Integrand size = 28, antiderivative size = 19 \[ \int (d+e x)^m \left (c d^2+2 c d e x+c e^2 x^2\right ) \, dx=\frac {c (d+e x)^{3+m}}{e (3+m)} \]
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Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {27, 12, 32} \[ \int (d+e x)^m \left (c d^2+2 c d e x+c e^2 x^2\right ) \, dx=\frac {c (d+e x)^{m+3}}{e (m+3)} \]
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Rule 12
Rule 27
Rule 32
Rubi steps \begin{align*} \text {integral}& = \int c (d+e x)^{2+m} \, dx \\ & = c \int (d+e x)^{2+m} \, dx \\ & = \frac {c (d+e x)^{3+m}}{e (3+m)} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int (d+e x)^m \left (c d^2+2 c d e x+c e^2 x^2\right ) \, dx=\frac {c (d+e x)^{3+m}}{e (3+m)} \]
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Time = 2.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.89
method | result | size |
gosper | \(\frac {c \left (e x +d \right )^{1+m} \left (x^{2} e^{2}+2 d e x +d^{2}\right )}{e \left (3+m \right )}\) | \(36\) |
risch | \(\frac {c \left (e^{3} x^{3}+3 d \,e^{2} x^{2}+3 d^{2} e x +d^{3}\right ) \left (e x +d \right )^{m}}{e \left (3+m \right )}\) | \(45\) |
parallelrisch | \(\frac {x^{3} \left (e x +d \right )^{m} c \,e^{3}+3 x^{2} \left (e x +d \right )^{m} c d \,e^{2}+3 x \left (e x +d \right )^{m} c \,d^{2} e +\left (e x +d \right )^{m} c \,d^{3}}{e \left (3+m \right )}\) | \(70\) |
norman | \(\frac {d^{3} c \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (3+m \right )}+\frac {e^{2} c \,x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{3+m}+\frac {3 d^{2} c x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{3+m}+\frac {3 c d e \,x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{3+m}\) | \(89\) |
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Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (19) = 38\).
Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.58 \[ \int (d+e x)^m \left (c d^2+2 c d e x+c e^2 x^2\right ) \, dx=\frac {{\left (c e^{3} x^{3} + 3 \, c d e^{2} x^{2} + 3 \, c d^{2} e x + c d^{3}\right )} {\left (e x + d\right )}^{m}}{e m + 3 \, e} \]
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Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (14) = 28\).
Time = 0.33 (sec) , antiderivative size = 116, normalized size of antiderivative = 6.11 \[ \int (d+e x)^m \left (c d^2+2 c d e x+c e^2 x^2\right ) \, dx=\begin {cases} \frac {c x}{d} & \text {for}\: e = 0 \wedge m = -3 \\c d^{2} d^{m} x & \text {for}\: e = 0 \\\frac {c \log {\left (\frac {d}{e} + x \right )}}{e} & \text {for}\: m = -3 \\\frac {c d^{3} \left (d + e x\right )^{m}}{e m + 3 e} + \frac {3 c d^{2} e x \left (d + e x\right )^{m}}{e m + 3 e} + \frac {3 c d e^{2} x^{2} \left (d + e x\right )^{m}}{e m + 3 e} + \frac {c e^{3} x^{3} \left (d + e x\right )^{m}}{e m + 3 e} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (19) = 38\).
Time = 0.21 (sec) , antiderivative size = 137, normalized size of antiderivative = 7.21 \[ \int (d+e x)^m \left (c d^2+2 c d e x+c e^2 x^2\right ) \, dx=\frac {2 \, {\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} c d}{{\left (m^{2} + 3 \, m + 2\right )} e} + \frac {{\left (e x + d\right )}^{m + 1} c d^{2}}{e {\left (m + 1\right )}} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} c}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e} \]
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Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (19) = 38\).
Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 3.68 \[ \int (d+e x)^m \left (c d^2+2 c d e x+c e^2 x^2\right ) \, dx=\frac {{\left (e x + d\right )}^{m} c e^{3} x^{3} + 3 \, {\left (e x + d\right )}^{m} c d e^{2} x^{2} + 3 \, {\left (e x + d\right )}^{m} c d^{2} e x + {\left (e x + d\right )}^{m} c d^{3}}{e m + 3 \, e} \]
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Time = 9.67 (sec) , antiderivative size = 60, normalized size of antiderivative = 3.16 \[ \int (d+e x)^m \left (c d^2+2 c d e x+c e^2 x^2\right ) \, dx={\left (d+e\,x\right )}^m\,\left (\frac {3\,c\,d^2\,x}{m+3}+\frac {c\,d^3}{e\,\left (m+3\right )}+\frac {c\,e^2\,x^3}{m+3}+\frac {3\,c\,d\,e\,x^2}{m+3}\right ) \]
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