3.1603 \(\int (d+e x) \left (9+12 x+4 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=50 \[ \frac{1}{24} (2 x+3) \left (4 x^2+12 x+9\right )^{5/2} (2 d-3 e)+\frac{1}{28} e \left (4 x^2+12 x+9\right )^{7/2} \]

[Out]

((2*d - 3*e)*(3 + 2*x)*(9 + 12*x + 4*x^2)^(5/2))/24 + (e*(9 + 12*x + 4*x^2)^(7/2
))/28

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Rubi [A]  time = 0.0464862, 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 \[ \frac{1}{24} (2 x+3) \left (4 x^2+12 x+9\right )^{5/2} (2 d-3 e)+\frac{1}{28} e \left (4 x^2+12 x+9\right )^{7/2} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)*(9 + 12*x + 4*x^2)^(5/2),x]

[Out]

((2*d - 3*e)*(3 + 2*x)*(9 + 12*x + 4*x^2)^(5/2))/24 + (e*(9 + 12*x + 4*x^2)^(7/2
))/28

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Rubi in Sympy [A]  time = 5.46597, size = 41, normalized size = 0.82 \[ \frac{e \left (4 x^{2} + 12 x + 9\right )^{\frac{7}{2}}}{28} + \left (\frac{d}{48} - \frac{e}{32}\right ) \left (8 x + 12\right ) \left (4 x^{2} + 12 x + 9\right )^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)*(4*x**2+12*x+9)**(5/2),x)

[Out]

e*(4*x**2 + 12*x + 9)**(7/2)/28 + (d/48 - e/32)*(8*x + 12)*(4*x**2 + 12*x + 9)**
(5/2)

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Mathematica [A]  time = 0.0595114, size = 81, normalized size = 1.62 \[ \frac{x \sqrt{(2 x+3)^2} \left (14 d \left (16 x^5+144 x^4+540 x^3+1080 x^2+1215 x+729\right )+3 e x \left (64 x^5+560 x^4+2016 x^3+3780 x^2+3780 x+1701\right )\right )}{42 (2 x+3)} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)*(9 + 12*x + 4*x^2)^(5/2),x]

[Out]

(x*Sqrt[(3 + 2*x)^2]*(14*d*(729 + 1215*x + 1080*x^2 + 540*x^3 + 144*x^4 + 16*x^5
) + 3*e*x*(1701 + 3780*x + 3780*x^2 + 2016*x^3 + 560*x^4 + 64*x^5)))/(42*(3 + 2*
x))

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Maple [B]  time = 0.008, size = 86, normalized size = 1.7 \[{\frac{x \left ( 192\,e{x}^{6}+224\,{x}^{5}d+1680\,{x}^{5}e+2016\,d{x}^{4}+6048\,e{x}^{4}+7560\,d{x}^{3}+11340\,{x}^{3}e+15120\,d{x}^{2}+11340\,e{x}^{2}+17010\,dx+5103\,ex+10206\,d \right ) }{42\, \left ( 2\,x+3 \right ) ^{5}} \left ( \left ( 2\,x+3 \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(4*x^2+12*x+9)^(5/2),x)

[Out]

1/42*x*(192*e*x^6+224*d*x^5+1680*e*x^5+2016*d*x^4+6048*e*x^4+7560*d*x^3+11340*e*
x^3+15120*d*x^2+11340*e*x^2+17010*d*x+5103*e*x+10206*d)*((2*x+3)^2)^(5/2)/(2*x+3
)^5

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Maxima [A]  time = 0.828891, size = 105, normalized size = 2.1 \[ \frac{1}{28} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{7}{2}} e + \frac{1}{6} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{5}{2}} d x - \frac{1}{4} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{5}{2}} e x + \frac{1}{4} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{5}{2}} d - \frac{3}{8} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{5}{2}} e \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(4*x^2 + 12*x + 9)^(5/2),x, algorithm="maxima")

[Out]

1/28*(4*x^2 + 12*x + 9)^(7/2)*e + 1/6*(4*x^2 + 12*x + 9)^(5/2)*d*x - 1/4*(4*x^2
+ 12*x + 9)^(5/2)*e*x + 1/4*(4*x^2 + 12*x + 9)^(5/2)*d - 3/8*(4*x^2 + 12*x + 9)^
(5/2)*e

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Fricas [A]  time = 0.201719, size = 93, normalized size = 1.86 \[ \frac{32}{7} \, e x^{7} + \frac{8}{3} \,{\left (2 \, d + 15 \, e\right )} x^{6} + 48 \,{\left (d + 3 \, e\right )} x^{5} + 90 \,{\left (2 \, d + 3 \, e\right )} x^{4} + 90 \,{\left (4 \, d + 3 \, e\right )} x^{3} + \frac{81}{2} \,{\left (10 \, d + 3 \, e\right )} x^{2} + 243 \, d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(4*x^2 + 12*x + 9)^(5/2),x, algorithm="fricas")

[Out]

32/7*e*x^7 + 8/3*(2*d + 15*e)*x^6 + 48*(d + 3*e)*x^5 + 90*(2*d + 3*e)*x^4 + 90*(
4*d + 3*e)*x^3 + 81/2*(10*d + 3*e)*x^2 + 243*d*x

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (d + e x\right ) \left (\left (2 x + 3\right )^{2}\right )^{\frac{5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)*(4*x**2+12*x+9)**(5/2),x)

[Out]

Integral((d + e*x)*((2*x + 3)**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.221967, size = 223, normalized size = 4.46 \[ \frac{32}{7} \, x^{7} e{\rm sign}\left (2 \, x + 3\right ) + \frac{16}{3} \, d x^{6}{\rm sign}\left (2 \, x + 3\right ) + 40 \, x^{6} e{\rm sign}\left (2 \, x + 3\right ) + 48 \, d x^{5}{\rm sign}\left (2 \, x + 3\right ) + 144 \, x^{5} e{\rm sign}\left (2 \, x + 3\right ) + 180 \, d x^{4}{\rm sign}\left (2 \, x + 3\right ) + 270 \, x^{4} e{\rm sign}\left (2 \, x + 3\right ) + 360 \, d x^{3}{\rm sign}\left (2 \, x + 3\right ) + 270 \, x^{3} e{\rm sign}\left (2 \, x + 3\right ) + 405 \, d x^{2}{\rm sign}\left (2 \, x + 3\right ) + \frac{243}{2} \, x^{2} e{\rm sign}\left (2 \, x + 3\right ) + 243 \, d x{\rm sign}\left (2 \, x + 3\right ) + \frac{243}{56} \,{\left (14 \, d - 3 \, e\right )}{\rm sign}\left (2 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(4*x^2 + 12*x + 9)^(5/2),x, algorithm="giac")

[Out]

32/7*x^7*e*sign(2*x + 3) + 16/3*d*x^6*sign(2*x + 3) + 40*x^6*e*sign(2*x + 3) + 4
8*d*x^5*sign(2*x + 3) + 144*x^5*e*sign(2*x + 3) + 180*d*x^4*sign(2*x + 3) + 270*
x^4*e*sign(2*x + 3) + 360*d*x^3*sign(2*x + 3) + 270*x^3*e*sign(2*x + 3) + 405*d*
x^2*sign(2*x + 3) + 243/2*x^2*e*sign(2*x + 3) + 243*d*x*sign(2*x + 3) + 243/56*(
14*d - 3*e)*sign(2*x + 3)