Optimal. Leaf size=32 \[ -\frac{x^{-n (p+1)} \left (a+b x^n\right )^{p+1}}{a n (p+1)} \]
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Rubi [A] time = 0.0092007, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {264} \[ -\frac{x^{-n (p+1)} \left (a+b x^n\right )^{p+1}}{a n (p+1)} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin{align*} \int x^{-1-n-n p} \left (a+b x^n\right )^p \, dx &=-\frac{x^{-n (1+p)} \left (a+b x^n\right )^{1+p}}{a n (1+p)}\\ \end{align*}
Mathematica [A] time = 0.019122, size = 32, normalized size = 1. \[ -\frac{x^{-n (p+1)} \left (a+b x^n\right )^{p+1}}{a n (p+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.079, size = 0, normalized size = 0. \begin{align*} \int{x}^{-np-n-1} \left ( a+b{x}^{n} \right ) ^{p}\, dx \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^{n} + a\right )}^{p} x^{-n p - n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.3014, size = 108, normalized size = 3.38 \begin{align*} -\frac{{\left (b x x^{-n p - n - 1} x^{n} + a x x^{-n p - n - 1}\right )}{\left (b x^{n} + a\right )}^{p}}{a n p + a n} \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^{n} + a\right )}^{p} x^{-n p - n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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