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

Optimal. Leaf size=34 \[ \frac{x^{1-m} (-a-b x)^{-n} (a+b x)^n}{1-m} \]

[Out]

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

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Rubi [A]  time = 0.0148539, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ \frac{x^{1-m} (-a-b x)^{-n} (a+b x)^n}{1-m} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.50403, size = 22, normalized size = 0.65 \[ \frac{x^{- m + 1} \left (- a - b x\right )^{- n} \left (a + b x\right )^{n}}{- m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**(-m + 1)*(-a - b*x)**(-n)*(a + b*x)**n/(-m + 1)

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Mathematica [A]  time = 0.010617, size = 34, normalized size = 1. \[ \frac{x^{1-m} (-a-b x)^{-n} (a+b x)^n}{1-m} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 33, normalized size = 1. \[ -{\frac{x \left ( bx+a \right ) ^{n}}{ \left ( -1+m \right ){x}^{m} \left ( -bx-a \right ) ^{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.37182, size = 28, normalized size = 0.82 \[ -\frac{x x^{-m}}{\left (-1\right )^{n} m - \left (-1\right )^{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x*x^(-m)/((-1)^n*m - (-1)^n)

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Fricas [A]  time = 0.256507, size = 23, normalized size = 0.68 \[ -\frac{x \cos \left (\pi n\right )}{{\left (m - 1\right )} x^{m}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x*cos(pi*n)/((m - 1)*x^m)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.287506, size = 1, normalized size = 0.03 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done