3.814 \(\int \frac{(-a-b x)^{-n} (a+b x)^n}{x^3} \, dx\)

Optimal. Leaf size=26 \[ -\frac{(-a-b x)^{-n} (a+b x)^n}{2 x^2} \]

[Out]

-(a + b*x)^n/(2*x^2*(-a - b*x)^n)

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Rubi [A]  time = 0.0110525, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ -\frac{(-a-b x)^{-n} (a+b x)^n}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^n/(x^3*(-a - b*x)^n),x]

[Out]

-(a + b*x)^n/(2*x^2*(-a - b*x)^n)

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Rubi in Sympy [A]  time = 3.7417, size = 20, normalized size = 0.77 \[ - \frac{\left (- a - b x\right )^{- n} \left (a + b x\right )^{n}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**n/x**3/((-b*x-a)**n),x)

[Out]

-(-a - b*x)**(-n)*(a + b*x)**n/(2*x**2)

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Mathematica [A]  time = 0.00409258, size = 26, normalized size = 1. \[ -\frac{(-a-b x)^{-n} (a+b x)^n}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^n/(x^3*(-a - b*x)^n),x]

[Out]

-(a + b*x)^n/(2*x^2*(-a - b*x)^n)

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Maple [A]  time = 0.003, size = 25, normalized size = 1. \[ -{\frac{ \left ( bx+a \right ) ^{n}}{2\,{x}^{2} \left ( -bx-a \right ) ^{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^n/x^3/((-b*x-a)^n),x)

[Out]

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

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Maxima [A]  time = 1.35798, size = 14, normalized size = 0.54 \[ -\frac{\left (-1\right )^{-n}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(-1)^(-n)/x^2

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Fricas [A]  time = 0.246726, size = 12, normalized size = 0.46 \[ -\frac{\cos \left (\pi n\right )}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*cos(pi*n)/x^2

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Sympy [A]  time = 47.7544, size = 20, normalized size = 0.77 \[ - \frac{\left (- a - b x\right )^{- n} \left (a + b x\right )^{n}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**n/x**3/((-b*x-a)**n),x)

[Out]

-(-a - b*x)**(-n)*(a + b*x)**n/(2*x**2)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{n}}{{\left (-b x - a\right )}^{n} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)^n/((-b*x - a)^n*x^3), x)