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

Optimal. Leaf size=22 \[ \log (x) (-a-b x)^{-n} (a+b x)^n \]

[Out]

((a + b*x)^n*Log[x])/(-a - b*x)^n

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Rubi [A]  time = 0.0129456, antiderivative size = 22, 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 \[ \log (x) (-a-b x)^{-n} (a+b x)^n \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^n*Log[x])/(-a - b*x)^n

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Rubi in Sympy [A]  time = 3.7012, size = 17, normalized size = 0.77 \[ \left (- a - b x\right )^{- n} \left (a + b x\right )^{n} \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-a - b*x)**(-n)*(a + b*x)**n*log(x)

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Mathematica [A]  time = 0.00451976, size = 22, normalized size = 1. \[ \log (x) (-a-b x)^{-n} (a+b x)^n \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^n*Log[x])/(-a - b*x)^n

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Maple [C]  time = 0.057, size = 56, normalized size = 2.6 \[ \ln \left ( x \right ) \left ( bx+a \right ) ^{n}{{\rm e}^{-n \left ( i\pi \, \left ({\it csgn} \left ( i \left ( bx+a \right ) \right ) \right ) ^{3}-i\pi \, \left ({\it csgn} \left ( i \left ( bx+a \right ) \right ) \right ) ^{2}+i\pi +\ln \left ( bx+a \right ) \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^n/x/((-b*x-a)^n),x)

[Out]

ln(x)*(b*x+a)^n*exp(-n*(I*Pi*csgn(I*(b*x+a))^3-I*Pi*csgn(I*(b*x+a))^2+I*Pi+ln(b*
x+a)))

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Maxima [A]  time = 1.35871, size = 11, normalized size = 0.5 \[ \left (-1\right )^{-n} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-1)^(-n)*log(x)

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Fricas [A]  time = 0.246739, size = 9, normalized size = 0.41 \[ \cos \left (\pi n\right ) \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

cos(pi*n)*log(x)

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Sympy [A]  time = 98.7377, size = 44, normalized size = 2. \[ \begin{cases} e^{- i \pi n} \log{\left (-1 + \frac{b \left (\frac{a}{b} + x\right )}{a} \right )} & \text{for}\: \left |{\frac{b \left (\frac{a}{b} + x\right )}{a}}\right | > 1 \\e^{- i \pi n} \log{\left (1 - \frac{b \left (\frac{a}{b} + x\right )}{a} \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((exp(-I*pi*n)*log(-1 + b*(a/b + x)/a), Abs(b*(a/b + x)/a) > 1), (exp(-
I*pi*n)*log(1 - b*(a/b + x)/a), True))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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