Integrand size = 19, antiderivative size = 171 \[ \int x^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=-\frac {a^3 x^4 \left (c x^n\right )^{-4/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{1+p}}{b^4 (1+p)}+\frac {3 a^2 x^4 \left (c x^n\right )^{-4/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{2+p}}{b^4 (2+p)}-\frac {3 a x^4 \left (c x^n\right )^{-4/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{3+p}}{b^4 (3+p)}+\frac {x^4 \left (c x^n\right )^{-4/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{4+p}}{b^4 (4+p)} \]
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Time = 0.06 (sec) , antiderivative size = 171, 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} \[ \int x^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=-\frac {a^3 x^4 \left (c x^n\right )^{-4/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{p+1}}{b^4 (p+1)}+\frac {3 a^2 x^4 \left (c x^n\right )^{-4/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{p+2}}{b^4 (p+2)}-\frac {3 a x^4 \left (c x^n\right )^{-4/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{p+3}}{b^4 (p+3)}+\frac {x^4 \left (c x^n\right )^{-4/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{p+4}}{b^4 (p+4)} \]
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Rule 45
Rule 375
Rubi steps \begin{align*} \text {integral}& = \left (x^4 \left (c x^n\right )^{-4/n}\right ) \text {Subst}\left (\int x^3 (a+b x)^p \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right ) \\ & = \left (x^4 \left (c x^n\right )^{-4/n}\right ) \text {Subst}\left (\int \left (-\frac {a^3 (a+b x)^p}{b^3}+\frac {3 a^2 (a+b x)^{1+p}}{b^3}-\frac {3 a (a+b x)^{2+p}}{b^3}+\frac {(a+b x)^{3+p}}{b^3}\right ) \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right ) \\ & = -\frac {a^3 x^4 \left (c x^n\right )^{-4/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{1+p}}{b^4 (1+p)}+\frac {3 a^2 x^4 \left (c x^n\right )^{-4/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{2+p}}{b^4 (2+p)}-\frac {3 a x^4 \left (c x^n\right )^{-4/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{3+p}}{b^4 (3+p)}+\frac {x^4 \left (c x^n\right )^{-4/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{4+p}}{b^4 (4+p)} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.66 \[ \int x^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\frac {x^4 \left (c x^n\right )^{-4/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{1+p} \left (-\frac {a^3}{1+p}+\frac {3 a^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{2+p}-\frac {3 a \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2}{3+p}+\frac {\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^3}{4+p}\right )}{b^4} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 5.93 (sec) , antiderivative size = 1127, normalized size of antiderivative = 6.59
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none
Time = 0.37 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.07 \[ \int x^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\frac {{\left (6 \, a^{3} b c^{\left (\frac {1}{n}\right )} p x + {\left (b^{4} p^{3} + 6 \, b^{4} p^{2} + 11 \, b^{4} p + 6 \, b^{4}\right )} c^{\frac {4}{n}} x^{4} + {\left (a b^{3} p^{3} + 3 \, a b^{3} p^{2} + 2 \, a b^{3} p\right )} c^{\frac {3}{n}} x^{3} - 6 \, a^{4} - 3 \, {\left (a^{2} b^{2} p^{2} + a^{2} b^{2} p\right )} c^{\frac {2}{n}} x^{2}\right )} {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p}}{{\left (b^{4} p^{4} + 10 \, b^{4} p^{3} + 35 \, b^{4} p^{2} + 50 \, b^{4} p + 24 \, b^{4}\right )} c^{\frac {4}{n}}} \]
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\[ \int x^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\int x^{3} \left (a + b \left (c x^{n}\right )^{\frac {1}{n}}\right )^{p}\, dx \]
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\[ \int x^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\int { {\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{p} x^{3} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (179) = 358\).
Time = 0.36 (sec) , antiderivative size = 384, normalized size of antiderivative = 2.25 \[ \int x^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\frac {{\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} b^{4} c^{\frac {4}{n}} p^{3} x^{4} + {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} a b^{3} c^{\frac {3}{n}} p^{3} x^{3} + 6 \, {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} b^{4} c^{\frac {4}{n}} p^{2} x^{4} + 3 \, {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} a b^{3} c^{\frac {3}{n}} p^{2} x^{3} + 11 \, {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} b^{4} c^{\frac {4}{n}} p x^{4} - 3 \, {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} a^{2} b^{2} c^{\frac {2}{n}} p^{2} x^{2} + 2 \, {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} a b^{3} c^{\frac {3}{n}} p x^{3} + 6 \, {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} b^{4} c^{\frac {4}{n}} x^{4} - 3 \, {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} a^{2} b^{2} c^{\frac {2}{n}} p x^{2} + 6 \, {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} a^{3} b c^{\left (\frac {1}{n}\right )} p x - 6 \, {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} a^{4}}{b^{4} c^{\frac {4}{n}} p^{4} + 10 \, b^{4} c^{\frac {4}{n}} p^{3} + 35 \, b^{4} c^{\frac {4}{n}} p^{2} + 50 \, b^{4} c^{\frac {4}{n}} p + 24 \, b^{4} c^{\frac {4}{n}}} \]
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Timed out. \[ \int x^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\int x^3\,{\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )}^p \,d x \]
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