Optimal. Leaf size=25 \[ \frac{x^{n+1} \left (b+c x+d x^2\right )^{n+1}}{n+1} \]
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Rubi [A] time = 0.0212676, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {1590} \[ \frac{x^{n+1} \left (b+c x+d x^2\right )^{n+1}}{n+1} \]
Antiderivative was successfully verified.
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Rule 1590
Rubi steps
\begin{align*} \int x^n \left (b+c x+d x^2\right )^n \left (b+2 c x+3 d x^2\right ) \, dx &=\frac{x^{1+n} \left (b+c x+d x^2\right )^{1+n}}{1+n}\\ \end{align*}
Mathematica [A] time = 0.0204268, size = 24, normalized size = 0.96 \[ \frac{x^{n+1} (b+x (c+d x))^{n+1}}{n+1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 26, normalized size = 1. \begin{align*}{\frac{{x}^{1+n} \left ( d{x}^{2}+cx+b \right ) ^{1+n}}{1+n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.20091, size = 53, normalized size = 2.12 \begin{align*} \frac{{\left (d x^{3} + c x^{2} + b x\right )} e^{\left (n \log \left (d x^{2} + c x + b\right ) + 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 = 1.35457, size = 74, normalized size = 2.96 \begin{align*} \frac{{\left (d x^{3} + c x^{2} + b x\right )}{\left (d x^{2} + c x + b\right )}^{n} x^{n}}{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 = 1.19264, size = 88, normalized size = 3.52 \begin{align*} \frac{{\left (d x^{2} + c x + b\right )}^{n} d x^{3} x^{n} +{\left (d x^{2} + c x + b\right )}^{n} c x^{2} x^{n} +{\left (d x^{2} + c x + b\right )}^{n} b x x^{n}}{n + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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