Optimal. Leaf size=31 \[ \frac {1}{28} (2 d-3 e) (3+2 x)^7+\frac {1}{32} e (3+2 x)^8 \]
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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, 45}
\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 45
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] Leaf count is larger than twice the leaf count of optimal. \(81\) vs. \(2(31)=62\).
time = 0.01, size = 81, normalized size = 2.61 \begin {gather*} 729 d x+\frac {729}{2} (4 d+e) x^2+324 (5 d+3 e) x^3+135 (8 d+9 e) x^4+432 (d+2 e) x^5+24 (4 d+15 e) x^6+\frac {64}{7} (d+9 e) x^7+8 e x^8 \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(83\) vs.
\(2(27)=54\).
time = 0.63, size = 84, normalized size = 2.71
method | result | size |
norman | \(8 e \,x^{8}+\left (\frac {64 d}{7}+\frac {576 e}{7}\right ) x^{7}+\left (96 d +360 e \right ) x^{6}+\left (432 d +864 e \right ) x^{5}+\left (1080 d +1215 e \right ) x^{4}+\left (1620 d +972 e \right ) x^{3}+\left (1458 d +\frac {729 e}{2}\right ) x^{2}+729 d x\) | \(78\) |
gosper | \(\frac {x \left (112 e \,x^{7}+128 d \,x^{6}+1152 x^{6} e +1344 x^{5} d +5040 x^{5} e +6048 d \,x^{4}+12096 x^{4} e +15120 d \,x^{3}+17010 e \,x^{3}+22680 d \,x^{2}+13608 e \,x^{2}+20412 d x +5103 e x +10206 d \right )}{14}\) | \(82\) |
default | \(8 e \,x^{8}+\frac {\left (64 d +576 e \right ) x^{7}}{7}+\frac {\left (576 d +2160 e \right ) x^{6}}{6}+\frac {\left (2160 d +4320 e \right ) x^{5}}{5}+\frac {\left (4320 d +4860 e \right ) x^{4}}{4}+\frac {\left (4860 d +2916 e \right ) x^{3}}{3}+\frac {\left (2916 d +729 e \right ) x^{2}}{2}+729 d x\) | \(84\) |
risch | \(8 e \,x^{8}+\frac {64}{7} x^{7} d +\frac {576}{7} e \,x^{7}+96 d \,x^{6}+360 x^{6} e +432 x^{5} d +864 x^{5} e +1080 d \,x^{4}+1215 x^{4} e +1620 d \,x^{3}+972 e \,x^{3}+1458 d \,x^{2}+\frac {729}{2} e \,x^{2}+729 d x\) | \(84\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 84 vs.
\(2 (29) = 58\).
time = 0.30, size = 84, normalized size = 2.71 \begin {gather*} 8 \, x^{8} e + \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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 81 vs.
\(2 (29) = 58\).
time = 2.67, size = 81, normalized size = 2.61 \begin {gather*} \frac {64}{7} \, d x^{7} + 96 \, d x^{6} + 432 \, d x^{5} + 1080 \, d x^{4} + 1620 \, d x^{3} + 1458 \, d x^{2} + 729 \, d x + \frac {1}{14} \, {\left (112 \, x^{8} + 1152 \, x^{7} + 5040 \, x^{6} + 12096 \, x^{5} + 17010 \, x^{4} + 13608 \, x^{3} + 5103 \, x^{2}\right )} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 76 vs.
\(2 (24) = 48\).
time = 0.02, size = 76, normalized size = 2.45 \begin {gather*} 729 d x + 8 e x^{8} + x^{7} \cdot \left (\frac {64 d}{7} + \frac {576 e}{7}\right ) + x^{6} \cdot \left (96 d + 360 e\right ) + x^{5} \cdot \left (432 d + 864 e\right ) + x^{4} \cdot \left (1080 d + 1215 e\right ) + x^{3} \cdot \left (1620 d + 972 e\right ) + x^{2} \cdot \left (1458 d + \frac {729 e}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 90 vs.
\(2 (29) = 58\).
time = 1.71, 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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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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