Optimal. Leaf size=50 \[ \frac {1}{8} (2 x+3) \sqrt {4 x^2+12 x+9} (2 d-3 e)+\frac {1}{12} e \left (4 x^2+12 x+9\right )^{3/2} \]
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Rubi [A] time = 0.01, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {640, 609} \begin {gather*} \frac {1}{8} (2 x+3) \sqrt {4 x^2+12 x+9} (2 d-3 e)+\frac {1}{12} e \left (4 x^2+12 x+9\right )^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 609
Rule 640
Rubi steps
\begin {align*} \int (d+e x) \sqrt {9+12 x+4 x^2} \, dx &=\frac {1}{12} e \left (9+12 x+4 x^2\right )^{3/2}+\frac {1}{2} (2 d-3 e) \int \sqrt {9+12 x+4 x^2} \, dx\\ &=\frac {1}{8} (2 d-3 e) (3+2 x) \sqrt {9+12 x+4 x^2}+\frac {1}{12} e \left (9+12 x+4 x^2\right )^{3/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 38, normalized size = 0.76 \begin {gather*} \frac {x \sqrt {(2 x+3)^2} (6 d (x+3)+e x (4 x+9))}{6 (2 x+3)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.25, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x) \sqrt {9+12 x+4 x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 23, normalized size = 0.46 \begin {gather*} \frac {2}{3} \, e x^{3} + \frac {1}{2} \, {\left (2 \, d + 3 \, e\right )} x^{2} + 3 \, d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 64, normalized size = 1.28 \begin {gather*} \frac {2}{3} \, x^{3} e \mathrm {sgn}\left (2 \, x + 3\right ) + d x^{2} \mathrm {sgn}\left (2 \, x + 3\right ) + \frac {3}{2} \, x^{2} e \mathrm {sgn}\left (2 \, x + 3\right ) + 3 \, d x \mathrm {sgn}\left (2 \, x + 3\right ) + \frac {9}{8} \, {\left (2 \, d - e\right )} \mathrm {sgn}\left (2 \, x + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 38, normalized size = 0.76 \begin {gather*} \frac {\left (4 e \,x^{2}+6 d x +9 e x +18 d \right ) \sqrt {\left (2 x +3\right )^{2}}\, x}{12 x +18} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.43, size = 78, normalized size = 1.56 \begin {gather*} \frac {1}{12} \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac {3}{2}} e + \frac {1}{2} \, \sqrt {4 \, x^{2} + 12 \, x + 9} d x - \frac {3}{4} \, \sqrt {4 \, x^{2} + 12 \, x + 9} e x + \frac {3}{4} \, \sqrt {4 \, x^{2} + 12 \, x + 9} d - \frac {9}{8} \, \sqrt {4 \, x^{2} + 12 \, x + 9} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.61, size = 30, normalized size = 0.60 \begin {gather*} \frac {\left (2\,x+3\right )\,\left (6\,d-3\,e+4\,e\,x\right )\,\sqrt {4\,x^2+12\,x+9}}{24} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d + e x\right ) \sqrt {\left (2 x + 3\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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