Optimal. Leaf size=31 \[ \frac {1}{12} (2 d-3 e) (3+2 x)^3+\frac {1}{16} e (3+2 x)^4 \]
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Rubi [A]
time = 0.01, 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, 45}
\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 45
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 = 0.00, size = 36, normalized size = 1.16 \begin {gather*} 9 d x+\frac {3}{2} (4 d+3 e) x^2+\frac {4}{3} (d+3 e) x^3+e x^4 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 35, normalized size = 1.13
method | result | size |
norman | \(x^{4} e +\left (\frac {4 d}{3}+4 e \right ) x^{3}+\left (6 d +\frac {9 e}{2}\right ) x^{2}+9 d x\) | \(33\) |
gosper | \(\frac {x \left (6 e \,x^{3}+8 d \,x^{2}+24 e \,x^{2}+36 d x +27 e x +54 d \right )}{6}\) | \(34\) |
default | \(x^{4} e +\frac {\left (4 d +12 e \right ) x^{3}}{3}+\frac {\left (12 d +9 e \right ) x^{2}}{2}+9 d x\) | \(35\) |
risch | \(x^{4} e +\frac {4}{3} d \,x^{3}+4 e \,x^{3}+6 d \,x^{2}+\frac {9}{2} e \,x^{2}+9 d x\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 35, normalized size = 1.13 \begin {gather*} x^{4} e + \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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Fricas [A]
time = 2.33, size = 37, normalized size = 1.19 \begin {gather*} \frac {4}{3} \, d x^{3} + 6 \, d x^{2} + 9 \, d x + \frac {1}{2} \, {\left (2 \, x^{4} + 8 \, x^{3} + 9 \, x^{2}\right )} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 32, normalized size = 1.03 \begin {gather*} 9 d x + e x^{4} + x^{3} \cdot \left (\frac {4 d}{3} + 4 e\right ) + x^{2} \cdot \left (6 d + \frac {9 e}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.44, 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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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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