Optimal. Leaf size=57 \[ 2 c f x \left (d+e x+f x^2\right )^{p+1}-\frac{(c e (p+2)-b f (2 p+3)) \left (d+e x+f x^2\right )^{p+1}}{p+1} \]
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Rubi [A] time = 0.121235, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 69, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.029, Rules used = {1661, 629} \[ 2 c f x \left (d+e x+f x^2\right )^{p+1}-\frac{(c e (p+2)-b f (2 p+3)) \left (d+e x+f x^2\right )^{p+1}}{p+1} \]
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+3 b e f-c e^2 p+2 b e f p+2 b f^2 (3+2 p) x+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 (-e f (3+2 p) (c e (2+p)-b f (3+2 p))-2 f^2 (3+2 p) (c e (2+p)-b f (3+2 p)) x\right ) \left (d+e x+f x^2\right )^p \, dx}{f (3+2 p)}\\ &=-\frac{(c e (2+p)-b f (3+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 = 0.300149, size = 43, normalized size = 0.75 \[ \frac{(d+x (e+f x))^{p+1} (b f (2 p+3)-c e (p+2)+2 c f (p+1) x)}{p+1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 51, normalized size = 0.9 \begin{align*}{\frac{ \left ( f{x}^{2}+ex+d \right ) ^{1+p} \left ( 2\,cfxp+2\,bfp-cep+2\,cfx+3\,bf-2\,ce \right ) }{1+p}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14622, size = 132, normalized size = 2.32 \begin{align*} \frac{{\left (2 \, c f^{2}{\left (p + 1\right )} x^{3} + b d f{\left (2 \, p + 3\right )} - c d e{\left (p + 2\right )} +{\left (b f^{2}{\left (2 \, p + 3\right )} + c e f p\right )} x^{2} +{\left (b e f{\left (2 \, p + 3\right )} -{\left (e^{2}{\left (p + 2\right )} - 2 \, d f{\left (p + 1\right )}\right )} c\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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Fricas [B] time = 1.37041, size = 269, normalized size = 4.72 \begin{align*} \frac{{\left (2 \,{\left (c f^{2} p + c f^{2}\right )} x^{3} - 2 \, c d e + 3 \, b d f +{\left (3 \, b f^{2} +{\left (c e f + 2 \, b f^{2}\right )} p\right )} x^{2} -{\left (c d e - 2 \, b d f\right )} p -{\left (2 \, c e^{2} -{\left (2 \, c d + 3 \, b e\right )} f +{\left (c e^{2} - 2 \,{\left (c d + b e\right )} 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 [B] time = 113.673, size = 483, normalized size = 8.47 \begin{align*} \begin{cases} \frac{2 b d f p \left (d + e x + f x^{2}\right )^{p}}{p + 1} + \frac{3 b d f \left (d + e x + f x^{2}\right )^{p}}{p + 1} + \frac{2 b e f p x \left (d + e x + f x^{2}\right )^{p}}{p + 1} + \frac{3 b e f x \left (d + e x + f x^{2}\right )^{p}}{p + 1} + \frac{2 b f^{2} p x^{2} \left (d + e x + f x^{2}\right )^{p}}{p + 1} + \frac{3 b f^{2} x^{2} \left (d + e x + f x^{2}\right )^{p}}{p + 1} - \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 \\b f \log{\left (\frac{e}{2 f} + x - \frac{\sqrt{- 4 d f + e^{2}}}{2 f} \right )} + b f \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 )} - 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 = 1.22389, size = 424, normalized size = 7.44 \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} b f^{2} p x^{2} + 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 + 3 \,{\left (f x^{2} + x e + d\right )}^{p} b f^{2} x^{2} + 2 \,{\left (f x^{2} + x e + d\right )}^{p} b f p x e + 2 \,{\left (f x^{2} + x e + d\right )}^{p} b d f p + 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 + 3 \,{\left (f x^{2} + x e + d\right )}^{p} b f x e + 3 \,{\left (f x^{2} + x e + d\right )}^{p} b d f - 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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