Optimal. Leaf size=22 \[ \frac {x^{2 (1+n)} (c+d x)^{1+n}}{1+n} \]
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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 = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {859}
\begin {gather*} \frac {x^{2 (n+1)} (c+d x)^{n+1}}{n+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 859
Rubi steps
\begin {align*} \int x^{2 n} (c+d x)^n \left (2 c x+3 d x^2\right ) \, dx &=\frac {x^{2 (1+n)} (c+d x)^{1+n}}{1+n}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 22, normalized size = 1.00 \begin {gather*} \frac {x^{2+2 n} (c+d x)^{1+n}}{1+n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 23, normalized size = 1.05
method | result | size |
gosper | \(\frac {x^{2+2 n} \left (d x +c \right )^{1+n}}{1+n}\) | \(23\) |
risch | \(\frac {\left (d x +c \right )^{n} x^{2 n} x^{2} \left (d x +c \right )}{1+n}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 32, normalized size = 1.45 \begin {gather*} \frac {{\left (d x^{3} + c x^{2}\right )} e^{\left (n \log \left (d x + c\right ) + 2 \, n \log \left (x\right )\right )}}{n + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 29, normalized size = 1.32 \begin {gather*} \frac {{\left (d x^{3} + c x^{2}\right )} {\left (d x + c\right )}^{n} x^{2 \, n}}{n + 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. 53 vs.
\(2 (17) = 34\).
time = 1.58, size = 53, normalized size = 2.41 \begin {gather*} \begin {cases} \frac {c x^{2} x^{2 n} \left (c + d x\right )^{n}}{n + 1} + \frac {d x^{3} x^{2 n} \left (c + d x\right )^{n}}{n + 1} & \text {for}\: n \neq -1 \\2 \log {\left (x \right )} + \log {\left (\frac {c}{d} + 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 = 4.27, size = 41, normalized size = 1.86 \begin {gather*} \frac {{\left (d x + c\right )}^{n} d x^{3} x^{2 \, n} + {\left (d x + c\right )}^{n} c x^{2} x^{2 \, n}}{n + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.19, size = 26, normalized size = 1.18 \begin {gather*} \frac {x^{2\,n}\,x^2\,{\left (c+d\,x\right )}^n\,\left (c+d\,x\right )}{n+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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