3.981 \(\int \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^3} \, dx\)

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

[Out]

(c^2*(d + e*x)^2)/(2*e)

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Rubi [A]  time = 0.0139772, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{c^2 (d+e x)^2}{2 e} \]

Antiderivative was successfully verified.

[In]  Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x)^3,x]

[Out]

(c^2*(d + e*x)^2)/(2*e)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ c^{2} e \int x\, dx + c^{2} \int d\, dx \]

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)**3,x)

[Out]

c**2*e*Integral(x, x) + c**2*Integral(d, x)

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Mathematica [A]  time = 0.00132441, size = 16, normalized size = 0.94 \[ c^2 \left (d x+\frac{e x^2}{2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x)^3,x]

[Out]

c^2*(d*x + (e*x^2)/2)

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Maple [A]  time = 0.001, size = 15, normalized size = 0.9 \[{c}^{2} \left ({\frac{e{x}^{2}}{2}}+dx \right ) \]

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)^3,x)

[Out]

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

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Maxima [A]  time = 0.697269, size = 22, normalized size = 1.29 \[ \frac{1}{2} \, c^{2} e x^{2} + c^{2} d x \]

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)^3,x, algorithm="maxima")

[Out]

1/2*c^2*e*x^2 + c^2*d*x

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Fricas [A]  time = 0.214892, size = 22, normalized size = 1.29 \[ \frac{1}{2} \, c^{2} e x^{2} + c^{2} d x \]

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)^3,x, algorithm="fricas")

[Out]

1/2*c^2*e*x^2 + c^2*d*x

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Sympy [A]  time = 0.177623, size = 15, normalized size = 0.88 \[ c^{2} d x + \frac{c^{2} e x^{2}}{2} \]

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/(e*x+d)**3,x)

[Out]

c**2*d*x + c**2*e*x**2/2

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GIAC/XCAS [A]  time = 0.211192, size = 31, normalized size = 1.82 \[ \frac{1}{2} \,{\left (c^{2} x^{2} e^{7} + 2 \, c^{2} d x e^{6}\right )} e^{\left (-6\right )} \]

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)^3,x, algorithm="giac")

[Out]

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