Optimal. Leaf size=25 \[ \frac{\left (a+b x+c x^2+d x^3\right )^{n+1}}{n+1} \]
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Rubi [A] time = 0.0239676, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033, Rules used = {1588} \[ \frac{\left (a+b x+c x^2+d x^3\right )^{n+1}}{n+1} \]
Antiderivative was successfully verified.
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Rule 1588
Rubi steps
\begin{align*} \int \left (b+2 c x+3 d x^2\right ) \left (a+b x+c x^2+d x^3\right )^n \, dx &=\frac{\left (a+b x+c x^2+d x^3\right )^{1+n}}{1+n}\\ \end{align*}
Mathematica [A] time = 0.0143151, size = 23, normalized size = 0.92 \[ \frac{(a+x (b+x (c+d x)))^{n+1}}{n+1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 26, normalized size = 1. \begin{align*}{\frac{ \left ( d{x}^{3}+c{x}^{2}+bx+a \right ) ^{1+n}}{1+n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.39835, size = 85, normalized size = 3.4 \begin{align*} \frac{{\left (d x^{3} + c x^{2} + b x + a\right )}{\left (d x^{3} + c x^{2} + b x + a\right )}^{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 = 1.29716, size = 122, normalized size = 4.88 \begin{align*} \frac{{\left (d x^{3} + c x^{2} + b x + a\right )}^{n} d x^{3} +{\left (d x^{3} + c x^{2} + b x + a\right )}^{n} c x^{2} +{\left (d x^{3} + c x^{2} + b x + a\right )}^{n} b x +{\left (d x^{3} + c x^{2} + b x + a\right )}^{n} a}{n + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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