3.1869 \(\int \frac{(d+e x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx\)

Optimal. Leaf size=18 \[ -\frac{1}{c d (a e+c d x)} \]

[Out]

-(1/(c*d*(a*e + c*d*x)))

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Rubi [A]  time = 0.0264392, antiderivative size = 18, 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 \[ -\frac{1}{c d (a e+c d x)} \]

Antiderivative was successfully verified.

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

[Out]

-(1/(c*d*(a*e + c*d*x)))

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Rubi in Sympy [A]  time = 10.1667, size = 14, normalized size = 0.78 \[ - \frac{1}{c d \left (a e + c d x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(c*d*(a*e + c*d*x))

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Mathematica [A]  time = 0.00643934, size = 18, normalized size = 1. \[ -\frac{1}{c d (a e+c d x)} \]

Antiderivative was successfully verified.

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

[Out]

-(1/(c*d*(a*e + c*d*x)))

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Maple [A]  time = 0.002, size = 19, normalized size = 1.1 \[ -{\frac{1}{cd \left ( cdx+ae \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.720574, size = 24, normalized size = 1.33 \[ -\frac{1}{c^{2} d^{2} x + a c d e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(c^2*d^2*x + a*c*d*e)

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Fricas [A]  time = 0.196037, size = 24, normalized size = 1.33 \[ -\frac{1}{c^{2} d^{2} x + a c d e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(c^2*d^2*x + a*c*d*e)

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Sympy [A]  time = 1.34224, size = 17, normalized size = 0.94 \[ - \frac{1}{a c d e + c^{2} d^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(a*c*d*e + c**2*d**2*x)

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GIAC/XCAS [A]  time = 0.219491, size = 147, normalized size = 8.17 \[ -\frac{c^{2} d^{4} x e + c^{2} d^{5} - 2 \, a c d^{2} x e^{3} - 2 \, a c d^{3} e^{2} + a^{2} x e^{5} + a^{2} d e^{4}}{{\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )}{\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(c^2*d^4*x*e + c^2*d^5 - 2*a*c*d^2*x*e^3 - 2*a*c*d^3*e^2 + a^2*x*e^5 + a^2*d*e^
4)/((c^3*d^5 - 2*a*c^2*d^3*e^2 + a^2*c*d*e^4)*(c*d*x^2*e + c*d^2*x + a*x*e^2 + a
*d*e))