Optimal. Leaf size=46 \[ \frac{x^{-m (p+1)} \left (a x^m+b x^{m p+m+n+1}\right )^{p+1}}{b (p+1) (m p+n+1)} \]
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Rubi [A] time = 0.0753067, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1980, 2014} \[ \frac{x^{-m (p+1)} \left (a x^m+b x^{m p+m+n+1}\right )^{p+1}}{b (p+1) (m p+n+1)} \]
Antiderivative was successfully verified.
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Rule 1980
Rule 2014
Rubi steps
\begin{align*} \int x^n \left (x^m \left (a+b x^{1+n+m p}\right )\right )^p \, dx &=\int x^n \left (a x^m+b x^{1+m+n+m p}\right )^p \, dx\\ &=\frac{x^{-m (1+p)} \left (a x^m+b x^{1+m+n+m p}\right )^{1+p}}{b (1+p) (1+n+m p)}\\ \end{align*}
Mathematica [A] time = 0.041979, size = 45, normalized size = 0.98 \[ \frac{x^{-m (p+1)} \left (x^m \left (a+b x^{m p+n+1}\right )\right )^{p+1}}{b (p+1) (m p+n+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.542, size = 0, normalized size = 0. \begin{align*} \int{x}^{n} \left ({x}^{m} \left ( a+b{x}^{mp+n+1} \right ) \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 ({\left (b x^{m p + n + 1} + a\right )} x^{m}\right )^{p} x^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.767607, size = 174, normalized size = 3.78 \begin{align*} \frac{{\left (b x x^{m p + n + 1} x^{n} + a x x^{n}\right )}{\left (b x^{m p + n + 1} x^{m} + a x^{m}\right )}^{p}}{{\left (b m p^{2} + b n +{\left (b m + b n + b\right )} p + b\right )} x^{m p + n + 1}} \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 ({\left (b x^{m p + n + 1} + a\right )} x^{m}\right )^{p} x^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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