Optimal. Leaf size=41 \[ \frac{c (d+e x)^{m+3}}{e^2 (m+3)}-\frac{c d (d+e x)^{m+2}}{e^2 (m+2)} \]
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Rubi [A] time = 0.0221012, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {626, 12, 43} \[ \frac{c (d+e x)^{m+3}}{e^2 (m+3)}-\frac{c d (d+e x)^{m+2}}{e^2 (m+2)} \]
Antiderivative was successfully verified.
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Rule 626
Rule 12
Rule 43
Rubi steps
\begin{align*} \int (d+e x)^m \left (c d x+c e x^2\right ) \, dx &=\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*}
Mathematica [A] time = 0.0199035, size = 34, normalized size = 0.83 \[ \frac{c (d+e x)^{m+2} (e (m+2) x-d)}{e^2 (m+2) (m+3)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 37, normalized size = 0.9 \begin{align*} -{\frac{c \left ( ex+d \right ) ^{2+m} \left ( -mex-2\,ex+d \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 [B] time = 1.23387, size = 154, normalized size = 3.76 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08918, size = 163, normalized size = 3.98 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.50261, size = 299, normalized size = 7.29 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.27748, size = 159, normalized size = 3.88 \begin{align*} \frac{{\left (x e + d\right )}^{m} c m x^{3} e^{3} + 2 \,{\left (x e + d\right )}^{m} c d m x^{2} e^{2} +{\left (x e + d\right )}^{m} c d^{2} m x e + 2 \,{\left (x e + d\right )}^{m} c x^{3} e^{3} + 3 \,{\left (x e + d\right )}^{m} c d x^{2} e^{2} -{\left (x e + d\right )}^{m} c d^{3}}{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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