Optimal. Leaf size=46 \[ 2 c f x \left (d+e x+f x^2\right )^{p+1}-\frac{c e (p+2) \left (d+e x+f x^2\right )^{p+1}}{p+1} \]
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Rubi [A] time = 0.0721821, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {1661, 629} \[ 2 c f x \left (d+e x+f x^2\right )^{p+1}-\frac{c e (p+2) \left (d+e x+f x^2\right )^{p+1}}{p+1} \]
Antiderivative was successfully verified.
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Rule 1661
Rule 629
Rubi steps
\begin{align*} \int \left (d+e x+f x^2\right )^p \left (-2 c e^2+2 c d f-c e^2 p+2 c f^2 (3+2 p) x^2\right ) \, dx &=2 c f x \left (d+e x+f x^2\right )^{1+p}+\frac{\int \left (-c e^2 f (2+p) (3+2 p)-2 c e f^2 (2+p) (3+2 p) x\right ) \left (d+e x+f x^2\right )^p \, dx}{f (3+2 p)}\\ &=-\frac{c e (2+p) \left (d+e x+f x^2\right )^{1+p}}{1+p}+2 c f x \left (d+e x+f x^2\right )^{1+p}\\ \end{align*}
Mathematica [A] time = 0.124169, size = 34, normalized size = 0.74 \[ \frac{c (2 f (p+1) x-e (p+2)) (d+x (e+f x))^{p+1}}{p+1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 39, normalized size = 0.9 \begin{align*} -{\frac{c \left ( f{x}^{2}+ex+d \right ) ^{1+p} \left ( -2\,fpx+ep-2\,fx+2\,e \right ) }{1+p}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11649, size = 89, normalized size = 1.93 \begin{align*} \frac{{\left (2 \, c f^{2}{\left (p + 1\right )} x^{3} + c e f p x^{2} - c d e{\left (p + 2\right )} -{\left (e^{2}{\left (p + 2\right )} - 2 \, d f{\left (p + 1\right )}\right )} c x\right )}{\left (f x^{2} + e x + d\right )}^{p}}{p + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37539, size = 182, normalized size = 3.96 \begin{align*} \frac{{\left (c e f p x^{2} - c d e p + 2 \,{\left (c f^{2} p + c f^{2}\right )} x^{3} - 2 \, c d e -{\left (2 \, c e^{2} - 2 \, c d f +{\left (c e^{2} - 2 \, c d f\right )} p\right )} x\right )}{\left (f x^{2} + e x + d\right )}^{p}}{p + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 95.9917, size = 280, normalized size = 6.09 \begin{align*} \begin{cases} - \frac{c d e p \left (d + e x + f x^{2}\right )^{p}}{p + 1} - \frac{2 c d e \left (d + e x + f x^{2}\right )^{p}}{p + 1} + \frac{2 c d f p x \left (d + e x + f x^{2}\right )^{p}}{p + 1} + \frac{2 c d f x \left (d + e x + f x^{2}\right )^{p}}{p + 1} - \frac{c e^{2} p x \left (d + e x + f x^{2}\right )^{p}}{p + 1} - \frac{2 c e^{2} x \left (d + e x + f x^{2}\right )^{p}}{p + 1} + \frac{c e f p x^{2} \left (d + e x + f x^{2}\right )^{p}}{p + 1} + \frac{2 c f^{2} p x^{3} \left (d + e x + f x^{2}\right )^{p}}{p + 1} + \frac{2 c f^{2} x^{3} \left (d + e x + f x^{2}\right )^{p}}{p + 1} & \text{for}\: p \neq -1 \\- c e \log{\left (\frac{e}{2 f} + x - \frac{\sqrt{- 4 d f + e^{2}}}{2 f} \right )} - c e \log{\left (\frac{e}{2 f} + x + \frac{\sqrt{- 4 d f + e^{2}}}{2 f} \right )} + 2 c f x & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17813, size = 258, normalized size = 5.61 \begin{align*} \frac{2 \,{\left (f x^{2} + x e + d\right )}^{p} c f^{2} p x^{3} + 2 \,{\left (f x^{2} + x e + d\right )}^{p} c f^{2} x^{3} +{\left (f x^{2} + x e + d\right )}^{p} c f p x^{2} e + 2 \,{\left (f x^{2} + x e + d\right )}^{p} c d f p x + 2 \,{\left (f x^{2} + x e + d\right )}^{p} c d f x -{\left (f x^{2} + x e + d\right )}^{p} c p x e^{2} -{\left (f x^{2} + x e + d\right )}^{p} c d p e - 2 \,{\left (f x^{2} + x e + d\right )}^{p} c x e^{2} - 2 \,{\left (f x^{2} + x e + d\right )}^{p} c d e}{p + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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