Optimal. Leaf size=44 \[ \frac{x^{-m (p+1)} \left (a x^m+b x^{m p+m+1}\right )^{p+1}}{b (p+1) (m p+1)} \]
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Rubi [A] time = 0.0154702, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {2000} \[ \frac{x^{-m (p+1)} \left (a x^m+b x^{m p+m+1}\right )^{p+1}}{b (p+1) (m p+1)} \]
Antiderivative was successfully verified.
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Rule 2000
Rubi steps
\begin{align*} \int \left (a x^m+b x^{1+m+m p}\right )^p \, dx &=\frac{x^{-m (1+p)} \left (a x^m+b x^{1+m+m p}\right )^{1+p}}{b (1+p) (1+m p)}\\ \end{align*}
Mathematica [A] time = 0.0350637, size = 43, normalized size = 0.98 \[ \frac{x^{-m (p+1)} \left (x^m \left (a+b x^{m p+1}\right )\right )^{p+1}}{b (p+1) (m p+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.547, size = 0, normalized size = 0. \begin{align*} \int \left ( a{x}^{m}+b{x}^{mp+m+1} \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^{m p + m + 1} + a x^{m}\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.754974, size = 147, normalized size = 3.34 \begin{align*} \frac{{\left (b x x^{m p + m + 1} + a x x^{m}\right )}{\left (b x^{m p + m + 1} + a x^{m}\right )}^{p}}{{\left (b m p^{2} +{\left (b m + b\right )} p + b\right )} x^{m p + m + 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 (b x^{m p + m + 1} + a x^{m}\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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