Optimal. Leaf size=57 \[ \frac {p x^{(1-n) (1+p)} \left (a x^{-((1-n) p)}+b x^{n-(1-n) p}\right )^{1+\frac {1}{p}}}{b n (1+p)} \]
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Rubi [A]
time = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2004, 2025}
\begin {gather*} \frac {p x^{(1-n) (p+1)} \left (a x^{-((1-n) p)}+b x^{n-(1-n) p}\right )^{\frac {1}{p}+1}}{b n (p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2004
Rule 2025
Rubi steps
\begin {align*} \int \left (x^{(-1+n) p} \left (a+b x^n\right )\right )^{\frac {1}{p}} \, dx &=\int \left (a x^{(-1+n) p}+b x^{n+(-1+n) p}\right )^{\frac {1}{p}} \, dx\\ &=\frac {p x^{(1-n) (1+p)} \left (a x^{-(1-n) p}+b x^{n-(1-n) p}\right )^{1+\frac {1}{p}}}{b n (1+p)}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 47, normalized size = 0.82 \begin {gather*} \frac {x^{1-n} \left (a+b x^n\right ) \left (x^{(-1+n) p} \left (a+b x^n\right )\right )^{\frac {1}{p}}}{b n \left (1+\frac {1}{p}\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \left (x^{\left (-1+n \right ) p} \left (a +b \,x^{n}\right )\right )^{\frac {1}{p}}\, dx\]
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 = 2.85, size = 47, normalized size = 0.82 \begin {gather*} \frac {{\left (b p x x^{n} + a p x\right )} \left ({\left (b x^{n} + a\right )} x^{{\left (n - 1\right )} p}\right )^{\left (\frac {1}{p}\right )}}{{\left (b n p + b n\right )} x^{n}} \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^{p \left (n - 1\right )} \left (a + b x^{n}\right )\right )^{\frac {1}{p}}\, dx \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 {\left (x^{p\,\left (n-1\right )}\,\left (a+b\,x^n\right )\right )}^{1/p} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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