Optimal. Leaf size=19 \[ \frac {\left (b x+c x^2\right )^{1+p}}{1+p} \]
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Rubi [A]
time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {643}
\begin {gather*} \frac {\left (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 (b x+c x^2\right )^p \, dx &=\frac {\left (b x+c x^2\right )^{1+p}}{1+p}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 17, normalized size = 0.89 \begin {gather*} \frac {(x (b+c x))^{1+p}}{1+p} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 20, normalized size = 1.05
method | result | size |
derivativedivides | \(\frac {\left (c \,x^{2}+b x \right )^{1+p}}{1+p}\) | \(20\) |
default | \(\frac {\left (c \,x^{2}+b x \right )^{1+p}}{1+p}\) | \(20\) |
risch | \(\frac {x \left (c x +b \right ) \left (x \left (c x +b \right )\right )^{p}}{1+p}\) | \(22\) |
gosper | \(\frac {x \left (c x +b \right ) \left (c \,x^{2}+b x \right )^{p}}{1+p}\) | \(24\) |
norman | \(\frac {b x \,{\mathrm e}^{p \ln \left (c \,x^{2}+b x \right )}}{1+p}+\frac {c \,x^{2} {\mathrm e}^{p \ln \left (c \,x^{2}+b x \right )}}{1+p}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 19, normalized size = 1.00 \begin {gather*} \frac {{\left (c x^{2} + b x\right )}^{p + 1}}{p + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 26, normalized size = 1.37 \begin {gather*} \frac {{\left (c x^{2} + b x\right )} {\left (c x^{2} + b x\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. 46 vs.
\(2 (14) = 28\).
time = 0.25, size = 46, normalized size = 2.42 \begin {gather*} \begin {cases} \frac {b x \left (b x + c x^{2}\right )^{p}}{p + 1} + \frac {c x^{2} \left (b x + c x^{2}\right )^{p}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (x \right )} + \log {\left (\frac {b}{c} + x \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.97, size = 19, normalized size = 1.00 \begin {gather*} \frac {{\left (c x^{2} + b x\right )}^{p + 1}}{p + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.03, size = 23, normalized size = 1.21 \begin {gather*} \frac {x\,{\left (c\,x^2+b\,x\right )}^p\,\left (b+c\,x\right )}{p+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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