3.1880 \(\int \frac{(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx\)

Optimal. Leaf size=20 \[ -\frac{1}{2 c d (a e+c d x)^2} \]

[Out]

-1/(2*c*d*(a*e + c*d*x)^2)

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Rubi [A]  time = 0.027724, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057 \[ -\frac{1}{2 c d (a e+c d x)^2} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*c*d*(a*e + c*d*x)^2)

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Rubi in Sympy [A]  time = 10.2334, size = 17, normalized size = 0.85 \[ - \frac{1}{2 c d \left (a e + c d x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

-1/(2*c*d*(a*e + c*d*x)**2)

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Mathematica [A]  time = 0.00612255, size = 20, normalized size = 1. \[ -\frac{1}{2 c d (a e+c d x)^2} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*c*d*(a*e + c*d*x)^2)

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Maple [A]  time = 0., size = 19, normalized size = 1. \[ -{\frac{1}{2\,cd \left ( cdx+ae \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.727916, size = 47, normalized size = 2.35 \[ -\frac{1}{2 \,{\left (c^{3} d^{3} x^{2} + 2 \, a c^{2} d^{2} e x + a^{2} c d e^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.196928, size = 47, normalized size = 2.35 \[ -\frac{1}{2 \,{\left (c^{3} d^{3} x^{2} + 2 \, a c^{2} d^{2} e x + a^{2} c d e^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.64128, size = 39, normalized size = 1.95 \[ - \frac{1}{2 a^{2} c d e^{2} + 4 a c^{2} d^{2} e x + 2 c^{3} d^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.252798, size = 1, normalized size = 0.05 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="giac")

[Out]

Done