Optimal. Leaf size=50 \[ \frac{1}{8} (2 x+3) \sqrt{4 x^2+12 x+9} (2 d-3 e)+\frac{1}{12} e \left (4 x^2+12 x+9\right )^{3/2} \]
[Out]
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Rubi [A] time = 0.0446594, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{1}{8} (2 x+3) \sqrt{4 x^2+12 x+9} (2 d-3 e)+\frac{1}{12} e \left (4 x^2+12 x+9\right )^{3/2} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)*Sqrt[9 + 12*x + 4*x^2],x]
[Out]
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Rubi in Sympy [A] time = 5.43339, size = 42, normalized size = 0.84 \[ \frac{e \left (4 x^{2} + 12 x + 9\right )^{\frac{3}{2}}}{12} + \left (\frac{d}{16} - \frac{3 e}{32}\right ) \left (8 x + 12\right ) \sqrt{4 x^{2} + 12 x + 9} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)*(4*x**2+12*x+9)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0244496, size = 38, normalized size = 0.76 \[ \frac{x \sqrt{(2 x+3)^2} (6 d (x+3)+e x (4 x+9))}{6 (2 x+3)} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)*Sqrt[9 + 12*x + 4*x^2],x]
[Out]
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Maple [A] time = 0.004, size = 38, normalized size = 0.8 \[{\frac{x \left ( 4\,e{x}^{2}+6\,dx+9\,ex+18\,d \right ) }{12\,x+18}\sqrt{ \left ( 2\,x+3 \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)*(4*x^2+12*x+9)^(1/2),x)
[Out]
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Maxima [A] time = 0.823591, size = 105, normalized size = 2.1 \[ \frac{1}{12} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{3}{2}} e + \frac{1}{2} \, \sqrt{4 \, x^{2} + 12 \, x + 9} d x - \frac{3}{4} \, \sqrt{4 \, x^{2} + 12 \, x + 9} e x + \frac{3}{4} \, \sqrt{4 \, x^{2} + 12 \, x + 9} d - \frac{9}{8} \, \sqrt{4 \, x^{2} + 12 \, x + 9} e \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)*sqrt(4*x^2 + 12*x + 9),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.199714, size = 31, normalized size = 0.62 \[ \frac{2}{3} \, e x^{3} + \frac{1}{2} \,{\left (2 \, d + 3 \, e\right )} x^{2} + 3 \, d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)*sqrt(4*x^2 + 12*x + 9),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (d + e x\right ) \sqrt{\left (2 x + 3\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)*(4*x**2+12*x+9)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.218515, size = 86, normalized size = 1.72 \[ \frac{2}{3} \, x^{3} e{\rm sign}\left (2 \, x + 3\right ) + d x^{2}{\rm sign}\left (2 \, x + 3\right ) + \frac{3}{2} \, x^{2} e{\rm sign}\left (2 \, x + 3\right ) + 3 \, d x{\rm sign}\left (2 \, x + 3\right ) + \frac{9}{8} \,{\left (2 \, d - e\right )}{\rm sign}\left (2 \, x + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)*sqrt(4*x^2 + 12*x + 9),x, algorithm="giac")
[Out]