Optimal. Leaf size=22 \[ \frac {x^{2 (n+1)} (a+b x)^{n+1}}{n+1} \]
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Rubi [A] time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 1.00 \[ \frac {x^{2 n+2} (a+b x)^{n+1}}{n+1} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 29, normalized size = 1.32 \[ \frac {{\left (b x^{2} + a x\right )} {\left (b x + a\right )}^{n} x^{2 \, n + 1}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.19, size = 47, normalized size = 2.14 \[ \frac {{\left (b x + a\right )}^{n} b x^{2} e^{\left (2 \, n \log \relax (x) + \log \relax (x)\right )} + {\left (b x + a\right )}^{n} a x e^{\left (2 \, n \log \relax (x) + \log \relax (x)\right )}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 23, normalized size = 1.05 \[ \frac {x^{2 n +2} \left (b x +a \right )^{n +1}}{n +1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.83, size = 32, normalized size = 1.45 \[ \frac {{\left (b x^{3} + a x^{2}\right )} e^{\left (n \log \left (b x + a\right ) + 2 \, n \log \relax (x)\right )}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.17, size = 41, normalized size = 1.86 \[ \left (\frac {a\,x\,x^{2\,n+1}}{n+1}+\frac {b\,x^{2\,n+1}\,x^2}{n+1}\right )\,{\left (a+b\,x\right )}^n \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 49.85, size = 53, normalized size = 2.41 \[ \begin {cases} \frac {a x^{2} x^{2 n} \left (a + b x\right )^{n}}{n + 1} + \frac {b x^{3} x^{2 n} \left (a + b x\right )^{n}}{n + 1} & \text {for}\: n \neq -1 \\2 \log {\relax (x )} + \log {\left (\frac {a}{b} + x \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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