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

Optimal. Leaf size=17 \[ \frac{c^2 (d+e x)^3}{3 e} \]

[Out]

(c^2*(d + e*x)^3)/(3*e)

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Rubi [A]  time = 0.0179194, 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 \[ \frac{c^2 (d+e x)^3}{3 e} \]

Antiderivative was successfully verified.

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

[Out]

(c^2*(d + e*x)^3)/(3*e)

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Rubi in Sympy [A]  time = 18.025, size = 12, normalized size = 0.71 \[ \frac{c^{2} \left (d + e x\right )^{3}}{3 e} \]

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)**2,x)

[Out]

c**2*(d + e*x)**3/(3*e)

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Mathematica [A]  time = 0.00305008, size = 17, normalized size = 1. \[ \frac{c^2 (d+e x)^3}{3 e} \]

Antiderivative was successfully verified.

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

[Out]

(c^2*(d + e*x)^3)/(3*e)

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Maple [A]  time = 0.001, size = 16, normalized size = 0.9 \[{\frac{{c}^{2} \left ( ex+d \right ) ^{3}}{3\,e}} \]

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)^2,x)

[Out]

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

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Maxima [A]  time = 0.694697, size = 39, normalized size = 2.29 \[ \frac{1}{3} \, c^{2} e^{2} x^{3} + c^{2} d e x^{2} + c^{2} d^{2} x \]

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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.201225, size = 39, normalized size = 2.29 \[ \frac{1}{3} \, c^{2} e^{2} x^{3} + c^{2} d e x^{2} + c^{2} d^{2} x \]

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)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.176458, size = 29, normalized size = 1.71 \[ c^{2} d^{2} x + c^{2} d e x^{2} + \frac{c^{2} e^{2} x^{3}}{3} \]

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)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.211763, size = 20, normalized size = 1.18 \[ \frac{1}{3} \,{\left (x e + d\right )}^{3} c^{2} e^{\left (-1\right )} \]

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)^2,x, algorithm="giac")

[Out]

1/3*(x*e + d)^3*c^2*e^(-1)