3.1162 \(\int \frac{b d+2 c d x}{\left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=15 \[ -\frac{d}{a+b x+c x^2} \]

[Out]

-(d/(a + b*x + c*x^2))

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Rubi [A]  time = 0.0129299, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ -\frac{d}{a+b x+c x^2} \]

Antiderivative was successfully verified.

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

[Out]

-(d/(a + b*x + c*x^2))

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Rubi in Sympy [A]  time = 4.84476, size = 12, normalized size = 0.8 \[ - \frac{d}{a + b x + c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d/(a + b*x + c*x**2)

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Mathematica [A]  time = 0.0072121, size = 14, normalized size = 0.93 \[ -\frac{d}{a+x (b+c x)} \]

Antiderivative was successfully verified.

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

[Out]

-(d/(a + x*(b + c*x)))

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Maple [A]  time = 0.001, size = 16, normalized size = 1.1 \[ -{\frac{d}{c{x}^{2}+bx+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.681004, size = 20, normalized size = 1.33 \[ -\frac{d}{c x^{2} + b x + a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d/(c*x^2 + b*x + a)

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Fricas [A]  time = 0.201956, size = 20, normalized size = 1.33 \[ -\frac{d}{c x^{2} + b x + a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d/(c*x^2 + b*x + a)

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Sympy [A]  time = 2.11874, size = 12, normalized size = 0.8 \[ - \frac{d}{a + b x + c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d/(a + b*x + c*x**2)

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GIAC/XCAS [A]  time = 0.215503, size = 20, normalized size = 1.33 \[ -\frac{d}{c x^{2} + b x + a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d/(c*x^2 + b*x + a)