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

Optimal. Leaf size=17 \[ \frac{c^2 (d+e x)^5}{5 e} \]

[Out]

(c^2*(d + e*x)^5)/(5*e)

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Rubi [A]  time = 0.0158964, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{c^2 (d+e x)^5}{5 e} \]

Antiderivative was successfully verified.

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

[Out]

(c^2*(d + e*x)^5)/(5*e)

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Rubi in Sympy [A]  time = 2.68926, size = 34, normalized size = 2. \[ \frac{\left (2 d + 2 e x\right ) \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{2}}{10 e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2*d + 2*e*x)*(c*d**2 + 2*c*d*e*x + c*e**2*x**2)**2/(10*e)

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Mathematica [A]  time = 0.00200149, size = 17, normalized size = 1. \[ \frac{c^2 (d+e x)^5}{5 e} \]

Antiderivative was successfully verified.

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

[Out]

(c^2*(d + e*x)^5)/(5*e)

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Maple [B]  time = 0.002, size = 58, normalized size = 3.4 \[{\frac{{x}^{5}{c}^{2}{e}^{4}}{5}}+{x}^{4}{c}^{2}d{e}^{3}+2\,{x}^{3}{c}^{2}{d}^{2}{e}^{2}+2\,{x}^{2}{c}^{2}{d}^{3}e+{c}^{2}{d}^{4}x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*x^5*c^2*e^4+x^4*c^2*d*e^3+2*x^3*c^2*d^2*e^2+2*x^2*c^2*d^3*e+c^2*d^4*x

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Maxima [A]  time = 0.700853, size = 92, normalized size = 5.41 \[ \frac{1}{5} \, c^{2} e^{4} x^{5} + c^{2} d e^{3} x^{4} + \frac{4}{3} \, c^{2} d^{2} e^{2} x^{3} + c^{2} d^{4} x + \frac{2}{3} \,{\left (c e^{2} x^{3} + 3 \, c d e x^{2}\right )} c d^{2} \]

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)^2,x, algorithm="maxima")

[Out]

1/5*c^2*e^4*x^5 + c^2*d*e^3*x^4 + 4/3*c^2*d^2*e^2*x^3 + c^2*d^4*x + 2/3*(c*e^2*x
^3 + 3*c*d*e*x^2)*c*d^2

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Fricas [A]  time = 0.191481, size = 1, normalized size = 0.06 \[ \frac{1}{5} x^{5} e^{4} c^{2} + x^{4} e^{3} d c^{2} + 2 x^{3} e^{2} d^{2} c^{2} + 2 x^{2} e d^{3} c^{2} + x d^{4} c^{2} \]

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)^2,x, algorithm="fricas")

[Out]

1/5*x^5*e^4*c^2 + x^4*e^3*d*c^2 + 2*x^3*e^2*d^2*c^2 + 2*x^2*e*d^3*c^2 + x*d^4*c^
2

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Sympy [A]  time = 0.116081, size = 60, normalized size = 3.53 \[ c^{2} d^{4} x + 2 c^{2} d^{3} e x^{2} + 2 c^{2} d^{2} e^{2} x^{3} + c^{2} d e^{3} x^{4} + \frac{c^{2} e^{4} x^{5}}{5} \]

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)**2,x)

[Out]

c**2*d**4*x + 2*c**2*d**3*e*x**2 + 2*c**2*d**2*e**2*x**3 + c**2*d*e**3*x**4 + c*
*2*e**4*x**5/5

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GIAC/XCAS [A]  time = 0.208906, size = 74, normalized size = 4.35 \[ \frac{1}{5} \, c^{2} x^{5} e^{4} + c^{2} d x^{4} e^{3} + 2 \, c^{2} d^{2} x^{3} e^{2} + 2 \, c^{2} d^{3} x^{2} e + c^{2} d^{4} x \]

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)^2,x, algorithm="giac")

[Out]

1/5*c^2*x^5*e^4 + c^2*d*x^4*e^3 + 2*c^2*d^2*x^3*e^2 + 2*c^2*d^3*x^2*e + c^2*d^4*
x