Optimal. Leaf size=31 \[ \frac {1}{12} (2 x+3)^3 (2 d-3 e)+\frac {1}{16} e (2 x+3)^4 \]
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Rubi [A] time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {27, 43} \begin {gather*} \frac {1}{12} (2 x+3)^3 (2 d-3 e)+\frac {1}{16} e (2 x+3)^4 \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int (d+e x) \left (9+12 x+4 x^2\right ) \, dx &=\int (3+2 x)^2 (d+e x) \, dx\\ &=\int \left (\frac {1}{2} (2 d-3 e) (3+2 x)^2+\frac {1}{2} e (3+2 x)^3\right ) \, dx\\ &=\frac {1}{12} (2 d-3 e) (3+2 x)^3+\frac {1}{16} e (3+2 x)^4\\ \end {align*}
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Mathematica [A] time = 0.01, size = 36, normalized size = 1.16 \begin {gather*} \frac {4}{3} x^3 (d+3 e)+\frac {3}{2} x^2 (4 d+3 e)+9 d x+e x^4 \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x) \left (9+12 x+4 x^2\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.35, size = 34, normalized size = 1.10 \begin {gather*} x^{4} e + 4 x^{3} e + \frac {4}{3} x^{3} d + \frac {9}{2} x^{2} e + 6 x^{2} d + 9 x d \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 37, normalized size = 1.19 \begin {gather*} x^{4} e + \frac {4}{3} \, d x^{3} + 4 \, x^{3} e + 6 \, d x^{2} + \frac {9}{2} \, x^{2} e + 9 \, d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 35, normalized size = 1.13 \begin {gather*} e \,x^{4}+\frac {\left (4 d +12 e \right ) x^{3}}{3}+9 d x +\frac {\left (12 d +9 e \right ) x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.23, size = 32, normalized size = 1.03 \begin {gather*} e x^{4} + \frac {4}{3} \, {\left (d + 3 \, e\right )} x^{3} + \frac {3}{2} \, {\left (4 \, d + 3 \, e\right )} x^{2} + 9 \, d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 32, normalized size = 1.03 \begin {gather*} e\,x^4+\left (\frac {4\,d}{3}+4\,e\right )\,x^3+\left (6\,d+\frac {9\,e}{2}\right )\,x^2+9\,d\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 32, normalized size = 1.03 \begin {gather*} 9 d x + e x^{4} + x^{3} \left (\frac {4 d}{3} + 4 e\right ) + x^{2} \left (6 d + \frac {9 e}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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