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 = 0.07, antiderivative size = 46, normalized size of antiderivative = 1.00, 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 629
Rule 1661
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*}
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Mathematica [A] time = 0.13, 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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fricas [A] time = 0.63, size = 83, normalized size = 1.80 \[ \frac {{\left (c e f p x^{2} - c d e p + 2 \, {\left (c f^{2} p + c f^{2}\right )} x^{3} - 2 \, c d e - {\left (2 \, c e^{2} - 2 \, c d f + {\left (c e^{2} - 2 \, c d f\right )} p\right )} x\right )} {\left (f x^{2} + e x + d\right )}^{p}}{p + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 191, normalized size = 4.15 \[ \frac {2 \, {\left (f x^{2} + x e + d\right )}^{p} c f^{2} p x^{3} + 2 \, {\left (f x^{2} + x e + d\right )}^{p} c f^{2} x^{3} + {\left (f x^{2} + x e + d\right )}^{p} c f p x^{2} e + 2 \, {\left (f x^{2} + x e + d\right )}^{p} c d f p x + 2 \, {\left (f x^{2} + x e + d\right )}^{p} c d f x - {\left (f x^{2} + x e + d\right )}^{p} c p x e^{2} - {\left (f x^{2} + x e + d\right )}^{p} c d p e - 2 \, {\left (f x^{2} + x e + d\right )}^{p} c x e^{2} - 2 \, {\left (f x^{2} + x e + d\right )}^{p} c d e}{p + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 39, normalized size = 0.85 \[ -\frac {\left (-2 f p x +e p -2 f x +2 e \right ) c \left (f \,x^{2}+e x +d \right )^{p +1}}{p +1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 66, normalized size = 1.43 \[ \frac {{\left (2 \, c f^{2} {\left (p + 1\right )} x^{3} + c e f p x^{2} - c d e {\left (p + 2\right )} - {\left (e^{2} {\left (p + 2\right )} - 2 \, d f {\left (p + 1\right )}\right )} c x\right )} {\left (f x^{2} + e x + d\right )}^{p}}{p + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.39, size = 78, normalized size = 1.70 \[ {\left (f\,x^2+e\,x+d\right )}^p\,\left (2\,c\,f^2\,x^3+\frac {c\,x\,\left (2\,d\,f-e^2\,p-2\,e^2+2\,d\,f\,p\right )}{p+1}-\frac {c\,d\,e\,\left (p+2\right )}{p+1}+\frac {c\,e\,f\,p\,x^2}{p+1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 173.95, size = 280, normalized size = 6.09 \[ \begin {cases} - \frac {c d e p \left (d + e x + f x^{2}\right )^{p}}{p + 1} - \frac {2 c d e \left (d + e x + f x^{2}\right )^{p}}{p + 1} + \frac {2 c d f p x \left (d + e x + f x^{2}\right )^{p}}{p + 1} + \frac {2 c d f x \left (d + e x + f x^{2}\right )^{p}}{p + 1} - \frac {c e^{2} p x \left (d + e x + f x^{2}\right )^{p}}{p + 1} - \frac {2 c e^{2} x \left (d + e x + f x^{2}\right )^{p}}{p + 1} + \frac {c e f p x^{2} \left (d + e x + f x^{2}\right )^{p}}{p + 1} + \frac {2 c f^{2} p x^{3} \left (d + e x + f x^{2}\right )^{p}}{p + 1} + \frac {2 c f^{2} x^{3} \left (d + e x + f x^{2}\right )^{p}}{p + 1} & \text {for}\: p \neq -1 \\- c e \log {\left (\frac {e}{2 f} + x - \frac {\sqrt {- 4 d f + e^{2}}}{2 f} \right )} - c e \log {\left (\frac {e}{2 f} + x + \frac {\sqrt {- 4 d f + e^{2}}}{2 f} \right )} + 2 c f x & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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