3.1441 \(\int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right ) \, dx\)

Optimal. Leaf size=65 \[ -\frac{2 b (d+e x)^5 (b d-a e)}{5 e^3}+\frac{(d+e x)^4 (b d-a e)^2}{4 e^3}+\frac{b^2 (d+e x)^6}{6 e^3} \]

[Out]

((b*d - a*e)^2*(d + e*x)^4)/(4*e^3) - (2*b*(b*d - a*e)*(d + e*x)^5)/(5*e^3) + (b
^2*(d + e*x)^6)/(6*e^3)

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Rubi [A]  time = 0.160673, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{2 b (d+e x)^5 (b d-a e)}{5 e^3}+\frac{(d+e x)^4 (b d-a e)^2}{4 e^3}+\frac{b^2 (d+e x)^6}{6 e^3} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

((b*d - a*e)^2*(d + e*x)^4)/(4*e^3) - (2*b*(b*d - a*e)*(d + e*x)^5)/(5*e^3) + (b
^2*(d + e*x)^6)/(6*e^3)

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Rubi in Sympy [A]  time = 36.1562, size = 56, normalized size = 0.86 \[ \frac{b^{2} \left (d + e x\right )^{6}}{6 e^{3}} + \frac{2 b \left (d + e x\right )^{5} \left (a e - b d\right )}{5 e^{3}} + \frac{\left (d + e x\right )^{4} \left (a e - b d\right )^{2}}{4 e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3*(b**2*x**2+2*a*b*x+a**2),x)

[Out]

b**2*(d + e*x)**6/(6*e**3) + 2*b*(d + e*x)**5*(a*e - b*d)/(5*e**3) + (d + e*x)**
4*(a*e - b*d)**2/(4*e**3)

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Mathematica [A]  time = 0.033572, size = 122, normalized size = 1.88 \[ \frac{1}{4} e x^4 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )+\frac{1}{3} d x^3 \left (3 a^2 e^2+6 a b d e+b^2 d^2\right )+a^2 d^3 x+\frac{1}{2} a d^2 x^2 (3 a e+2 b d)+\frac{1}{5} b e^2 x^5 (2 a e+3 b d)+\frac{1}{6} b^2 e^3 x^6 \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

a^2*d^3*x + (a*d^2*(2*b*d + 3*a*e)*x^2)/2 + (d*(b^2*d^2 + 6*a*b*d*e + 3*a^2*e^2)
*x^3)/3 + (e*(3*b^2*d^2 + 6*a*b*d*e + a^2*e^2)*x^4)/4 + (b*e^2*(3*b*d + 2*a*e)*x
^5)/5 + (b^2*e^3*x^6)/6

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Maple [B]  time = 0.001, size = 125, normalized size = 1.9 \[{\frac{{e}^{3}{b}^{2}{x}^{6}}{6}}+{\frac{ \left ( 2\,ab{e}^{3}+3\,{b}^{2}d{e}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ({a}^{2}{e}^{3}+6\,d{e}^{2}ab+3\,{d}^{2}e{b}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,d{e}^{2}{a}^{2}+6\,{d}^{2}eab+{d}^{3}{b}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,{d}^{2}e{a}^{2}+2\,{d}^{3}ab \right ){x}^{2}}{2}}+{a}^{2}{d}^{3}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3*(b^2*x^2+2*a*b*x+a^2),x)

[Out]

1/6*e^3*b^2*x^6+1/5*(2*a*b*e^3+3*b^2*d*e^2)*x^5+1/4*(a^2*e^3+6*a*b*d*e^2+3*b^2*d
^2*e)*x^4+1/3*(3*a^2*d*e^2+6*a*b*d^2*e+b^2*d^3)*x^3+1/2*(3*a^2*d^2*e+2*a*b*d^3)*
x^2+a^2*d^3*x

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Maxima [A]  time = 0.688484, size = 167, normalized size = 2.57 \[ \frac{1}{6} \, b^{2} e^{3} x^{6} + a^{2} d^{3} x + \frac{1}{5} \,{\left (3 \, b^{2} d e^{2} + 2 \, a b e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (3 \, b^{2} d^{2} e + 6 \, a b d e^{2} + a^{2} e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (b^{2} d^{3} + 6 \, a b d^{2} e + 3 \, a^{2} d e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (2 \, a b d^{3} + 3 \, a^{2} d^{2} e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)*(e*x + d)^3,x, algorithm="maxima")

