3.187 \(\int \frac{1}{(a+b x)^3} \, dx\)

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

[Out]

-1/(2*b*(a + b*x)^2)

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Rubi [A]  time = 0.00692411, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{1}{2 b (a+b x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(-3),x]

[Out]

-1/(2*b*(a + b*x)^2)

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Rubi in Sympy [A]  time = 1.26706, size = 12, normalized size = 0.86 \[ - \frac{1}{2 b \left (a + b x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**3,x)

[Out]

-1/(2*b*(a + b*x)**2)

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Mathematica [A]  time = 0.00333774, size = 14, normalized size = 1. \[ -\frac{1}{2 b (a+b x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(-3),x]

[Out]

-1/(2*b*(a + b*x)^2)

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Maple [A]  time = 0.002, size = 13, normalized size = 0.9 \[ -{\frac{1}{2\,b \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^3,x)

[Out]

-1/2/b/(b*x+a)^2

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Maxima [A]  time = 1.48088, size = 16, normalized size = 1.14 \[ -\frac{1}{2 \,{\left (b x + a\right )}^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(-3),x, algorithm="maxima")

[Out]

-1/2/((b*x + a)^2*b)

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Fricas [A]  time = 0.206673, size = 32, normalized size = 2.29 \[ -\frac{1}{2 \,{\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(-3),x, algorithm="fricas")

[Out]

-1/2/(b^3*x^2 + 2*a*b^2*x + a^2*b)

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Sympy [A]  time = 1.30741, size = 26, normalized size = 1.86 \[ - \frac{1}{2 a^{2} b + 4 a b^{2} x + 2 b^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)**3,x)

[Out]

-1/(2*a**2*b + 4*a*b**2*x + 2*b**3*x**2)

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GIAC/XCAS [A]  time = 0.22274, size = 16, normalized size = 1.14 \[ -\frac{1}{2 \,{\left (b x + a\right )}^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(-3),x, algorithm="giac")

[Out]

-1/2/((b*x + a)^2*b)