3.755 \(\int x^{2 n-3 (1+n)} (a+b x)^n \, dx\)

Optimal. Leaf size=58 \[ \frac{b x^{-n-1} (a+b x)^{n+1}}{a^2 (n+1) (n+2)}-\frac{x^{-n-2} (a+b x)^{n+1}}{a (n+2)} \]

[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 = 0.0398184, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{b x^{-n-1} (a+b x)^{n+1}}{a^2 (n+1) (n+2)}-\frac{x^{-n-2} (a+b x)^{n+1}}{a (n+2)} \]

Antiderivative was successfully verified.

[In]  Int[x^(2*n - 3*(1 + 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 = 6.96497, size = 46, normalized size = 0.79 \[ - \frac{x^{- n - 2} \left (a + b x\right )^{n + 1}}{a \left (n + 2\right )} + \frac{b x^{- n - 1} \left (a + b x\right )^{n + 1}}{a^{2} \left (n + 1\right ) \left (n + 2\right )} \]

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 = 0.026372, size = 40, normalized size = 0.69 \[ -\frac{x^{-n-2} (a n+a-b x) (a+b x)^{n+1}}{a^2 (n+1) (n+2)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(2*n - 3*(1 + n))*(a + b*x)^n,x]

[Out]

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

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Maple [A]  time = 0., size = 41, normalized size = 0.7 \[ -{\frac{ \left ( bx+a \right ) ^{1+n}{x}^{-2-n} \left ( an-bx+a \right ) }{ \left ( 2+n \right ) \left ( 1+n \right ){a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-3-n)*(b*x+a)^n,x)

[Out]

-(b*x+a)^(1+n)*x^(-2-n)*(a*n-b*x+a)/(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((b*x + a)^n*x^(-n - 3),x, algorithm="maxima")

[Out]

integrate((b*x + a)^n*x^(-n - 3), x)

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Fricas [A]  time = 0.220853, size = 86, normalized size = 1.48 \[ -\frac{{\left (a b n x^{2} - b^{2} x^{3} +{\left (a^{2} n + a^{2}\right )} x\right )}{\left (b x + a\right )}^{n} x^{-n - 3}}{a^{2} n^{2} + 3 \, a^{2} n + 2 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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. \[ \int{\left (b x + a\right )}^{n} x^{-n - 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)^n*x^(-n - 3), x)