Optimal. Leaf size=19 \[ \frac{c (d+e x)^{m+3}}{e (m+3)} \]
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Rubi [A] time = 0.0077997, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {27, 12, 32} \[ \frac{c (d+e x)^{m+3}}{e (m+3)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 12
Rule 32
Rubi steps
\begin{align*} \int (d+e x)^m \left (c d^2+2 c d e x+c e^2 x^2\right ) \, dx &=\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*}
Mathematica [A] time = 0.0140483, size = 20, normalized size = 1.05 \[ \frac{c (d+e x)^{m+3}}{e m+3 e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 36, normalized size = 1.9 \begin{align*}{\frac{ \left ( ex+d \right ) ^{1+m}c \left ({e}^{2}{x}^{2}+2\,dex+{d}^{2} \right ) }{e \left ( 3+m \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 = 2.48514, size = 101, normalized size = 5.32 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.70723, size = 116, normalized size = 6.11 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.27151, size = 101, normalized size = 5.32 \begin{align*} \frac{{\left (x e + d\right )}^{m} c x^{3} e^{3} + 3 \,{\left (x e + d\right )}^{m} c d x^{2} e^{2} + 3 \,{\left (x e + d\right )}^{m} c d^{2} x e +{\left (x e + d\right )}^{m} c d^{3}}{m e + 3 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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