Optimal. Leaf size=52 \[ \frac{c d (d+e x)^{m+3}}{e^2 (m+3)}-\frac{\left (c d^2-a e^2\right ) (d+e x)^{m+2}}{e^2 (m+2)} \]
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Rubi [A] time = 0.025133, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {626, 43} \[ \frac{c d (d+e x)^{m+3}}{e^2 (m+3)}-\frac{\left (c d^2-a e^2\right ) (d+e x)^{m+2}}{e^2 (m+2)} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \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 &=\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*}
Mathematica [A] time = 0.0361307, size = 45, normalized size = 0.87 \[ \frac{(d+e x)^{m+2} \left (a e^2 (m+3)+c d (e (m+2) x-d)\right )}{e^2 (m+2) (m+3)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 55, normalized size = 1.1 \begin{align*}{\frac{ \left ( ex+d \right ) ^{2+m} \left ( cdemx+a{e}^{2}m+2\,cdex+3\,a{e}^{2}-c{d}^{2} \right ) }{{e}^{2} \left ({m}^{2}+5\,m+6 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.93392, size = 277, normalized size = 5.33 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.3552, size = 556, normalized size = 10.69 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.12733, size = 296, normalized size = 5.69 \begin{align*} \frac{{\left (x e + d\right )}^{m} c d m x^{3} e^{3} + 2 \,{\left (x e + d\right )}^{m} c d^{2} m x^{2} e^{2} +{\left (x e + d\right )}^{m} c d^{3} m x e + 2 \,{\left (x e + d\right )}^{m} c d x^{3} e^{3} + 3 \,{\left (x e + d\right )}^{m} c d^{2} x^{2} e^{2} -{\left (x e + d\right )}^{m} c d^{4} +{\left (x e + d\right )}^{m} a m x^{2} e^{4} + 2 \,{\left (x e + d\right )}^{m} a d m x e^{3} +{\left (x e + d\right )}^{m} a d^{2} m e^{2} + 3 \,{\left (x e + d\right )}^{m} a x^{2} e^{4} + 6 \,{\left (x e + d\right )}^{m} a d x e^{3} + 3 \,{\left (x e + d\right )}^{m} a d^{2} e^{2}}{m^{2} e^{2} + 5 \, m e^{2} + 6 \, e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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