Integrand size = 21, antiderivative size = 69 \[ \int (d+e x)^m \left (c d x+c e x^2\right )^2 \, dx=\frac {c^2 d^2 (d+e x)^{3+m}}{e^3 (3+m)}-\frac {2 c^2 d (d+e x)^{4+m}}{e^3 (4+m)}+\frac {c^2 (d+e x)^{5+m}}{e^3 (5+m)} \]
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Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {640, 12, 45} \[ \int (d+e x)^m \left (c d x+c e x^2\right )^2 \, dx=\frac {c^2 d^2 (d+e x)^{m+3}}{e^3 (m+3)}-\frac {2 c^2 d (d+e x)^{m+4}}{e^3 (m+4)}+\frac {c^2 (d+e x)^{m+5}}{e^3 (m+5)} \]
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Rule 12
Rule 45
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int c^2 x^2 (d+e x)^{2+m} \, dx \\ & = c^2 \int x^2 (d+e x)^{2+m} \, dx \\ & = c^2 \int \left (\frac {d^2 (d+e x)^{2+m}}{e^2}-\frac {2 d (d+e x)^{3+m}}{e^2}+\frac {(d+e x)^{4+m}}{e^2}\right ) \, dx \\ & = \frac {c^2 d^2 (d+e x)^{3+m}}{e^3 (3+m)}-\frac {2 c^2 d (d+e x)^{4+m}}{e^3 (4+m)}+\frac {c^2 (d+e x)^{5+m}}{e^3 (5+m)} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.87 \[ \int (d+e x)^m \left (c d x+c e x^2\right )^2 \, dx=\frac {c^2 (d+e x)^{3+m} \left (2 d^2-2 d e (3+m) x+e^2 \left (12+7 m+m^2\right ) x^2\right )}{e^3 (3+m) (4+m) (5+m)} \]
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Time = 2.10 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.10
method | result | size |
gosper | \(\frac {c^{2} \left (e x +d \right )^{3+m} \left (e^{2} m^{2} x^{2}+7 e^{2} m \,x^{2}-2 d e m x +12 x^{2} e^{2}-6 d e x +2 d^{2}\right )}{e^{3} \left (m^{3}+12 m^{2}+47 m +60\right )}\) | \(76\) |
risch | \(\frac {c^{2} \left (e^{5} m^{2} x^{5}+3 d \,e^{4} m^{2} x^{4}+7 e^{5} m \,x^{5}+3 d^{2} e^{3} m^{2} x^{3}+19 d \,e^{4} m \,x^{4}+12 e^{5} x^{5}+d^{3} e^{2} m^{2} x^{2}+15 d^{2} e^{3} m \,x^{3}+30 x^{4} d \,e^{4}+d^{3} e^{2} m \,x^{2}+20 x^{3} d^{2} e^{3}-2 d^{4} m x e +2 d^{5}\right ) \left (e x +d \right )^{m}}{\left (4+m \right ) \left (5+m \right ) \left (3+m \right ) e^{3}}\) | \(163\) |
norman | \(\frac {c^{2} e^{2} x^{5} {\mathrm e}^{m \ln \left (e x +d \right )}}{5+m}+\frac {c^{2} d^{2} \left (3 m^{2}+15 m +20\right ) x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{m^{3}+12 m^{2}+47 m +60}+\frac {\left (3 m +10\right ) c^{2} d e \,x^{4} {\mathrm e}^{m \ln \left (e x +d \right )}}{m^{2}+9 m +20}+\frac {m \left (1+m \right ) c^{2} d^{3} x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (m^{3}+12 m^{2}+47 m +60\right )}+\frac {2 c^{2} d^{5} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{3} \left (m^{3}+12 m^{2}+47 m +60\right )}-\frac {2 m \,c^{2} d^{4} x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e^{2} \left (m^{3}+12 m^{2}+47 m +60\right )}\) | \(216\) |
parallelrisch | \(\frac {x^{5} \left (e x +d \right )^{m} c^{2} d \,e^{5} m^{2}+7 x^{5} \left (e x +d \right )^{m} c^{2} d \,e^{5} m +3 x^{4} \left (e x +d \right )^{m} c^{2} d^{2} e^{4} m^{2}+12 x^{5} \left (e x +d \right )^{m} c^{2} d \,e^{5}+19 x^{4} \left (e x +d \right )^{m} c^{2} d^{2} e^{4} m +3 x^{3} \left (e x +d \right )^{m} c^{2} d^{3} e^{3} m^{2}+30 x^{4} \left (e x +d \right )^{m} c^{2} d^{2} e^{4}+15 x^{3} \left (e x +d \right )^{m} c^{2} d^{3} e^{3} m +x^{2} \left (e x +d \right )^{m} c^{2} d^{4} e^{2} m^{2}+20 x^{3} \left (e x +d \right )^{m} c^{2} d^{3} e^{3}+x^{2} \left (e x +d \right )^{m} c^{2} d^{4} e^{2} m -2 x \left (e x +d \right )^{m} c^{2} d^{5} e m +2 c^{2} d^{6} \left (e x +d \right )^{m}}{\left (5+m \right ) \left (4+m \right ) d \left (3+m \right ) e^{3}}\) | \(295\) |
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Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (69) = 138\).
