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 = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, 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 32
Rule 254
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*}
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Mathematica [A] time = 0.15, 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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fricas [A] time = 0.90, size = 37, normalized size = 0.97 \[ \frac {{\left (b c^{\left (\frac {1}{n}\right )} x + a\right )} {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p}}{{\left (b p + b\right )} c^{\left (\frac {1}{n}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 54, normalized size = 1.42 \[ \frac {{\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} b c^{\left (\frac {1}{n}\right )} x + {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} a}{b c^{\left (\frac {1}{n}\right )} p + b c^{\left (\frac {1}{n}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.16, size = 147, normalized size = 3.87 \[ \frac {x \,c^{-\frac {1}{n}} \left (x^{n}\right )^{-\frac {1}{n}} \left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}}+a \right )^{p +1} {\mathrm e}^{-\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (-\mathrm {csgn}\left (i x^{n}\right )+\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 n}}}{\left (p +1\right ) b} \]
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 \[ \int {\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int {\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )}^p \,d x \]
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 \[ \int \left (a + b \left (c x^{n}\right )^{\frac {1}{n}}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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