Optimal. Leaf size=38 \[ \frac{x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+1}}{b (p+1)} \]
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Rubi [A] time = 0.0120628, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {254, 32} \[ \frac{x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+1}}{b (p+1)} \]
Antiderivative was successfully verified.
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Rule 254
Rule 32
Rubi steps
\begin{align*} \int \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^p \, dx &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int (a+b x)^p \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=\frac{x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{1+p}}{b (1+p)}\\ \end{align*}
Mathematica [A] time = 0.131238, size = 64, normalized size = 1.68 \[ \frac{x \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^p \left (\frac{a \left (c x^n\right )^{-1/n} \left (1-\left (\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a}+1\right )^{-p}\right )}{b}+1\right )}{p+1} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.23, size = 336, normalized size = 8.8 \begin{align*}{\frac{x}{1+p} \left ( b{{\rm e}^{-{\frac{i\pi \,{\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( i{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( i{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+2\,n\ln \left ( x \right ) -2\,\ln \left ( c \right ) -2\,\ln \left ({x}^{n} \right ) }{2\,n}}}}x+a \right ) ^{p}}+{\frac{a}{ \left ( 1+p \right ) \sqrt [n]{c}b} \left ( b{{\rm e}^{-{\frac{i\pi \,{\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( i{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( i{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+2\,n\ln \left ( x \right ) -2\,\ln \left ( c \right ) -2\,\ln \left ({x}^{n} \right ) }{2\,n}}}}x+a \right ) ^{p}{{\rm e}^{{\frac{i\pi \,{\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( i{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( i{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+2\,n\ln \left ( x \right ) -2\,\ln \left ({x}^{n} \right ) }{2\,n}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55481, size = 80, normalized size = 2.11 \begin{align*} \frac{{\left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{\left (b c^{\left (\frac{1}{n}\right )} x + a\right )}^{p}}{{\left (b p + b\right )} c^{\left (\frac{1}{n}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2327, size = 73, normalized size = 1.92 \begin{align*} \frac{{\left (b c^{\left (\frac{1}{n}\right )} x + a\right )}^{p} b c^{\left (\frac{1}{n}\right )} x +{\left (b c^{\left (\frac{1}{n}\right )} x + a\right )}^{p} a}{b c^{\left (\frac{1}{n}\right )} p + b c^{\left (\frac{1}{n}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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