Time = 0.46 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.91 \[ \int (d+e x)^m \left (c d x+c e x^2\right )^2 \, dx=-\frac {{\left (2 \, c^{2} d^{4} e m x - 2 \, c^{2} d^{5} - {\left (c^{2} e^{5} m^{2} + 7 \, c^{2} e^{5} m + 12 \, c^{2} e^{5}\right )} x^{5} - {\left (3 \, c^{2} d e^{4} m^{2} + 19 \, c^{2} d e^{4} m + 30 \, c^{2} d e^{4}\right )} x^{4} - {\left (3 \, c^{2} d^{2} e^{3} m^{2} + 15 \, c^{2} d^{2} e^{3} m + 20 \, c^{2} d^{2} e^{3}\right )} x^{3} - {\left (c^{2} d^{3} e^{2} m^{2} + c^{2} d^{3} e^{2} m\right )} x^{2}\right )} {\left (e x + d\right )}^{m}}{e^{3} m^{3} + 12 \, e^{3} m^{2} + 47 \, e^{3} m + 60 \, e^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 993 vs. \(2 (61) = 122\).
Time = 0.82 (sec) , antiderivative size = 993, normalized size of antiderivative = 14.39 \[ \int (d+e x)^m \left (c d x+c e x^2\right )^2 \, dx=\begin {cases} \frac {c^{2} d^{2} d^{m} x^{3}}{3} & \text {for}\: e = 0 \\\frac {2 c^{2} d^{2} \log {\left (\frac {d}{e} + x \right )}}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} + \frac {3 c^{2} d^{2}}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} + \frac {4 c^{2} d e x \log {\left (\frac {d}{e} + x \right )}}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} + \frac {4 c^{2} d e x}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} + \frac {2 c^{2} e^{2} x^{2} \log {\left (\frac {d}{e} + x \right )}}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} & \text {for}\: m = -5 \\- \frac {2 c^{2} d^{2} \log {\left (\frac {d}{e} + x \right )}}{d e^{3} + e^{4} x} - \frac {4 c^{2} d^{2}}{d e^{3} + e^{4} x} - \frac {2 c^{2} d e x \log {\left (\frac {d}{e} + x \right )}}{d e^{3} + e^{4} x} - \frac {2 c^{2} d e x}{d e^{3} + e^{4} x} + \frac {c^{2} e^{2} x^{2}}{d e^{3} + e^{4} x} & \text {for}\: m = -4 \\\frac {c^{2} d^{2} \log {\left (\frac {d}{e} + x \right )}}{e^{3}} - \frac {c^{2} d x}{e^{2}} + \frac {c^{2} x^{2}}{2 e} & \text {for}\: m = -3 \\\frac {2 c^{2} d^{5} \left (d + e x\right )^{m}}{e^{3} m^{3} + 12 e^{3} m^{2} + 47 e^{3} m + 60 e^{3}} - \frac {2 c^{2} d^{4} e m x \left (d + e x\right )^{m}}{e^{3} m^{3} + 12 e^{3} m^{2} + 47 e^{3} m + 60 e^{3}} + \frac {c^{2} d^{3} e^{2} m^{2} x^{2} \left (d + e x\right )^{m}}{e^{3} m^{3} + 12 e^{3} m^{2} + 47 e^{3} m + 60 e^{3}} + \frac {c^{2} d^{3} e^{2} m x^{2} \left (d + e x\right )^{m}}{e^{3} m^{3} + 12 e^{3} m^{2} + 47 e^{3} m + 60 e^{3}} + \frac {3 c^{2} d^{2} e^{3} m^{2} x^{3} \left (d + e x\right )^{m}}{e^{3} m^{3} + 12 e^{3} m^{2} + 47 e^{3} m + 60 e^{3}} + \frac {15 c^{2} d^{2} e^{3} m x^{3} \left (d + e x\right )^{m}}{e^{3} m^{3} + 12 e^{3} m^{2} + 47 e^{3} m + 60 e^{3}} + \frac {20 c^{2} d^{2} e^{3} x^{3} \left (d + e x\right )^{m}}{e^{3} m^{3} + 12 e^{3} m^{2} + 47 e^{3} m + 60 e^{3}} + \frac {3 c^{2} d e^{4} m^{2} x^{4} \left (d + e x\right )^{m}}{e^{3} m^{3} + 12 e^{3} m^{2} + 47 e^{3} m + 60 e^{3}} + \frac {19 c^{2} d e^{4} m x^{4} \left (d + e x\right )^{m}}{e^{3} m^{3} + 12 e^{3} m^{2} + 47 e^{3} m + 60 e^{3}} + \frac {30 c^{2} d e^{4} x^{4} \left (d + e x\right )^{m}}{e^{3} m^{3} + 12 e^{3} m^{2} + 47 e^{3} m + 60 e^{3}} + \frac {c^{2} e^{5} m^{2} x^{5} \left (d + e x\right )^{m}}{e^{3} m^{3} + 12 e^{3} m^{2} + 47 e^{3} m + 60 e^{3}} + \frac {7 c^{2} e^{5} m x^{5} \left (d + e x\right )^{m}}{e^{3} m^{3} + 12 e^{3} m^{2} + 47 e^{3} m + 60 e^{3}} + \frac {12 c^{2} e^{5} x^{5} \left (d + e x\right )^{m}}{e^{3} m^{3} + 12 e^{3} m^{2} + 47 e^{3} m + 60 e^{3}} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 323 vs. \(2 (69) = 138\).
