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 = 0.08, antiderivative size = 46, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.05, 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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fricas [A] time = 0.41, size = 76, normalized size = 1.65 \[ \frac {{\left (b x x^{m p + n + 1} x^{n} + a x x^{n}\right )} {\left (b x^{m p + n + 1} x^{m} + a x^{m}\right )}^{p}}{{\left (b m p^{2} + b n + {\left (b m + b n + b\right )} p + b\right )} x^{m p + n + 1}} \]
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 \[ \int \left ({\left (b x^{m p + n + 1} + a\right )} x^{m}\right )^{p} x^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.91, size = 0, normalized size = 0.00 \[ \int x^{n} \left (\left (b \,x^{m p +n +1}+a \right ) x^{m}\right )^{p}\, dx \]
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 \[ \int \left ({\left (b x^{m p + n + 1} + a\right )} x^{m}\right )^{p} x^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^n\,{\left (x^m\,\left (a+b\,x^{n+m\,p+1}\right )\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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