Optimal. Leaf size=31 \[ \frac {1}{28} (2 x+3)^7 (2 d-3 e)+\frac {1}{32} e (2 x+3)^8 \]
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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 = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {27, 43} \begin {gather*} \frac {1}{28} (2 x+3)^7 (2 d-3 e)+\frac {1}{32} e (2 x+3)^8 \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 )^3 \, dx &=\int (3+2 x)^6 (d+e x) \, dx\\ &=\int \left (\frac {1}{2} (2 d-3 e) (3+2 x)^6+\frac {1}{2} e (3+2 x)^7\right ) \, dx\\ &=\frac {1}{28} (2 d-3 e) (3+2 x)^7+\frac {1}{32} e (3+2 x)^8\\ \end {align*}
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Mathematica [B] time = 0.02, size = 81, normalized size = 2.61 \begin {gather*} \frac {64}{7} x^7 (d+9 e)+24 x^6 (4 d+15 e)+432 x^5 (d+2 e)+135 x^4 (8 d+9 e)+324 x^3 (5 d+3 e)+\frac {729}{2} x^2 (4 d+e)+729 d x+8 e x^8 \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 )^3 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.34, size = 83, normalized size = 2.68 \begin {gather*} 8 x^{8} e + \frac {576}{7} x^{7} e + \frac {64}{7} x^{7} d + 360 x^{6} e + 96 x^{6} d + 864 x^{5} e + 432 x^{5} d + 1215 x^{4} e + 1080 x^{4} d + 972 x^{3} e + 1620 x^{3} d + \frac {729}{2} x^{2} e + 1458 x^{2} d + 729 x d \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 90, normalized size = 2.90 \begin {gather*} 8 \, x^{8} e + \frac {64}{7} \, d x^{7} + \frac {576}{7} \, x^{7} e + 96 \, d x^{6} + 360 \, x^{6} e + 432 \, d x^{5} + 864 \, x^{5} e + 1080 \, d x^{4} + 1215 \, x^{4} e + 1620 \, d x^{3} + 972 \, x^{3} e + 1458 \, d x^{2} + \frac {729}{2} \, x^{2} e + 729 \, d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 84, normalized size = 2.71 \begin {gather*} 8 e \,x^{8}+\frac {\left (64 d +576 e \right ) x^{7}}{7}+\frac {\left (576 d +2160 e \right ) x^{6}}{6}+\frac {\left (2160 d +4320 e \right ) x^{5}}{5}+\frac {\left (4320 d +4860 e \right ) x^{4}}{4}+\frac {\left (4860 d +2916 e \right ) x^{3}}{3}+729 d x +\frac {\left (2916 d +729 e \right ) x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.30, size = 77, normalized size = 2.48 \begin {gather*} 8 \, e x^{8} + \frac {64}{7} \, {\left (d + 9 \, e\right )} x^{7} + 24 \, {\left (4 \, d + 15 \, e\right )} x^{6} + 432 \, {\left (d + 2 \, e\right )} x^{5} + 135 \, {\left (8 \, d + 9 \, e\right )} x^{4} + 324 \, {\left (5 \, d + 3 \, e\right )} x^{3} + \frac {729}{2} \, {\left (4 \, d + e\right )} x^{2} + 729 \, d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 77, normalized size = 2.48 \begin {gather*} 8\,e\,x^8+\left (\frac {64\,d}{7}+\frac {576\,e}{7}\right )\,x^7+\left (96\,d+360\,e\right )\,x^6+\left (432\,d+864\,e\right )\,x^5+\left (1080\,d+1215\,e\right )\,x^4+\left (1620\,d+972\,e\right )\,x^3+\left (1458\,d+\frac {729\,e}{2}\right )\,x^2+729\,d\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.09, size = 76, normalized size = 2.45 \begin {gather*} 729 d x + 8 e x^{8} + x^{7} \left (\frac {64 d}{7} + \frac {576 e}{7}\right ) + x^{6} \left (96 d + 360 e\right ) + x^{5} \left (432 d + 864 e\right ) + x^{4} \left (1080 d + 1215 e\right ) + x^{3} \left (1620 d + 972 e\right ) + x^{2} \left (1458 d + \frac {729 e}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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