Optimal. Leaf size=24 \[ \frac {x^{n+1} \left (c x+d x^2\right )^{n+1}}{n+1} \]
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Rubi [A] time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1584, 763} \[ \frac {x^{n+1} \left (c x+d x^2\right )^{n+1}}{n+1} \]
Antiderivative was successfully verified.
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Rule 763
Rule 1584
Rubi steps
\begin {align*} \int x^n \left (c x+d x^2\right )^n \left (2 c x+3 d x^2\right ) \, dx &=\int x^{1+n} (2 c+3 d x) \left (c x+d x^2\right )^n \, dx\\ &=\frac {x^{1+n} \left (c x+d x^2\right )^{1+n}}{1+n}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 0.92 \[ \frac {x^{n+1} (x (c+d x))^{n+1}}{n+1} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 31, normalized size = 1.29 \[ \frac {{\left (d x^{3} + c x^{2}\right )} {\left (d x^{2} + c x\right )}^{n} x^{n}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.46, size = 51, normalized size = 2.12 \[ \frac {d x^{3} x^{n} e^{\left (n \log \left (d x + c\right ) + n \log \relax (x)\right )} + c x^{2} x^{n} e^{\left (n \log \left (d x + c\right ) + n \log \relax (x)\right )}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 28, normalized size = 1.17 \[ \frac {\left (d x +c \right ) x^{n +2} \left (d \,x^{2}+c x \right )^{n}}{n +1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.83, size = 32, normalized size = 1.33 \[ \frac {{\left (d x^{3} + c x^{2}\right )} e^{\left (n \log \left (d x + c\right ) + 2 \, n \log \relax (x)\right )}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.20, size = 28, normalized size = 1.17 \[ \frac {x^n\,x^2\,{\left (d\,x^2+c\,x\right )}^n\,\left (c+d\,x\right )}{n+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.17, size = 56, normalized size = 2.33 \[ \begin {cases} \frac {c x^{2} x^{n} \left (c x + d x^{2}\right )^{n}}{n + 1} + \frac {d x^{3} x^{n} \left (c x + d x^{2}\right )^{n}}{n + 1} & \text {for}\: n \neq -1 \\2 \log {\relax (x )} + \log {\left (\frac {c}{d} + x \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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