3.1891 \(\int \frac{(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx\)

Optimal. Leaf size=20 \[ -\frac{1}{3 c d (a e+c d x)^3} \]

[Out]

-1/(3*c*d*(a*e + c*d*x)^3)

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Rubi [A]  time = 0.0278788, 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 \[ -\frac{1}{3 c d (a e+c d x)^3} \]

Antiderivative was successfully verified.

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

[Out]

-1/(3*c*d*(a*e + c*d*x)^3)

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Rubi in Sympy [A]  time = 10.2929, size = 17, normalized size = 0.85 \[ - \frac{1}{3 c d \left (a e + c d x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(3*c*d*(a*e + c*d*x)**3)

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Mathematica [A]  time = 0.00748696, size = 20, normalized size = 1. \[ -\frac{1}{3 c d (a e+c d x)^3} \]

Antiderivative was successfully verified.

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

[Out]

-1/(3*c*d*(a*e + c*d*x)^3)

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Maple [A]  time = 0.002, size = 19, normalized size = 1. \[ -{\frac{1}{3\,cd \left ( cdx+ae \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.732468, size = 70, normalized size = 3.5 \[ -\frac{1}{3 \,{\left (c^{4} d^{4} x^{3} + 3 \, a c^{3} d^{3} e x^{2} + 3 \, a^{2} c^{2} d^{2} e^{2} x + a^{3} c d e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.195856, size = 70, normalized size = 3.5 \[ -\frac{1}{3 \,{\left (c^{4} d^{4} x^{3} + 3 \, a c^{3} d^{3} e x^{2} + 3 \, a^{2} c^{2} d^{2} e^{2} x + a^{3} c d e^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.05297, size = 58, normalized size = 2.9 \[ - \frac{1}{3 a^{3} c d e^{3} + 9 a^{2} c^{2} d^{2} e^{2} x + 9 a c^{3} d^{3} e x^{2} + 3 c^{4} d^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(3*a**3*c*d*e**3 + 9*a**2*c**2*d**2*e**2*x + 9*a*c**3*d**3*e*x**2 + 3*c**4*d*
*4*x**3)

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GIAC/XCAS [A]  time = 0.251277, size = 1, normalized size = 0.05 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done