Optimal. Leaf size=26 \[ \frac {(1+x)^{1+m} \left (1+2 x+x^2\right )^n}{1+m+2 n} \]
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Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {658, 32}
\begin {gather*} \frac {(x+1)^{m+1} \left (x^2+2 x+1\right )^n}{m+2 n+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 658
Rubi steps
\begin {align*} \int (1+x)^m \left (1+2 x+x^2\right )^n \, dx &=\left ((1+x)^{-2 n} \left (1+2 x+x^2\right )^n\right ) \int (1+x)^{m+2 n} \, dx\\ &=\frac {(1+x)^{1+m} \left (1+2 x+x^2\right )^n}{1+m+2 n}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 23, normalized size = 0.88 \begin {gather*} \frac {(1+x)^{1+m} \left ((1+x)^2\right )^n}{1+m+2 n} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
3.
time = 0.77, size = 18, normalized size = 0.69
method | result | size |
meijerg | \(x \hypergeom \left (\left [1, -m -2 n \right ], \left [2\right ], -x \right )\) | \(18\) |
gosper | \(\frac {\left (x +1\right )^{1+m} \left (x^{2}+2 x +1\right )^{n}}{1+m +2 n}\) | \(27\) |
norman | \(\frac {{\mathrm e}^{m \ln \left (x +1\right )} {\mathrm e}^{n \ln \left (x^{2}+2 x +1\right )}}{1+m +2 n}+\frac {x \,{\mathrm e}^{m \ln \left (x +1\right )} {\mathrm e}^{n \ln \left (x^{2}+2 x +1\right )}}{1+m +2 n}\) | \(59\) |
risch | \(\frac {\left (x +1\right )^{m} \left (x +1\right ) {\mathrm e}^{\frac {n \left (-i \pi \mathrm {csgn}\left (i \left (x +1\right )^{2}\right )^{3}+2 i \pi \mathrm {csgn}\left (i \left (x +1\right )^{2}\right )^{2} \mathrm {csgn}\left (i \left (x +1\right )\right )-i \pi \,\mathrm {csgn}\left (i \left (x +1\right )^{2}\right ) \mathrm {csgn}\left (i \left (x +1\right )\right )^{2}+4 \ln \left (x +1\right )\right )}{2}}}{1+m +2 n}\) | \(88\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 27, normalized size = 1.04 \begin {gather*} \frac {{\left (x + 1\right )} e^{\left (m \log \left (x + 1\right ) + 2 \, n \log \left (x + 1\right )\right )}}{m + 2 \, n + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.89, size = 24, normalized size = 0.92 \begin {gather*} \frac {{\left (x + 1\right )}^{m} {\left (x + 1\right )}^{2 \, n} {\left (x + 1\right )}}{m + 2 \, n + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} \frac {x \left (x + 1\right )^{m} \left (x^{2} + 2 x + 1\right )^{n}}{m + 2 n + 1} + \frac {\left (x + 1\right )^{m} \left (x^{2} + 2 x + 1\right )^{n}}{m + 2 n + 1} & \text {for}\: m \neq - 2 n - 1 \\\int \left (x + 1\right )^{- 2 n - 1} \left (\left (x + 1\right )^{2}\right )^{n}\, dx & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.96, size = 37, normalized size = 1.42 \begin {gather*} \frac {{\left (x + 1\right )}^{m} {\left (x + 1\right )}^{2 \, n} x + {\left (x + 1\right )}^{m} {\left (x + 1\right )}^{2 \, n}}{m + 2 \, n + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.59, size = 26, normalized size = 1.00 \begin {gather*} \frac {{\left (x+1\right )}^{m+1}\,{\left (x^2+2\,x+1\right )}^n}{m+2\,n+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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