3.810 \(\int x (-a-b x)^{-n} (a+b x)^n \, dx\)

Optimal. Leaf size=26 \[ \frac{1}{2} x^2 (-a-b x)^{-n} (a+b x)^n \]

[Out]

(x^2*(a + b*x)^n)/(2*(-a - b*x)^n)

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Rubi [A]  time = 0.0114058, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{1}{2} x^2 (-a-b x)^{-n} (a+b x)^n \]

Antiderivative was successfully verified.

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

[Out]

(x^2*(a + b*x)^n)/(2*(-a - b*x)^n)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \left (- a - b x\right )^{- n} \left (a + b x\right )^{n} \int x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-a - b*x)**(-n)*(a + b*x)**n*Integral(x, x)

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Mathematica [A]  time = 0.00467303, size = 26, normalized size = 1. \[ \frac{1}{2} x^2 (-a-b x)^{-n} (a+b x)^n \]

Antiderivative was successfully verified.

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

[Out]

(x^2*(a + b*x)^n)/(2*(-a - b*x)^n)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x+a)^n/((-b*x-a)^n),x)

[Out]

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

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Maxima [A]  time = 1.35256, size = 14, normalized size = 0.54 \[ \frac{1}{2} \, \left (-1\right )^{-n} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.23917, size = 12, normalized size = 0.46 \[ \frac{1}{2} \, x^{2} \cos \left (\pi n\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 41.3416, size = 19, normalized size = 0.73 \[ \frac{x^{2} \left (- a - b x\right )^{- n} \left (a + b x\right )^{n}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2*(-a - b*x)**(-n)*(a + b*x)**n/2

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GIAC/XCAS [A]  time = 0.224153, size = 7, normalized size = 0.27 \[ \frac{1}{2} \, x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2