Optimal. Leaf size=35 \[ \frac{x \left (a+d x^3\right )^{n+1} \, _2F_1\left (1,n+\frac{4}{3};\frac{4}{3};-\frac{d x^3}{a}\right )}{a} \]
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Rubi [A] time = 0.0088327, antiderivative size = 44, normalized size of antiderivative = 1.26, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {246, 245} \[ x \left (a+d x^3\right )^n \left (\frac{d x^3}{a}+1\right )^{-n} \, _2F_1\left (\frac{1}{3},-n;\frac{4}{3};-\frac{d x^3}{a}\right ) \]
Antiderivative was successfully verified.
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Rule 246
Rule 245
Rubi steps
\begin{align*} \int \left (a+d x^3\right )^n \, dx &=\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*}
Mathematica [C] time = 0.150824, size = 196, normalized size = 5.6 \[ \frac{2^{-n} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{d} x\right ) \left (\frac{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{d} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}\right )^{-n} \left (\frac{i \left (\frac{\sqrt [3]{d} x}{\sqrt [3]{a}}+1\right )}{\sqrt{3}+3 i}\right )^{-n} \left (a+d x^3\right )^n F_1\left (n+1;-n,-n;n+2;-\frac{i \left (\sqrt [3]{d} x+(-1)^{2/3} \sqrt [3]{a}\right )}{\sqrt{3} \sqrt [3]{a}},\frac{-\frac{2 i \sqrt [3]{d} x}{\sqrt [3]{a}}+\sqrt{3}+i}{3 i+\sqrt{3}}\right )}{\sqrt [3]{d} (n+1)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.029, size = 0, normalized size = 0. \begin{align*} \int \left ( d{x}^{3}+a \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x^{3} + a\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (d x^{3} + a\right )}^{n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 16.0086, size = 34, normalized size = 0.97 \begin{align*} \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x^{3} + a\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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