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 = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 1.02, size = 35, normalized size = 1.40 \[ \frac {{\left (d x^{3} + c x^{2} + b x\right )} {\left (d x^{2} + c x + b\right )}^{n} x^{n}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.40, size = 65, normalized size = 2.60 \[ \frac {{\left (d x^{2} + c x + b\right )}^{n} d x^{3} x^{n} + {\left (d x^{2} + c x + b\right )}^{n} c x^{2} x^{n} + {\left (d x^{2} + c x + b\right )}^{n} b x x^{n}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 26, normalized size = 1.04 \[ \frac {x^{n +1} \left (d \,x^{2}+c x +b \right )^{n +1}}{n +1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.86, size = 39, normalized size = 1.56 \[ \frac {{\left (d x^{3} + c x^{2} + b x\right )} e^{\left (n \log \left (d x^{2} + c x + b\right ) + n \log \relax (x)\right )}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.15, size = 51, normalized size = 2.04 \[ \left (\frac {c\,x^n\,x^2}{n+1}+\frac {d\,x^n\,x^3}{n+1}+\frac {b\,x\,x^n}{n+1}\right )\,{\left (d\,x^2+c\,x+b\right )}^n \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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