3.979 \(\int \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{d+e x} \, dx\)

Optimal. Leaf size=17 \[ \frac{c^2 (d+e x)^4}{4 e} \]

[Out]

(c^2*(d + e*x)^4)/(4*e)

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Rubi [A]  time = 0.0169869, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{c^2 (d+e x)^4}{4 e} \]

Antiderivative was successfully verified.

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

[Out]

(c^2*(d + e*x)^4)/(4*e)

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Rubi in Sympy [A]  time = 18.1288, size = 12, normalized size = 0.71 \[ \frac{c^{2} \left (d + e x\right )^{4}}{4 e} \]

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/(e*x+d),x)

[Out]

c**2*(d + e*x)**4/(4*e)

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Mathematica [A]  time = 0.00291473, size = 17, normalized size = 1. \[ \frac{c^2 (d+e x)^4}{4 e} \]

Antiderivative was successfully verified.

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

[Out]

(c^2*(d + e*x)^4)/(4*e)

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Maple [B]  time = 0.002, size = 36, normalized size = 2.1 \[{c}^{2} \left ({\frac{{e}^{3}{x}^{4}}{4}}+d{e}^{2}{x}^{3}+{\frac{3\,{d}^{2}e{x}^{2}}{2}}+{d}^{3}x \right ) \]

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/(e*x+d),x)

[Out]

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

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Maxima [A]  time = 0.704348, size = 58, normalized size = 3.41 \[ \frac{1}{4} \, c^{2} e^{3} x^{4} + c^{2} d e^{2} x^{3} + \frac{3}{2} \, c^{2} d^{2} e x^{2} + c^{2} d^{3} x \]

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)^2/(e*x + d),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.202643, size = 58, normalized size = 3.41 \[ \frac{1}{4} \, c^{2} e^{3} x^{4} + c^{2} d e^{2} x^{3} + \frac{3}{2} \, c^{2} d^{2} e x^{2} + c^{2} d^{3} x \]

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)^2/(e*x + d),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.166407, size = 46, normalized size = 2.71 \[ c^{2} d^{3} x + \frac{3 c^{2} d^{2} e x^{2}}{2} + c^{2} d e^{2} x^{3} + \frac{c^{2} e^{3} x^{4}}{4} \]

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/(e*x+d),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.209816, size = 66, normalized size = 3.88 \[ \frac{1}{4} \,{\left (c^{2} x^{4} e^{7} + 4 \, c^{2} d x^{3} e^{6} + 6 \, c^{2} d^{2} x^{2} e^{5} + 4 \, c^{2} d^{3} x e^{4}\right )} e^{\left (-4\right )} \]

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/(e*x + d),x, algorithm="giac")

[Out]

1/4*(c^2*x^4*e^7 + 4*c^2*d*x^3*e^6 + 6*c^2*d^2*x^2*e^5 + 4*c^2*d^3*x*e^4)*e^(-4)