Integrand size = 9, antiderivative size = 35 \[ \int \left (a+d x^3\right )^n \, dx=\frac {x \left (a+d x^3\right )^{1+n} \operatorname {Hypergeometric2F1}\left (1,\frac {4}{3}+n,\frac {4}{3},-\frac {d x^3}{a}\right )}{a} \]
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Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.26, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {252, 251} \[ \int \left (a+d x^3\right )^n \, dx=x \left (a+d x^3\right )^n \left (\frac {d x^3}{a}+1\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},-n,\frac {4}{3},-\frac {d x^3}{a}\right ) \]
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Rule 251
Rule 252
Rubi steps \begin{align*} \text {integral}& = \left (\left (a+d x^3\right )^n \left (1+\frac {d x^3}{a}\right )^{-n}\right ) \int \left (1+\frac {d x^3}{a}\right )^n \, dx \\ & = x \left (a+d x^3\right )^n \left (1+\frac {d x^3}{a}\right )^{-n} \, _2F_1\left (\frac {1}{3},-n;\frac {4}{3};-\frac {d x^3}{a}\right ) \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 0.14 (sec) , antiderivative size = 203, normalized size of antiderivative = 5.80 \[ \int \left (a+d x^3\right )^n \, dx=\frac {2^{-n} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{d} x\right ) \left (\frac {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{d} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}\right )^{-n} \left (\frac {i \left (1+\frac {\sqrt [3]{d} x}{\sqrt [3]{a}}\right )}{3 i+\sqrt {3}}\right )^{-n} \left (a+d x^3\right )^n \operatorname {AppellF1}\left (1+n,-n,-n,2+n,-\frac {(-1)^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{d} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}},\frac {i+\sqrt {3}-\frac {2 i \sqrt [3]{d} x}{\sqrt [3]{a}}}{3 i+\sqrt {3}}\right )}{\sqrt [3]{d} (1+n)} \]
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\[\int \left (x^{3} d +a \right )^{n}d x\]
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\[ \int \left (a+d x^3\right )^n \, dx=\int { {\left (d x^{3} + a\right )}^{n} \,d x } \]
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Result contains complex when optimal does not.
Time = 5.13 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.97 \[ \int \left (a+d x^3\right )^n \, dx=\frac {a^{n} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, - n \\ \frac {4}{3} \end {matrix}\middle | {\frac {d x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} \]
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\[ \int \left (a+d x^3\right )^n \, dx=\int { {\left (d x^{3} + a\right )}^{n} \,d x } \]
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\[ \int \left (a+d x^3\right )^n \, dx=\int { {\left (d x^{3} + a\right )}^{n} \,d x } \]
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Time = 10.48 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.17 \[ \int \left (a+d x^3\right )^n \, dx=\frac {x\,{\left (d\,x^3+a\right )}^n\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},-n;\ \frac {4}{3};\ -\frac {d\,x^3}{a}\right )}{{\left (\frac {d\,x^3}{a}+1\right )}^n} \]
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