3.1604 \(\int (d+e x) \left (9+12 x+4 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=50 \[ \frac{1}{16} (2 x+3) \left (4 x^2+12 x+9\right )^{3/2} (2 d-3 e)+\frac{1}{20} e \left (4 x^2+12 x+9\right )^{5/2} \]

[Out]

((2*d - 3*e)*(3 + 2*x)*(9 + 12*x + 4*x^2)^(3/2))/16 + (e*(9 + 12*x + 4*x^2)^(5/2
))/20

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Rubi [A]  time = 0.045971, 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 \[ \frac{1}{16} (2 x+3) \left (4 x^2+12 x+9\right )^{3/2} (2 d-3 e)+\frac{1}{20} e \left (4 x^2+12 x+9\right )^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)*(9 + 12*x + 4*x^2)^(3/2),x]

[Out]

((2*d - 3*e)*(3 + 2*x)*(9 + 12*x + 4*x^2)^(3/2))/16 + (e*(9 + 12*x + 4*x^2)^(5/2
))/20

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Rubi in Sympy [A]  time = 5.43295, size = 42, normalized size = 0.84 \[ \frac{e \left (4 x^{2} + 12 x + 9\right )^{\frac{5}{2}}}{20} + \left (\frac{d}{32} - \frac{3 e}{64}\right ) \left (8 x + 12\right ) \left (4 x^{2} + 12 x + 9\right )^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)*(4*x**2+12*x+9)**(3/2),x)

[Out]

e*(4*x**2 + 12*x + 9)**(5/2)/20 + (d/32 - 3*e/64)*(8*x + 12)*(4*x**2 + 12*x + 9)
**(3/2)

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Mathematica [A]  time = 0.0393909, size = 57, normalized size = 1.14 \[ \frac{x \sqrt{(2 x+3)^2} \left (10 d \left (2 x^3+12 x^2+27 x+27\right )+e x \left (16 x^3+90 x^2+180 x+135\right )\right )}{20 x+30} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)*(9 + 12*x + 4*x^2)^(3/2),x]

[Out]

(x*Sqrt[(3 + 2*x)^2]*(10*d*(27 + 27*x + 12*x^2 + 2*x^3) + e*x*(135 + 180*x + 90*
x^2 + 16*x^3)))/(30 + 20*x)

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Maple [A]  time = 0.006, size = 62, normalized size = 1.2 \[{\frac{x \left ( 16\,e{x}^{4}+20\,d{x}^{3}+90\,{x}^{3}e+120\,d{x}^{2}+180\,e{x}^{2}+270\,dx+135\,ex+270\,d \right ) }{10\, \left ( 2\,x+3 \right ) ^{3}} \left ( \left ( 2\,x+3 \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(4*x^2+12*x+9)^(3/2),x)

[Out]

1/10*x*(16*e*x^4+20*d*x^3+90*e*x^3+120*d*x^2+180*e*x^2+270*d*x+135*e*x+270*d)*((
2*x+3)^2)^(3/2)/(2*x+3)^3

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Maxima [A]  time = 0.831412, size = 105, normalized size = 2.1 \[ \frac{1}{20} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{5}{2}} e + \frac{1}{4} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{3}{2}} d x - \frac{3}{8} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{3}{2}} e x + \frac{3}{8} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{3}{2}} d - \frac{9}{16} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{3}{2}} e \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(4*x^2 + 12*x + 9)^(3/2),x, algorithm="maxima")

[Out]

1/20*(4*x^2 + 12*x + 9)^(5/2)*e + 1/4*(4*x^2 + 12*x + 9)^(3/2)*d*x - 3/8*(4*x^2
+ 12*x + 9)^(3/2)*e*x + 3/8*(4*x^2 + 12*x + 9)^(3/2)*d - 9/16*(4*x^2 + 12*x + 9)
^(3/2)*e

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Fricas [A]  time = 0.200639, size = 59, normalized size = 1.18 \[ \frac{8}{5} \, e x^{5} +{\left (2 \, d + 9 \, e\right )} x^{4} + 6 \,{\left (2 \, d + 3 \, e\right )} x^{3} + \frac{27}{2} \,{\left (2 \, d + e\right )} x^{2} + 27 \, d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(4*x^2 + 12*x + 9)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (d + e x\right ) \left (\left (2 x + 3\right )^{2}\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)*(4*x**2+12*x+9)**(3/2),x)

[Out]

Integral((d + e*x)*((2*x + 3)**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.218393, size = 155, normalized size = 3.1 \[ \frac{8}{5} \, x^{5} e{\rm sign}\left (2 \, x + 3\right ) + 2 \, d x^{4}{\rm sign}\left (2 \, x + 3\right ) + 9 \, x^{4} e{\rm sign}\left (2 \, x + 3\right ) + 12 \, d x^{3}{\rm sign}\left (2 \, x + 3\right ) + 18 \, x^{3} e{\rm sign}\left (2 \, x + 3\right ) + 27 \, d x^{2}{\rm sign}\left (2 \, x + 3\right ) + \frac{27}{2} \, x^{2} e{\rm sign}\left (2 \, x + 3\right ) + 27 \, d x{\rm sign}\left (2 \, x + 3\right ) + \frac{81}{80} \,{\left (10 \, d - 3 \, e\right )}{\rm sign}\left (2 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(4*x^2 + 12*x + 9)^(3/2),x, algorithm="giac")

[Out]

8/5*x^5*e*sign(2*x + 3) + 2*d*x^4*sign(2*x + 3) + 9*x^4*e*sign(2*x + 3) + 12*d*x
^3*sign(2*x + 3) + 18*x^3*e*sign(2*x + 3) + 27*d*x^2*sign(2*x + 3) + 27/2*x^2*e*
sign(2*x + 3) + 27*d*x*sign(2*x + 3) + 81/80*(10*d - 3*e)*sign(2*x + 3)