3.1812 \(\int \frac{(a+b x)^3}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx\)

Optimal. Leaf size=14 \[ -\frac{1}{2 d (c+d x)^2} \]

[Out]

-1/(2*d*(c + d*x)^2)

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Rubi [A]  time = 0.0223294, antiderivative size = 14, 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 \[ -\frac{1}{2 d (c+d x)^2} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*d*(c + d*x)^2)

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Rubi in Sympy [A]  time = 7.31774, size = 12, normalized size = 0.86 \[ - \frac{1}{2 d \left (c + d x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(2*d*(c + d*x)**2)

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Mathematica [A]  time = 0.0056845, size = 14, normalized size = 1. \[ -\frac{1}{2 d (c+d x)^2} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*d*(c + d*x)^2)

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Maple [A]  time = 0.001, size = 13, normalized size = 0.9 \[ -{\frac{1}{2\,d \left ( dx+c \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.785846, size = 32, normalized size = 2.29 \[ -\frac{1}{2 \,{\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/(d^3*x^2 + 2*c*d^2*x + c^2*d)

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Fricas [A]  time = 0.192536, size = 32, normalized size = 2.29 \[ -\frac{1}{2 \,{\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/(d^3*x^2 + 2*c*d^2*x + c^2*d)

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Sympy [A]  time = 1.46925, size = 26, normalized size = 1.86 \[ - \frac{1}{2 c^{2} d + 4 c d^{2} x + 2 d^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.214112, size = 16, normalized size = 1.14 \[ -\frac{1}{2 \,{\left (d x + c\right )}^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/((d*x + c)^2*d)