3.1527 \(\int (d+e x) \left (9+12 x+4 x^2\right )^2 \, dx\)

Optimal. Leaf size=31 \[ \frac{1}{20} (2 x+3)^5 (2 d-3 e)+\frac{1}{24} e (2 x+3)^6 \]

[Out]

((2*d - 3*e)*(3 + 2*x)^5)/20 + (e*(3 + 2*x)^6)/24

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Rubi [A]  time = 0.0395301, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{1}{20} (2 x+3)^5 (2 d-3 e)+\frac{1}{24} e (2 x+3)^6 \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)*(9 + 12*x + 4*x^2)^2,x]

[Out]

((2*d - 3*e)*(3 + 2*x)^5)/20 + (e*(3 + 2*x)^6)/24

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Rubi in Sympy [A]  time = 12.1766, size = 24, normalized size = 0.77 \[ \frac{e \left (2 x + 3\right )^{6}}{24} + \left (\frac{d}{10} - \frac{3 e}{20}\right ) \left (2 x + 3\right )^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)*(4*x**2+12*x+9)**2,x)

[Out]

e*(2*x + 3)**6/24 + (d/10 - 3*e/20)*(2*x + 3)**5

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Mathematica [A]  time = 0.0171204, size = 59, normalized size = 1.9 \[ \frac{16}{5} x^5 (d+6 e)+6 x^4 (4 d+9 e)+72 x^3 (d+e)+\frac{27}{2} x^2 (8 d+3 e)+81 d x+\frac{8 e x^6}{3} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)*(9 + 12*x + 4*x^2)^2,x]

[Out]

81*d*x + (27*(8*d + 3*e)*x^2)/2 + 72*(d + e)*x^3 + 6*(4*d + 9*e)*x^4 + (16*(d +
6*e)*x^5)/5 + (8*e*x^6)/3

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Maple [B]  time = 0., size = 60, normalized size = 1.9 \[{\frac{8\,e{x}^{6}}{3}}+{\frac{ \left ( 16\,d+96\,e \right ){x}^{5}}{5}}+{\frac{ \left ( 96\,d+216\,e \right ){x}^{4}}{4}}+{\frac{ \left ( 216\,d+216\,e \right ){x}^{3}}{3}}+{\frac{ \left ( 216\,d+81\,e \right ){x}^{2}}{2}}+81\,dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(4*x^2+12*x+9)^2,x)

[Out]

8/3*e*x^6+1/5*(16*d+96*e)*x^5+1/4*(96*d+216*e)*x^4+1/3*(216*d+216*e)*x^3+1/2*(21
6*d+81*e)*x^2+81*d*x

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Maxima [A]  time = 0.684278, size = 72, normalized size = 2.32 \[ \frac{8}{3} \, e x^{6} + \frac{16}{5} \,{\left (d + 6 \, e\right )} x^{5} + 6 \,{\left (4 \, d + 9 \, e\right )} x^{4} + 72 \,{\left (d + e\right )} x^{3} + \frac{27}{2} \,{\left (8 \, d + 3 \, e\right )} x^{2} + 81 \, d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(4*x^2 + 12*x + 9)^2,x, algorithm="maxima")

[Out]

8/3*e*x^6 + 16/5*(d + 6*e)*x^5 + 6*(4*d + 9*e)*x^4 + 72*(d + e)*x^3 + 27/2*(8*d
+ 3*e)*x^2 + 81*d*x

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Fricas [A]  time = 0.17761, size = 1, normalized size = 0.03 \[ \frac{8}{3} x^{6} e + \frac{96}{5} x^{5} e + \frac{16}{5} x^{5} d + 54 x^{4} e + 24 x^{4} d + 72 x^{3} e + 72 x^{3} d + \frac{81}{2} x^{2} e + 108 x^{2} d + 81 x d \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(4*x^2 + 12*x + 9)^2,x, algorithm="fricas")

[Out]

8/3*x^6*e + 96/5*x^5*e + 16/5*x^5*d + 54*x^4*e + 24*x^4*d + 72*x^3*e + 72*x^3*d
+ 81/2*x^2*e + 108*x^2*d + 81*x*d

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Sympy [A]  time = 0.127898, size = 58, normalized size = 1.87 \[ 81 d x + \frac{8 e x^{6}}{3} + x^{5} \left (\frac{16 d}{5} + \frac{96 e}{5}\right ) + x^{4} \left (24 d + 54 e\right ) + x^{3} \left (72 d + 72 e\right ) + x^{2} \left (108 d + \frac{81 e}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)*(4*x**2+12*x+9)**2,x)

[Out]

81*d*x + 8*e*x**6/3 + x**5*(16*d/5 + 96*e/5) + x**4*(24*d + 54*e) + x**3*(72*d +
72*e) + x**2*(108*d + 81*e/2)

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GIAC/XCAS [A]  time = 0.209153, size = 86, normalized size = 2.77 \[ \frac{8}{3} \, x^{6} e + \frac{16}{5} \, d x^{5} + \frac{96}{5} \, x^{5} e + 24 \, d x^{4} + 54 \, x^{4} e + 72 \, d x^{3} + 72 \, x^{3} e + 108 \, d x^{2} + \frac{81}{2} \, x^{2} e + 81 \, d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(4*x^2 + 12*x + 9)^2,x, algorithm="giac")

[Out]

8/3*x^6*e + 16/5*d*x^5 + 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 + 81/2*x^2*e + 81*d*x