Optimal. Leaf size=20 \[ \frac {\left (a+b x+c x^2\right )^{p+1}}{p+1} \]
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Rubi [A] time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {629} \begin {gather*} \frac {\left (a+b x+c x^2\right )^{p+1}}{p+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 629
Rubi steps
\begin {align*} \int (b+2 c x) \left (a+b x+c x^2\right )^p \, dx &=\frac {\left (a+b x+c x^2\right )^{1+p}}{1+p}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 19, normalized size = 0.95 \begin {gather*} \frac {(a+x (b+c x))^{p+1}}{p+1} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.03, size = 0, normalized size = 0.00 \begin {gather*} \int (b+2 c x) \left (a+b x+c x^2\right )^p \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.79, size = 28, normalized size = 1.40 \begin {gather*} \frac {{\left (c x^{2} + b x + a\right )} {\left (c x^{2} + b x + a\right )}^{p}}{p + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 20, normalized size = 1.00 \begin {gather*} \frac {{\left (c x^{2} + b x + a\right )}^{p + 1}}{p + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 21, normalized size = 1.05 \begin {gather*} \frac {\left (c \,x^{2}+b x +a \right )^{p +1}}{p +1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 20, normalized size = 1.00 \begin {gather*} \frac {{\left (c x^{2} + b x + a\right )}^{p + 1}}{p + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.04, size = 39, normalized size = 1.95 \begin {gather*} \left (\frac {a}{p+1}+\frac {b\,x}{p+1}+\frac {c\,x^2}{p+1}\right )\,{\left (c\,x^2+b\,x+a\right )}^p \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 57.11, size = 104, normalized size = 5.20 \begin {gather*} \begin {cases} \frac {a \left (a + b x + c x^{2}\right )^{p}}{p + 1} + \frac {b x \left (a + b x + c x^{2}\right )^{p}}{p + 1} + \frac {c x^{2} \left (a + b x + c x^{2}\right )^{p}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (\frac {b}{2 c} + x - \frac {\sqrt {- 4 a c + b^{2}}}{2 c} \right )} + \log {\left (\frac {b}{2 c} + x + \frac {\sqrt {- 4 a c + b^{2}}}{2 c} \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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