3.1803 \(\int \frac{(a+b x)^2}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx\)

Optimal. Leaf size=12 \[ -\frac{1}{d (c+d x)} \]

[Out]

-(1/(d*(c + d*x)))

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Rubi [A]  time = 0.0222987, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -\frac{1}{d (c+d x)} \]

Antiderivative was successfully verified.

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

[Out]

-(1/(d*(c + d*x)))

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Rubi in Sympy [A]  time = 7.15977, size = 8, normalized size = 0.67 \[ - \frac{1}{d \left (c + d x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(d*(c + d*x))

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Mathematica [A]  time = 0.00493894, size = 12, normalized size = 1. \[ -\frac{1}{d (c+d x)} \]

Antiderivative was successfully verified.

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

[Out]

-(1/(d*(c + d*x)))

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Maple [A]  time = 0.001, size = 13, normalized size = 1.1 \[ -{\frac{1}{d \left ( dx+c \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.727956, size = 18, normalized size = 1.5 \[ -\frac{1}{d^{2} x + c d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(d^2*x + c*d)

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Fricas [A]  time = 0.21833, size = 18, normalized size = 1.5 \[ -\frac{1}{d^{2} x + c d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(d^2*x + c*d)

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Sympy [A]  time = 1.23324, size = 10, normalized size = 0.83 \[ - \frac{1}{c d + d^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(c*d + d**2*x)

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GIAC/XCAS [A]  time = 0.210433, size = 16, normalized size = 1.33 \[ -\frac{1}{{\left (d x + c\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/((d*x + c)*d)