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 = 0.02, antiderivative size = 44, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {x^{-m (p+1)} \left (a x^m+b x^{m p+m+1}\right )^{p+1}}{b (p+1) (m p+1)} \end {gather*}
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*}
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Mathematica [A] time = 0.04, size = 43, normalized size = 0.98 \begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.08, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a x^m+b x^{1+m+m p}\right )^p \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.65, size = 64, normalized size = 1.45 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (b x^{m p + m + 1} + a x^{m}\right )}^{p}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.86, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a \,x^{m}+b \,x^{m p +m +1}\right )^{p}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (b x^{m p + m + 1} + a x^{m}\right )}^{p}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.30, size = 76, normalized size = 1.73 \begin {gather*} \frac {a\,{\left (a\,x^m+b\,x^{m+m\,p+1}\right )}^p\,\left (\frac {b\,x^{m\,p+1}}{a}-\frac {1}{{\left (\frac {b\,x^{m\,p+1}}{a}+1\right )}^p}+1\right )}{b\,x^{m\,p}\,\left (m\,p+1\right )\,\left (p+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a x^{m} + b x^{m p + m + 1}\right )^{p}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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