Optimal. Leaf size=22 \[ \frac{x^{2 (n+1)} (a+b x)^{n+1}}{n+1} \]
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Rubi [A] time = 0.0045588, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {74} \[ \frac{x^{2 (n+1)} (a+b x)^{n+1}}{n+1} \]
Antiderivative was successfully verified.
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Rule 74
Rubi steps
\begin{align*} \int x^{1+2 n} (a+b x)^n (2 a+3 b x) \, dx &=\frac{x^{2 (1+n)} (a+b x)^{1+n}}{1+n}\\ \end{align*}
Mathematica [A] time = 0.0108333, size = 22, normalized size = 1. \[ \frac{x^{2 n+2} (a+b x)^{n+1}}{n+1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 23, normalized size = 1.1 \begin{align*}{\frac{{x}^{2+2\,n} \left ( bx+a \right ) ^{1+n}}{1+n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.99243, size = 43, normalized size = 1.95 \begin{align*} \frac{{\left (b x^{3} + a x^{2}\right )} e^{\left (n \log \left (b x + a\right ) + 2 \, n \log \left (x\right )\right )}}{n + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00643, size = 63, normalized size = 2.86 \begin{align*} \frac{{\left (b x^{2} + a x\right )}{\left (b x + a\right )}^{n} x^{2 \, n + 1}}{n + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.13459, size = 63, normalized size = 2.86 \begin{align*} \frac{{\left (b x + a\right )}^{n} b x^{2} e^{\left (2 \, n \log \left (x\right ) + \log \left (x\right )\right )} +{\left (b x + a\right )}^{n} a x e^{\left (2 \, n \log \left (x\right ) + \log \left (x\right )\right )}}{n + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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