Optimal. Leaf size=46 \[ \frac {x^{-m (1+p)} \left (a x^m+b x^{1+m+n+m p}\right )^{1+p}}{b (1+p) (1+n+m p)} \]
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Rubi [A]
time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2039}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2039
Rubi steps
\begin {align*} \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 = 0.00, size = 45, normalized size = 0.98 \begin {gather*} \frac {x^{-m (1+p)} \left (x^m \left (a+b x^{1+n+m p}\right )\right )^{1+p}}{b (1+p) (1+n+m p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.21, size = 0, normalized size = 0.00 \[\int x^{n} \left (a \,x^{m}+b \,x^{m p +m +n +1}\right )^{p}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.45, size = 79, normalized size = 1.72 \begin {gather*} \frac {{\left (b x x^{m p + m + n + 1} x^{n} + a x x^{m} x^{n}\right )} {\left (b x^{m p + m + n + 1} + 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 + m + n + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^n\,{\left (a\,x^m+b\,x^{m+n+m\,p+1}\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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