Optimal. Leaf size=13 \[ x^m \left (a+b x^n\right )^p \]
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Rubi [A] time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {449} \begin {gather*} x^m \left (a+b x^n\right )^p \end {gather*}
Antiderivative was successfully verified.
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Rule 449
Rubi steps
\begin {align*} \int x^{-1+m} \left (a+b x^n\right )^{-1+p} \left (a m+b (m+n p) x^n\right ) \, dx &=x^m \left (a+b x^n\right )^p\\ \end {align*}
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Mathematica [C] time = 0.18, size = 107, normalized size = 8.23 \begin {gather*} \frac {x^m \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left (b x^n (m+n p) \, _2F_1\left (\frac {m+n}{n},1-p;\frac {m}{n}+2;-\frac {b x^n}{a}\right )+a (m+n) \, _2F_1\left (\frac {m}{n},1-p;\frac {m+n}{n};-\frac {b x^n}{a}\right )\right )}{a (m+n)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.10, size = 0, normalized size = 0.00 \begin {gather*} \int x^{-1+m} \left (a+b x^n\right )^{-1+p} \left (a m+b (m+n p) x^n\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.43, size = 32, normalized size = 2.46 \begin {gather*} {\left (b x x^{m - 1} x^{n} + a x x^{m - 1}\right )} {\left (b x^{n} + a\right )}^{p - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.28, size = 70, normalized size = 5.38 \begin {gather*} b x x^{n} e^{\left (p \log \left (b x^{n} + a\right ) + m \log \relax (x) - \log \left (b x^{n} + a\right ) - \log \relax (x)\right )} + a x e^{\left (p \log \left (b x^{n} + a\right ) + m \log \relax (x) - \log \left (b x^{n} + a\right ) - \log \relax (x)\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.68, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a m +\left (n p +m \right ) b \,x^{n}\right ) x^{m -1} \left (b \,x^{n}+a \right )^{p -1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.84, size = 16, normalized size = 1.23 \begin {gather*} e^{\left (p \log \left (b x^{n} + a\right ) + m \log \relax (x)\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.81, size = 25, normalized size = 1.92 \begin {gather*} \left (a\,x^m+b\,x^{m+n}\right )\,{\left (a+b\,x^n\right )}^{p-1} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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