3.1820 \(\int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx\)

Optimal. Leaf size=39 \[ \frac{1}{3} (d+e x)^3 \left (a-\frac{c d^2}{e^2}\right )+\frac{c d (d+e x)^4}{4 e^2} \]

[Out]

((a - (c*d^2)/e^2)*(d + e*x)^3)/3 + (c*d*(d + e*x)^4)/(4*e^2)

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Rubi [A]  time = 0.0953101, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{1}{3} (d+e x)^3 \left (a-\frac{c d^2}{e^2}\right )+\frac{c d (d+e x)^4}{4 e^2} \]

Antiderivative was successfully verified.

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

[Out]

((a - (c*d^2)/e^2)*(d + e*x)^3)/3 + (c*d*(d + e*x)^4)/(4*e^2)

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Rubi in Sympy [A]  time = 19.3631, size = 36, normalized size = 0.92 \[ \frac{c d \left (d + e x\right )^{4}}{4 e^{2}} + \frac{\left (d + e x\right )^{3} \left (a e^{2} - c d^{2}\right )}{3 e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*d*(d + e*x)**4/(4*e**2) + (d + e*x)**3*(a*e**2 - c*d**2)/(3*e**2)

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Mathematica [A]  time = 0.0212942, size = 51, normalized size = 1.31 \[ \frac{1}{12} x \left (4 a e \left (3 d^2+3 d e x+e^2 x^2\right )+c d x \left (6 d^2+8 d e x+3 e^2 x^2\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*(4*a*e*(3*d^2 + 3*d*e*x + e^2*x^2) + c*d*x*(6*d^2 + 8*d*e*x + 3*e^2*x^2)))/12

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Maple [A]  time = 0.001, size = 69, normalized size = 1.8 \[{\frac{d{e}^{2}c{x}^{4}}{4}}+{\frac{ \left ({d}^{2}ec+e \left ( a{e}^{2}+c{d}^{2} \right ) \right ){x}^{3}}{3}}+{\frac{ \left ( d \left ( a{e}^{2}+c{d}^{2} \right ) +ad{e}^{2} \right ){x}^{2}}{2}}+ae{d}^{2}x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*d*e^2*c*x^4+1/3*(d^2*e*c+e*(a*e^2+c*d^2))*x^3+1/2*(d*(a*e^2+c*d^2)+a*d*e^2)*
x^2+a*e*d^2*x

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Maxima [A]  time = 0.715933, size = 73, normalized size = 1.87 \[ \frac{1}{4} \, c d e^{2} x^{4} + a d^{2} e x + \frac{1}{3} \,{\left (2 \, c d^{2} e + a e^{3}\right )} x^{3} + \frac{1}{2} \,{\left (c d^{3} + 2 \, a d e^{2}\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*c*d*e^2*x^4 + a*d^2*e*x + 1/3*(2*c*d^2*e + a*e^3)*x^3 + 1/2*(c*d^3 + 2*a*d*e
^2)*x^2

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Fricas [A]  time = 0.184198, size = 1, normalized size = 0.03 \[ \frac{1}{4} x^{4} e^{2} d c + \frac{2}{3} x^{3} e d^{2} c + \frac{1}{3} x^{3} e^{3} a + \frac{1}{2} x^{2} d^{3} c + x^{2} e^{2} d a + x e d^{2} a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*x^4*e^2*d*c + 2/3*x^3*e*d^2*c + 1/3*x^3*e^3*a + 1/2*x^2*d^3*c + x^2*e^2*d*a
+ x*e*d^2*a

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Sympy [A]  time = 0.109864, size = 56, normalized size = 1.44 \[ a d^{2} e x + \frac{c d e^{2} x^{4}}{4} + x^{3} \left (\frac{a e^{3}}{3} + \frac{2 c d^{2} e}{3}\right ) + x^{2} \left (a d e^{2} + \frac{c d^{3}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*d**2*e*x + c*d*e**2*x**4/4 + x**3*(a*e**3/3 + 2*c*d**2*e/3) + x**2*(a*d*e**2 +
c*d**3/2)

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GIAC/XCAS [A]  time = 0.209509, size = 73, normalized size = 1.87 \[ \frac{1}{4} \, c d x^{4} e^{2} + \frac{2}{3} \, c d^{2} x^{3} e + \frac{1}{2} \, c d^{3} x^{2} + \frac{1}{3} \, a x^{3} e^{3} + a d x^{2} e^{2} + a d^{2} x e \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*c*d*x^4*e^2 + 2/3*c*d^2*x^3*e + 1/2*c*d^3*x^2 + 1/3*a*x^3*e^3 + a*d*x^2*e^2
+ a*d^2*x*e