (2 \, c d e^{2} m + 3 \, c d e^{2}\right )} x^{2}\right )} {\left (e x + d\right )}^{m}}{e^{2} m^{2} + 5 \, e^{2} m + 6 \, e^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (34) = 68\).
Time = 0.41 (sec) , antiderivative size = 299, normalized size of antiderivative = 7.29 \[ \int (d+e x)^m \left (c d x+c e x^2\right ) \, dx=\begin {cases} \frac {c d d^{m} x^{2}}{2} & \text {for}\: e = 0 \\\frac {c d \log {\left (\frac {d}{e} + x \right )}}{d e^{2} + e^{3} x} + \frac {c d}{d e^{2} + e^{3} x} + \frac {c e x \log {\left (\frac {d}{e} + x \right )}}{d e^{2} + e^{3} x} & \text {for}\: m = -3 \\- \frac {c d \log {\left (\frac {d}{e} + x \right )}}{e^{2}} + \frac {c x}{e} & \text {for}\: m = -2 \\- \frac {c d^{3} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {c d^{2} e m x \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {2 c d e^{2} m x^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {3 c d e^{2} x^{2} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {c e^{3} m x^{3} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} + \frac {2 c e^{3} x^{3} \left (d + e x\right )^{m}}{e^{2} m^{2} + 5 e^{2} m + 6 e^{2}} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (41) = 82\).
Time = 0.21 (sec) , antiderivative size = 114, normalized size of antiderivative = 2.78 \[ \int (d+e x)^m \left (c d x+c e x^2\right ) \, dx=\frac {{\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} c d}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} c}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (41) = 82\).
Time = 0.28 (sec) , antiderivative size = 118, normalized size of antiderivative = 2.88 \[ \int (d+e x)^m \left (c d x+c e x^2\right ) \, dx=\frac {{\left (e x + d\right )}^{m} c e^{3} m x^{3} + 2 \, {\left (e x + d\right )}^{m} c d e^{2} m x^{2} + 2 \, {\left (e x + d\right )}^{m} c e^{3} x^{3} + {\left (e x + d\right )}^{m} c d^{2} e m x + 3 \, {\left (e x + d\right )}^{m} c d e^{2} x^{2} - {\left (e x + d\right )}^{m} c d^{3}}{e^{2} m^{2} + 5 \, e^{2} m + 6 \, e^{2}} \]
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Time = 9.54 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.15 \[ \int (d+e x)^m \left (c d x+c e x^2\right ) \, dx={\left (d+e\,x\right )}^m\,\left (\frac {c\,e\,x^3\,\left (m+2\right )}{m^2+5\,m+6}-\frac {c\,d^3}{e^2\,\left (m^2+5\,m+6\right )}+\frac {c\,d\,x^2\,\left (2\,m+3\right )}{m^2+5\,m+6}+\frac {c\,d^2\,m\,x}{e\,\left (m^2+5\,m+6\right )}\right ) \]
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