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

Optimal. Leaf size=21 \[ x (-a-b x)^{-n} (a+b x)^n \]

[Out]

(x*(a + b*x)^n)/(-a - b*x)^n

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Rubi [A]  time = 0.0100334, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ x (-a-b x)^{-n} (a+b x)^n \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^n/(-a - b*x)^n,x]

[Out]

(x*(a + b*x)^n)/(-a - b*x)^n

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Rubi in Sympy [A]  time = 3.55898, size = 15, normalized size = 0.71 \[ x \left (- a - b x\right )^{- n} \left (a + b x\right )^{n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(-a - b*x)**(-n)*(a + b*x)**n

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Mathematica [A]  time = 0.00311663, size = 21, normalized size = 1. \[ x (-a-b x)^{-n} (a+b x)^n \]

Antiderivative was successfully verified.

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

[Out]

(x*(a + b*x)^n)/(-a - b*x)^n

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Maple [A]  time = 0.033, size = 26, normalized size = 1.2 \[{\frac{x{{\rm e}^{n\ln \left ( bx+a \right ) }}}{{{\rm e}^{n\ln \left ( -bx-a \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^n/((-b*x-a)^n),x)

[Out]

x*exp(n*ln(b*x+a))/exp(n*ln(-b*x-a))

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Maxima [A]  time = 1.35964, size = 9, normalized size = 0.43 \[ \left (-1\right )^{-n} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-1)^(-n)*x

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Fricas [A]  time = 0.237317, size = 8, normalized size = 0.38 \[ x \cos \left (\pi n\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*cos(pi*n)

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Sympy [A]  time = 30.5269, size = 44, normalized size = 2.1 \[ \begin{cases} - \frac{a \left (- a - b x\right )^{- n} \left (a + b x\right )^{n}}{b} + x \left (- a - b x\right )^{- n} \left (a + b x\right )^{n} & \text{for}\: b \neq 0 \\a^{n} x \left (- a\right )^{- n} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-a*(-a - b*x)**(-n)*(a + b*x)**n/b + x*(-a - b*x)**(-n)*(a + b*x)**n,
Ne(b, 0)), (a**n*x*(-a)**(-n), True))

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GIAC/XCAS [A]  time = 0.215549, size = 1, normalized size = 0.05 \[ x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x