[Out]

1/6*b^2*e^3*x^6 + a^2*d^3*x + 1/5*(3*b^2*d*e^2 + 2*a*b*e^3)*x^5 + 1/4*(3*b^2*d^2
*e + 6*a*b*d*e^2 + a^2*e^3)*x^4 + 1/3*(b^2*d^3 + 6*a*b*d^2*e + 3*a^2*d*e^2)*x^3
+ 1/2*(2*a*b*d^3 + 3*a^2*d^2*e)*x^2

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Fricas [A]  time = 0.178819, size = 1, normalized size = 0.02 \[ \frac{1}{6} x^{6} e^{3} b^{2} + \frac{3}{5} x^{5} e^{2} d b^{2} + \frac{2}{5} x^{5} e^{3} b a + \frac{3}{4} x^{4} e d^{2} b^{2} + \frac{3}{2} x^{4} e^{2} d b a + \frac{1}{4} x^{4} e^{3} a^{2} + \frac{1}{3} x^{3} d^{3} b^{2} + 2 x^{3} e d^{2} b a + x^{3} e^{2} d a^{2} + x^{2} d^{3} b a + \frac{3}{2} x^{2} e d^{2} a^{2} + x d^{3} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)*(e*x + d)^3,x, algorithm="fricas")

[Out]

1/6*x^6*e^3*b^2 + 3/5*x^5*e^2*d*b^2 + 2/5*x^5*e^3*b*a + 3/4*x^4*e*d^2*b^2 + 3/2*
x^4*e^2*d*b*a + 1/4*x^4*e^3*a^2 + 1/3*x^3*d^3*b^2 + 2*x^3*e*d^2*b*a + x^3*e^2*d*
a^2 + x^2*d^3*b*a + 3/2*x^2*e*d^2*a^2 + x*d^3*a^2

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Sympy [A]  time = 0.159245, size = 133, normalized size = 2.05 \[ a^{2} d^{3} x + \frac{b^{2} e^{3} x^{6}}{6} + x^{5} \left (\frac{2 a b e^{3}}{5} + \frac{3 b^{2} d e^{2}}{5}\right ) + x^{4} \left (\frac{a^{2} e^{3}}{4} + \frac{3 a b d e^{2}}{2} + \frac{3 b^{2} d^{2} e}{4}\right ) + x^{3} \left (a^{2} d e^{2} + 2 a b d^{2} e + \frac{b^{2} d^{3}}{3}\right ) + x^{2} \left (\frac{3 a^{2} d^{2} e}{2} + a b d^{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3*(b**2*x**2+2*a*b*x+a**2),x)

[Out]

a**2*d**3*x + b**2*e**3*x**6/6 + x**5*(2*a*b*e**3/5 + 3*b**2*d*e**2/5) + x**4*(a
**2*e**3/4 + 3*a*b*d*e**2/2 + 3*b**2*d**2*e/4) + x**3*(a**2*d*e**2 + 2*a*b*d**2*
e + b**2*d**3/3) + x**2*(3*a**2*d**2*e/2 + a*b*d**3)

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GIAC/XCAS [A]  time = 0.209201, size = 171, normalized size = 2.63 \[ \frac{1}{6} \, b^{2} x^{6} e^{3} + \frac{3}{5} \, b^{2} d x^{5} e^{2} + \frac{3}{4} \, b^{2} d^{2} x^{4} e + \frac{1}{3} \, b^{2} d^{3} x^{3} + \frac{2}{5} \, a b x^{5} e^{3} + \frac{3}{2} \, a b d x^{4} e^{2} + 2 \, a b d^{2} x^{3} e + a b d^{3} x^{2} + \frac{1}{4} \, a^{2} x^{4} e^{3} + a^{2} d x^{3} e^{2} + \frac{3}{2} \, a^{2} d^{2} x^{2} e + a^{2} d^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)*(e*x + d)^3,x, algorithm="giac")

[Out]

1/6*b^2*x^6*e^3 + 3/5*b^2*d*x^5*e^2 + 3/4*b^2*d^2*x^4*e + 1/3*b^2*d^3*x^3 + 2/5*
a*b*x^5*e^3 + 3/2*a*b*d*x^4*e^2 + 2*a*b*d^2*x^3*e + a*b*d^3*x^2 + 1/4*a^2*x^4*e^
3 + a^2*d*x^3*e^2 + 3/2*a^2*d^2*x^2*e + a^2*d^3*x