3.988 \(\int \frac{(d+e x)^4}{c d^2+2 c d e x+c e^2 x^2} \, dx\)

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

[Out]

(d + e*x)^3/(3*c*e)

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Rubi [A]  time = 0.0164337, 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 \[ \frac{(d+e x)^3}{3 c e} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^4/(c*d^2 + 2*c*d*e*x + c*e^2*x^2),x]

[Out]

(d + e*x)^3/(3*c*e)

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Rubi in Sympy [A]  time = 17.9555, size = 10, normalized size = 0.59 \[ \frac{\left (d + e x\right )^{3}}{3 c e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**4/(c*e**2*x**2+2*c*d*e*x+c*d**2),x)

[Out]

(d + e*x)**3/(3*c*e)

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Mathematica [A]  time = 0.00194518, size = 17, normalized size = 1. \[ \frac{(d+e x)^3}{3 c e} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^4/(c*d^2 + 2*c*d*e*x + c*e^2*x^2),x]

[Out]

(d + e*x)^3/(3*c*e)

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Maple [A]  time = 0.002, size = 16, normalized size = 0.9 \[{\frac{ \left ( ex+d \right ) ^{3}}{3\,ce}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^4/(c*e^2*x^2+2*c*d*e*x+c*d^2),x)

[Out]

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

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Maxima [A]  time = 0.697081, size = 35, normalized size = 2.06 \[ \frac{e^{2} x^{3} + 3 \, d e x^{2} + 3 \, d^{2} x}{3 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/(c*e^2*x^2 + 2*c*d*e*x + c*d^2),x, algorithm="maxima")

[Out]

1/3*(e^2*x^3 + 3*d*e*x^2 + 3*d^2*x)/c

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Fricas [A]  time = 0.20507, size = 35, normalized size = 2.06 \[ \frac{e^{2} x^{3} + 3 \, d e x^{2} + 3 \, d^{2} x}{3 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/(c*e^2*x^2 + 2*c*d*e*x + c*d^2),x, algorithm="fricas")

[Out]

1/3*(e^2*x^3 + 3*d*e*x^2 + 3*d^2*x)/c

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Sympy [A]  time = 0.169475, size = 24, normalized size = 1.41 \[ \frac{d^{2} x}{c} + \frac{d e x^{2}}{c} + \frac{e^{2} x^{3}}{3 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**4/(c*e**2*x**2+2*c*d*e*x+c*d**2),x)

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/(c*e^2*x^2 + 2*c*d*e*x + c*d^2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError