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