Time = 0.21 (sec) , antiderivative size = 323, normalized size of antiderivative = 4.68 \[ \int (d+e x)^m \left (c d x+c e x^2\right )^2 \, dx=\frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} c^{2} d^{2}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {2 \, {\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} c^{2} d}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{3}} + \frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{5} x^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d e^{4} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} e^{3} x^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} e^{2} x^{2} - 24 \, d^{4} e m x + 24 \, d^{5}\right )} {\left (e x + d\right )}^{m} c^{2}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 293 vs. \(2 (69) = 138\).
Time = 0.27 (sec) , antiderivative size = 293, normalized size of antiderivative = 4.25 \[ \int (d+e x)^m \left (c d x+c e x^2\right )^2 \, dx=\frac {{\left (e x + d\right )}^{m} c^{2} e^{5} m^{2} x^{5} + 3 \, {\left (e x + d\right )}^{m} c^{2} d e^{4} m^{2} x^{4} + 7 \, {\left (e x + d\right )}^{m} c^{2} e^{5} m x^{5} + 3 \, {\left (e x + d\right )}^{m} c^{2} d^{2} e^{3} m^{2} x^{3} + 19 \, {\left (e x + d\right )}^{m} c^{2} d e^{4} m x^{4} + 12 \, {\left (e x + d\right )}^{m} c^{2} e^{5} x^{5} + {\left (e x + d\right )}^{m} c^{2} d^{3} e^{2} m^{2} x^{2} + 15 \, {\left (e x + d\right )}^{m} c^{2} d^{2} e^{3} m x^{3} + 30 \, {\left (e x + d\right )}^{m} c^{2} d e^{4} x^{4} + {\left (e x + d\right )}^{m} c^{2} d^{3} e^{2} m x^{2} + 20 \, {\left (e x + d\right )}^{m} c^{2} d^{2} e^{3} x^{3} - 2 \, {\left (e x + d\right )}^{m} c^{2} d^{4} e m x + 2 \, {\left (e x + d\right )}^{m} c^{2} d^{5}}{e^{3} m^{3} + 12 \, e^{3} m^{2} + 47 \, e^{3} m + 60 \, e^{3}} \]
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Time = 9.75 (sec) , antiderivative size = 197, normalized size of antiderivative = 2.86 \[ \int (d+e x)^m \left (c d x+c e x^2\right )^2 \, dx={\left (d+e\,x\right )}^m\,\left (\frac {2\,c^2\,d^5}{e^3\,\left (m^3+12\,m^2+47\,m+60\right )}+\frac {c^2\,e^2\,x^5\,\left (m^2+7\,m+12\right )}{m^3+12\,m^2+47\,m+60}+\frac {c^2\,d^2\,x^3\,\left (3\,m^2+15\,m+20\right )}{m^3+12\,m^2+47\,m+60}-\frac {2\,c^2\,d^4\,m\,x}{e^2\,\left (m^3+12\,m^2+47\,m+60\right )}+\frac {c^2\,d\,e\,x^4\,\left (3\,m^2+19\,m+30\right )}{m^3+12\,m^2+47\,m+60}+\frac {c^2\,d^3\,m\,x^2\,\left (m+1\right )}{e\,\left (m^3+12\,m^2+47\,m+60\right )}\right ) \]
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