Optimal. Leaf size=34 \[ \frac {x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^3}{3 b} \]
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Rubi [A] time = 0.01, antiderivative size = 34, 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} \[ \frac {x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^3}{3 b} \]
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 )^2 \, dx &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int (a+b x)^2 \, 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 )^3}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 38, normalized size = 1.12 \[ a^2 x+a b x \left (c x^n\right )^{\frac {1}{n}}+\frac {1}{3} b^2 x \left (c x^n\right )^{2/n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 32, normalized size = 0.94 \[ \frac {1}{3} \, b^{2} c^{\frac {2}{n}} x^{3} + a b c^{\left (\frac {1}{n}\right )} x^{2} + a^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 32, normalized size = 0.94 \[ \frac {1}{3} \, b^{2} c^{\frac {2}{n}} x^{3} + a b c^{\left (\frac {1}{n}\right )} x^{2} + a^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.21, size = 0, normalized size = 0.00 \[ \int \left (b \left (c \,x^{n}\right )^{\frac {1}{n}}+a \right )^{2}\, dx \]
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 \[ b^{2} c^{\frac {2}{n}} \int {\left (x^{n}\right )}^{\frac {2}{n}}\,{d x} + 2 \, a b c^{\left (\frac {1}{n}\right )} \int {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )}\,{d x} + a^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.23, size = 36, normalized size = 1.06 \[ a^2\,x+\frac {b^2\,x\,{\left (c\,x^n\right )}^{2/n}}{3}+a\,b\,x\,{\left (c\,x^n\right )}^{1/n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.43, size = 39, normalized size = 1.15 \[ a^{2} x + a b c^{\frac {1}{n}} x \left (x^{n}\right )^{\frac {1}{n}} + \frac {b^{2} c^{\frac {2}{n}} x \left (x^{n}\right )^{\frac {2}{n}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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