Integrand size = 19, antiderivative size = 126 \[ \int x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\frac {a^2 x^3 \left (c x^n\right )^{-3/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{1+p}}{b^3 (1+p)}-\frac {2 a x^3 \left (c x^n\right )^{-3/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{2+p}}{b^3 (2+p)}+\frac {x^3 \left (c x^n\right )^{-3/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{3+p}}{b^3 (3+p)} \]
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Time = 0.04 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {375, 45} \[ \int x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\frac {a^2 x^3 \left (c x^n\right )^{-3/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{p+1}}{b^3 (p+1)}-\frac {2 a x^3 \left (c x^n\right )^{-3/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{p+2}}{b^3 (p+2)}+\frac {x^3 \left (c x^n\right )^{-3/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{p+3}}{b^3 (p+3)} \]
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Rule 45
Rule 375
Rubi steps \begin{align*} \text {integral}& = \left (x^3 \left (c x^n\right )^{-3/n}\right ) \text {Subst}\left (\int x^2 (a+b x)^p \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right ) \\ & = \left (x^3 \left (c x^n\right )^{-3/n}\right ) \text {Subst}\left (\int \left (\frac {a^2 (a+b x)^p}{b^2}-\frac {2 a (a+b x)^{1+p}}{b^2}+\frac {(a+b x)^{2+p}}{b^2}\right ) \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right ) \\ & = \frac {a^2 x^3 \left (c x^n\right )^{-3/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{1+p}}{b^3 (1+p)}-\frac {2 a x^3 \left (c x^n\right )^{-3/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{2+p}}{b^3 (2+p)}+\frac {x^3 \left (c x^n\right )^{-3/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{3+p}}{b^3 (3+p)} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.75 \[ \int x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\frac {x^3 \left (c x^n\right )^{-3/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{1+p} \left (2 a^2-2 a b (1+p) \left (c x^n\right )^{\frac {1}{n}}+b^2 \left (2+3 p+p^2\right ) \left (c x^n\right )^{2/n}\right )}{b^3 (1+p) (2+p) (3+p)} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 14.90 (sec) , antiderivative size = 787, normalized size of antiderivative = 6.25
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Time = 0.44 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.02 \[ \int x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=-\frac {{\left (2 \, a^{2} b c^{\left (\frac {1}{n}\right )} p x - {\left (b^{3} p^{2} + 3 \, b^{3} p + 2 \, b^{3}\right )} c^{\frac {3}{n}} x^{3} - {\left (a b^{2} p^{2} + a b^{2} p\right )} c^{\frac {2}{n}} x^{2} - 2 \, a^{3}\right )} {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p}}{{\left (b^{3} p^{3} + 6 \, b^{3} p^{2} + 11 \, b^{3} p + 6 \, b^{3}\right )} c^{\frac {3}{n}}} \]
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\[ \int x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\int x^{2} \left (a + b \left (c x^{n}\right )^{\frac {1}{n}}\right )^{p}\, dx \]
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\[ \int x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\int { {\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{p} x^{2} \,d x } \]
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Time = 0.35 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.93 \[ \int x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\frac {{\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} b^{3} c^{\frac {3}{n}} p^{2} x^{3} + {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} a b^{2} c^{\frac {2}{n}} p^{2} x^{2} + 3 \, {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} b^{3} c^{\frac {3}{n}} p x^{3} + {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} a b^{2} c^{\frac {2}{n}} p x^{2} + 2 \, {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} b^{3} c^{\frac {3}{n}} x^{3} - 2 \, {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} a^{2} b c^{\left (\frac {1}{n}\right )} p x + 2 \, {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} a^{3}}{b^{3} c^{\frac {3}{n}} p^{3} + 6 \, b^{3} c^{\frac {3}{n}} p^{2} + 11 \, b^{3} c^{\frac {3}{n}} p + 6 \, b^{3} c^{\frac {3}{n}}} \]
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Timed out. \[ \int x^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\int x^2\,{\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )}^p \,d x \]
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