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 = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1979, 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 1979
Rule 2000
Rubi steps
\begin {align*} \int \left (x^m \left (a+b x^{1+m p}\right )\right )^p \, dx &=\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 = 0.00, 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 = 0.02, size = 0, normalized size = 0.00 \begin {gather*} \int \left (x^m \left (a+b x^{1+m p}\right )\right )^p \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 61, normalized size = 1.39 \begin {gather*} \frac {{\left (b x x^{m p + 1} + a x\right )} {\left (b x^{m p + 1} x^{m} + a x^{m}\right )}^{p}}{{\left (b m p^{2} + {\left (b m + b\right )} p + b\right )} x^{m p + 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 ({\left (b x^{m p + 1} + a\right )} 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 = 0.85, size = 0, normalized size = 0.00 \begin {gather*} \int \left (\left (b \,x^{m p +1}+a \right ) x^{m}\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 ({\left (b x^{m p + 1} + a\right )} x^{m}\right )^{p}\,{d x} \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 {\left (x^m\,\left (a+b\,x^{m\,p+1}\right )\right )}^p \,d x \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 (x^{m} \left (a + b x^{m p + 1}\right )\right )^{p}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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