Optimal. Leaf size=21 \[ \frac {d \left (a+b x+c x^2\right )^{p+1}}{p+1} \]
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Rubi [A] time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {629} \[ \frac {d \left (a+b x+c x^2\right )^{p+1}}{p+1} \]
Antiderivative was successfully verified.
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Rule 629
Rubi steps
\begin {align*} \int (b d+2 c d x) \left (a+b x+c x^2\right )^p \, dx &=\frac {d \left (a+b x+c x^2\right )^{1+p}}{1+p}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 20, normalized size = 0.95 \[ \frac {d (a+x (b+c x))^{p+1}}{p+1} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 32, normalized size = 1.52 \[ \frac {{\left (c d x^{2} + b d x + a d\right )} {\left (c x^{2} + b x + a\right )}^{p}}{p + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 21, normalized size = 1.00 \[ \frac {{\left (c x^{2} + b x + a\right )}^{p + 1} d}{p + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 22, normalized size = 1.05 \[ \frac {d \left (c \,x^{2}+b x +a \right )^{p +1}}{p +1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 21, normalized size = 1.00 \[ \frac {{\left (c x^{2} + b x + a\right )}^{p + 1} d}{p + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.52, size = 42, normalized size = 2.00 \[ {\left (c\,x^2+b\,x+a\right )}^p\,\left (\frac {a\,d}{p+1}+\frac {c\,d\,x^2}{p+1}+\frac {b\,d\,x}{p+1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 58.41, size = 112, normalized size = 5.33 \[ \begin {cases} \frac {a d \left (a + b x + c x^{2}\right )^{p}}{p + 1} + \frac {b d x \left (a + b x + c x^{2}\right )^{p}}{p + 1} + \frac {c d x^{2} \left (a + b x + c x^{2}\right )^{p}}{p + 1} & \text {for}\: p \neq -1 \\d \log {\left (\frac {b}{2 c} + x - \frac {\sqrt {- 4 a c + b^{2}}}{2 c} \right )} + d \log {\left (\frac {b}{2 c} + x + \frac {\sqrt {- 4 a c + b^{2}}}{2 c} \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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