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 = 0.0111185, 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, Rules used = {45, 37} \[ \frac{b x^{-n-1} (a+b x)^{n+1}}{a^2 (n+1) (n+2)}-\frac{x^{-n-2} (a+b x)^{n+1}}{a (n+2)} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
Rubi steps
\begin{align*} \int x^{2 n-3 (1+n)} (a+b x)^n \, dx &=-\frac{x^{-2-n} (a+b x)^{1+n}}{a (2+n)}-\frac{b \int x^{-2-n} (a+b x)^n \, dx}{a (2+n)}\\ &=-\frac{x^{-2-n} (a+b x)^{1+n}}{a (2+n)}+\frac{b x^{-1-n} (a+b x)^{1+n}}{a^2 (1+n) (2+n)}\\ \end{align*}
Mathematica [A] time = 0.0021056, 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.
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Maple [A] time = 0., size = 41, normalized size = 0.7 \begin{align*} -{\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}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{n} x^{-n - 3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56186, size = 126, normalized size = 2.17 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{n} x^{-n - 3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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