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

Optimal. Leaf size=64 \[ \frac{b x^{n-1} (a+b x)^{1-n}}{a^2 (1-n) (2-n)}-\frac{x^{n-2} (a+b x)^{1-n}}{a (2-n)} \]

[Out]

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

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Rubi [A]  time = 0.0382108, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{b x^{n-1} (a+b x)^{1-n}}{a^2 (1-n) (2-n)}-\frac{x^{n-2} (a+b x)^{1-n}}{a (2-n)} \]

Antiderivative was successfully verified.

[In]  Int[x^(-3 + n)/(a + b*x)^n,x]

[Out]

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

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Rubi in Sympy [A]  time = 7.21266, size = 42, normalized size = 0.66 \[ - \frac{x^{n - 2} \left (a + b x\right )^{- n + 1}}{a \left (- n + 2\right )} + \frac{b x^{n - 1} \left (a + b x\right )^{- n + 1}}{a^{2} \left (- n + 1\right ) \left (- n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0470845, size = 39, normalized size = 0.61 \[ \frac{x^{n-2} (a+b x)^{1-n} (a (n-1)+b x)}{a^2 (n-2) (n-1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 44, normalized size = 0.7 \[{\frac{{x}^{-2+n} \left ( an+bx-a \right ) \left ( bx+a \right ) }{ \left ( bx+a \right ) ^{n} \left ( -2+n \right ) \left ( -1+n \right ){a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-3+n)/((b*x+a)^n),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.222416, size = 86, normalized size = 1.34 \[ \frac{{\left (a b n x^{2} + b^{2} x^{3} +{\left (a^{2} n - a^{2}\right )} x\right )} x^{n - 3}}{{\left (a^{2} n^{2} - 3 \, a^{2} n + 2 \, a^{2}\right )}{\left (b x + a\right )}^{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a*b*n*x^2 + b^2*x^3 + (a^2*n - a^2)*x)*x^(n - 3)/((a^2*n^2 - 3*a^2*n + 2*a^2)*(
b*x + a)^n)

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^(n - 3)/(b*x + a)^n, x)