3.1747 \(\int (d+e x) \left (9+12 x+4 x^2\right )^p \, dx\)

Optimal. Leaf size=60 \[ \frac{(2 x+3) (2 d-3 e) \left (4 x^2+12 x+9\right )^p}{4 (2 p+1)}+\frac{e \left (4 x^2+12 x+9\right )^{p+1}}{8 (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0475853, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{(2 x+3) (2 d-3 e) \left (4 x^2+12 x+9\right )^p}{4 (2 p+1)}+\frac{e \left (4 x^2+12 x+9\right )^{p+1}}{8 (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.5793, size = 48, normalized size = 0.8 \[ \frac{e \left (4 x^{2} + 12 x + 9\right )^{p + 1}}{8 \left (p + 1\right )} + \frac{\left (2 d - 3 e\right ) \left (8 x + 12\right ) \left (4 x^{2} + 12 x + 9\right )^{p}}{16 \left (2 p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0428713, size = 48, normalized size = 0.8 \[ \frac{(2 x+3) \left ((2 x+3)^2\right )^p (4 d (p+1)+e ((4 p+2) x-3))}{8 (p+1) (2 p+1)} \]

Antiderivative was successfully verified.

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

[Out]

((3 + 2*x)*((3 + 2*x)^2)^p*(4*d*(1 + p) + e*(-3 + (2 + 4*p)*x)))/(8*(1 + p)*(1 +
2*p))

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Maple [A]  time = 0.006, size = 52, normalized size = 0.9 \[{\frac{ \left ( 4\,{x}^{2}+12\,x+9 \right ) ^{p} \left ( 4\,epx+4\,dp+2\,ex+4\,d-3\,e \right ) \left ( 2\,x+3 \right ) }{16\,{p}^{2}+24\,p+8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(4*x^2+12*x+9)^p,x)

[Out]

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

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Maxima [A]  time = 0.756033, size = 88, normalized size = 1.47 \[ \frac{{\left (4 \,{\left (2 \, p + 1\right )} x^{2} + 12 \, p x - 9\right )} e{\left (2 \, x + 3\right )}^{2 \, p}}{8 \,{\left (2 \, p^{2} + 3 \, p + 1\right )}} + \frac{d{\left (2 \, x + 3\right )}^{2 \, p}{\left (2 \, x + 3\right )}}{2 \,{\left (2 \, p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.217798, size = 86, normalized size = 1.43 \[ \frac{{\left (4 \,{\left (2 \, e p + e\right )} x^{2} + 12 \, d p + 4 \,{\left ({\left (2 \, d + 3 \, e\right )} p + 2 \, d\right )} x + 12 \, d - 9 \, e\right )}{\left (4 \, x^{2} + 12 \, x + 9\right )}^{p}}{8 \,{\left (2 \, p^{2} + 3 \, p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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GIAC/XCAS [A]  time = 0.215129, size = 227, normalized size = 3.78 \[ \frac{8 \, p x^{2} e^{\left (p{\rm ln}\left (4 \, x^{2} + 12 \, x + 9\right ) + 1\right )} + 8 \, d p x e^{\left (p{\rm ln}\left (4 \, x^{2} + 12 \, x + 9\right )\right )} + 12 \, p x e^{\left (p{\rm ln}\left (4 \, x^{2} + 12 \, x + 9\right ) + 1\right )} + 4 \, x^{2} e^{\left (p{\rm ln}\left (4 \, x^{2} + 12 \, x + 9\right ) + 1\right )} + 12 \, d p e^{\left (p{\rm ln}\left (4 \, x^{2} + 12 \, x + 9\right )\right )} + 8 \, d x e^{\left (p{\rm ln}\left (4 \, x^{2} + 12 \, x + 9\right )\right )} + 12 \, d e^{\left (p{\rm ln}\left (4 \, x^{2} + 12 \, x + 9\right )\right )} - 9 \, e^{\left (p{\rm ln}\left (4 \, x^{2} + 12 \, x + 9\right ) + 1\right )}}{8 \,{\left (2 \, p^{2} + 3 \, p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(8*p*x^2*e^(p*ln(4*x^2 + 12*x + 9) + 1) + 8*d*p*x*e^(p*ln(4*x^2 + 12*x + 9))
+ 12*p*x*e^(p*ln(4*x^2 + 12*x + 9) + 1) + 4*x^2*e^(p*ln(4*x^2 + 12*x + 9) + 1)
+ 12*d*p*e^(p*ln(4*x^2 + 12*x + 9)) + 8*d*x*e^(p*ln(4*x^2 + 12*x + 9)) + 12*d*e^
(p*ln(4*x^2 + 12*x + 9)) - 9*e^(p*ln(4*x^2 + 12*x + 9) + 1))/(2*p^2 + 3*p + 1)