Optimal. Leaf size=39 \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{p+1}}{2 c e (p+1)} \]
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Rubi [A] time = 0.0103326, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {629} \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{p+1}}{2 c e (p+1)} \]
Antiderivative was successfully verified.
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Rule 629
Rubi steps
\begin{align*} \int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^p \, dx &=\frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{1+p}}{2 c e (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0143138, size = 28, normalized size = 0.72 \[ \frac{\left (c (d+e x)^2\right )^{p+1}}{2 c e (p+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 40, normalized size = 1. \begin{align*}{\frac{ \left ( ex+d \right ) ^{2} \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{p}}{2\,e \left ( 1+p \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 [A] time = 2.622, size = 101, normalized size = 2.59 \begin{align*} \frac{{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )}{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{p}}{2 \,{\left (e p + e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.472735, size = 139, normalized size = 3.56 \begin{align*} \begin{cases} \frac{x}{c d} & \text{for}\: e = 0 \wedge p = -1 \\d x \left (c d^{2}\right )^{p} & \text{for}\: e = 0 \\\frac{\log{\left (\frac{d}{e} + x \right )}}{c e} & \text{for}\: p = -1 \\\frac{d^{2} \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 e p + 2 e} + \frac{2 d e x \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 e p + 2 e} + \frac{e^{2} x^{2} \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 e p + 2 e} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19729, size = 127, normalized size = 3.26 \begin{align*} \frac{{\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{p} x^{2} e^{2} + 2 \,{\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{p} d x e +{\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{p} d^{2}}{2 \,{\left (p e + e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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