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

Optimal. Leaf size=15 \[ \frac{c (d+e x)^4}{4 e} \]

[Out]

(c*(d + e*x)^4)/(4*e)

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Rubi [A]  time = 0.0150821, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{c (d+e x)^4}{4 e} \]

Antiderivative was successfully verified.

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

[Out]

(c*(d + e*x)^4)/(4*e)

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Rubi in Sympy [A]  time = 14.4362, size = 10, normalized size = 0.67 \[ \frac{c \left (d + e x\right )^{4}}{4 e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*(d + e*x)**4/(4*e)

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Mathematica [A]  time = 0.00331726, size = 15, normalized size = 1. \[ \frac{c (d+e x)^4}{4 e} \]

Antiderivative was successfully verified.

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

[Out]

(c*(d + e*x)^4)/(4*e)

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Maple [B]  time = 0., size = 36, normalized size = 2.4 \[{\frac{c{e}^{3}{x}^{4}}{4}}+dc{e}^{2}{x}^{3}+{\frac{3\,{d}^{2}ec{x}^{2}}{2}}+c{d}^{3}x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.697588, size = 41, normalized size = 2.73 \[ \frac{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{2}}{4 \, c e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.194046, size = 1, normalized size = 0.07 \[ \frac{1}{4} x^{4} e^{3} c + x^{3} e^{2} d c + \frac{3}{2} x^{2} e d^{2} c + x d^{3} c \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.094103, size = 39, normalized size = 2.6 \[ c d^{3} x + \frac{3 c d^{2} e x^{2}}{2} + c d e^{2} x^{3} + \frac{c e^{3} x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*d**3*x + 3*c*d**2*e*x**2/2 + c*d*e**2*x**3 + c*e**3*x**4/4

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GIAC/XCAS [A]  time = 0.207413, size = 46, normalized size = 3.07 \[ \frac{1}{4} \, c x^{4} e^{3} + c d x^{3} e^{2} + \frac{3}{2} \, c d^{2} x^{2} e + c d^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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