Optimal. Leaf size=22 \[ \frac {\left (-a+b x+c x^2\right )^{p+1}}{p+1} \]
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Rubi [A] time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {629} \[ \frac {\left (-a+b x+c x^2\right )^{p+1}}{p+1} \]
Antiderivative was successfully verified.
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Rule 629
Rubi steps
\begin {align*} \int (b+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 = 21, normalized size = 0.95 \[ \frac {(x (b+c x)-a)^{p+1}}{p+1} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 32, normalized size = 1.45 \[ \frac {{\left (c x^{2} + b x - a\right )} {\left (c x^{2} + b x - a\right )}^{p}}{p + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 22, normalized size = 1.00 \[ \frac {{\left (c x^{2} + b x - a\right )}^{p + 1}}{p + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 23, normalized size = 1.05 \[ \frac {\left (c \,x^{2}+b x -a \right )^{p +1}}{p +1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 22, normalized size = 1.00 \[ \frac {{\left (c x^{2} + b x - a\right )}^{p + 1}}{p + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.05, size = 42, normalized size = 1.91 \[ \left (\frac {b\,x}{p+1}-\frac {a}{p+1}+\frac {c\,x^2}{p+1}\right )\,{\left (c\,x^2+b\,x-a\right )}^p \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 56.66, size = 104, normalized size = 4.73 \[ \begin {cases} - \frac {a \left (- a + b x + c x^{2}\right )^{p}}{p + 1} + \frac {b x \left (- a + b x + c x^{2}\right )^{p}}{p + 1} + \frac {c x^{2} \left (- a + b x + c x^{2}\right )^{p}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (\frac {b}{2 c} + x - \frac {\sqrt {4 a c + b^{2}}}{2 c} \right )} + \log {\left (\frac {b}{2 c} + x + \frac {\sqrt {4 a c + b^{2}}}{2 c} \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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