Optimal. Leaf size=23 \[ \frac{\left (a+b x^n\right )^{p+1}}{b n (p+1)} \]
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Rubi [A] time = 0.0083898, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {261} \[ \frac{\left (a+b x^n\right )^{p+1}}{b n (p+1)} \]
Antiderivative was successfully verified.
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Rule 261
Rubi steps
\begin{align*} \int x^{-1+n} \left (a+b x^n\right )^p \, dx &=\frac{\left (a+b x^n\right )^{1+p}}{b n (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0066652, size = 23, normalized size = 1. \[ \frac{\left (a+b x^n\right )^{p+1}}{b n (p+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 29, normalized size = 1.3 \begin{align*}{\frac{ \left ( a+b{x}^{n} \right ) \left ( a+b{x}^{n} \right ) ^{p}}{b \left ( 1+p \right ) n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33997, size = 55, normalized size = 2.39 \begin{align*} \frac{{\left (b x^{n} + a\right )}{\left (b x^{n} + a\right )}^{p}}{b n p + b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 132.362, size = 75, normalized size = 3.26 \begin{align*} \begin{cases} \frac{\log{\left (x \right )}}{a} & \text{for}\: b = 0 \wedge n = 0 \wedge p = -1 \\\frac{a^{p} x^{n}}{n} & \text{for}\: b = 0 \\\left (a + b\right )^{p} \log{\left (x \right )} & \text{for}\: n = 0 \\\frac{\log{\left (\frac{a}{b} + x^{n} \right )}}{b n} & \text{for}\: p = -1 \\\frac{a \left (a + b x^{n}\right )^{p}}{b n p + b n} + \frac{b x^{n} \left (a + b x^{n}\right )^{p}}{b n p + b n} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11277, size = 31, normalized size = 1.35 \begin{align*} \frac{{\left (b x^{n} + a\right )}^{p + 1}}{b n{\left (p + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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