Optimal. Leaf size=22 \[ \frac {\left (-a+b x+c x^2\right )^{1+p}}{1+p} \]
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Rubi [A]
time = 0.00, 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 = {643}
\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 643
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 = 0.05, size = 21, normalized size = 0.95 \begin {gather*} \frac {(-a+x (b+c x))^{1+p}}{1+p} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.27, size = 23, normalized size = 1.05
method | result | size |
gosper | \(\frac {\left (c \,x^{2}+b x -a \right )^{1+p}}{1+p}\) | \(23\) |
derivativedivides | \(\frac {\left (c \,x^{2}+b x -a \right )^{1+p}}{1+p}\) | \(23\) |
default | \(\frac {\left (c \,x^{2}+b x -a \right )^{1+p}}{1+p}\) | \(23\) |
risch | \(-\frac {\left (-c \,x^{2}-b x +a \right ) \left (c \,x^{2}+b x -a \right )^{p}}{1+p}\) | \(34\) |
norman | \(\frac {b x \,{\mathrm e}^{p \ln \left (c \,x^{2}+b x -a \right )}}{1+p}+\frac {c \,x^{2} {\mathrm e}^{p \ln \left (c \,x^{2}+b x -a \right )}}{1+p}-\frac {a \,{\mathrm e}^{p \ln \left (c \,x^{2}+b x -a \right )}}{1+p}\) | \(76\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 22, 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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Fricas [A]
time = 0.37, size = 32, normalized size = 1.45 \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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 104 vs.
\(2 (15) = 30\).
time = 34.76, size = 104, normalized size = 4.73 \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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Giac [A]
time = 4.52, size = 22, 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 = 2.05, size = 42, normalized size = 1.91 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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