Optimal. Leaf size=31 \[ \frac {1}{20} (2 x+3)^5 (2 d-3 e)+\frac {1}{24} e (2 x+3)^6 \]
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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}{20} (2 x+3)^5 (2 d-3 e)+\frac {1}{24} e (2 x+3)^6 \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 )^2 \, dx &=\int (3+2 x)^4 (d+e x) \, dx\\ &=\int \left (\frac {1}{2} (2 d-3 e) (3+2 x)^4+\frac {1}{2} e (3+2 x)^5\right ) \, dx\\ &=\frac {1}{20} (2 d-3 e) (3+2 x)^5+\frac {1}{24} e (3+2 x)^6\\ \end {align*}
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Mathematica [A] time = 0.01, size = 59, normalized size = 1.90 \begin {gather*} \frac {16}{5} x^5 (d+6 e)+6 x^4 (4 d+9 e)+72 x^3 (d+e)+\frac {27}{2} x^2 (8 d+3 e)+81 d x+\frac {8 e x^6}{3} \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 )^2 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.35, size = 59, normalized size = 1.90 \begin {gather*} \frac {8}{3} x^{6} e + \frac {96}{5} x^{5} e + \frac {16}{5} x^{5} d + 54 x^{4} e + 24 x^{4} d + 72 x^{3} e + 72 x^{3} d + \frac {81}{2} x^{2} e + 108 x^{2} d + 81 x d \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 64, normalized size = 2.06 \begin {gather*} \frac {8}{3} \, x^{6} e + \frac {16}{5} \, d x^{5} + \frac {96}{5} \, x^{5} e + 24 \, d x^{4} + 54 \, x^{4} e + 72 \, d x^{3} + 72 \, x^{3} e + 108 \, d x^{2} + \frac {81}{2} \, x^{2} e + 81 \, d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 60, normalized size = 1.94 \begin {gather*} \frac {8 e \,x^{6}}{3}+\frac {\left (16 d +96 e \right ) x^{5}}{5}+\frac {\left (96 d +216 e \right ) x^{4}}{4}+\frac {\left (216 d +216 e \right ) x^{3}}{3}+81 d x +\frac {\left (216 d +81 e \right ) x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 53, normalized size = 1.71 \begin {gather*} \frac {8}{3} \, e x^{6} + \frac {16}{5} \, {\left (d + 6 \, e\right )} x^{5} + 6 \, {\left (4 \, d + 9 \, e\right )} x^{4} + 72 \, {\left (d + e\right )} x^{3} + \frac {27}{2} \, {\left (8 \, d + 3 \, e\right )} x^{2} + 81 \, d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.51, size = 55, normalized size = 1.77 \begin {gather*} \frac {8\,e\,x^6}{3}+\left (\frac {16\,d}{5}+\frac {96\,e}{5}\right )\,x^5+\left (24\,d+54\,e\right )\,x^4+\left (72\,d+72\,e\right )\,x^3+\left (108\,d+\frac {81\,e}{2}\right )\,x^2+81\,d\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.08, size = 58, normalized size = 1.87 \begin {gather*} 81 d x + \frac {8 e x^{6}}{3} + x^{5} \left (\frac {16 d}{5} + \frac {96 e}{5}\right ) + x^{4} \left (24 d + 54 e\right ) + x^{3} \left (72 d + 72 e\right ) + x^{2} \left (108 d + \frac {81 e}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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