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 = 0.01, 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 245
Rule 246
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*}
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Mathematica [C] time = 0.20, size = 196, normalized size = 5.60 \[ \frac {2^{-n} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{d} x\right ) \left (\frac {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{d} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}\right )^{-n} \left (\frac {i \left (\frac {\sqrt [3]{d} x}{\sqrt [3]{a}}+1\right )}{\sqrt {3}+3 i}\right )^{-n} \left (a+d x^3\right )^n F_1\left (n+1;-n,-n;n+2;-\frac {i \left (\sqrt [3]{d} x+(-1)^{2/3} \sqrt [3]{a}\right )}{\sqrt {3} \sqrt [3]{a}},\frac {-\frac {2 i \sqrt [3]{d} x}{\sqrt [3]{a}}+\sqrt {3}+i}{3 i+\sqrt {3}}\right )}{\sqrt [3]{d} (n+1)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (d x^{3} + a\right )}^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x^{3} + a\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \left (d \,x^{3}+a \right )^{n}\, dx \]
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 \[ \int {\left (d x^{3} + a\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.18, size = 41, normalized size = 1.17 \[ \frac {x\,{\left (d\,x^3+a\right )}^n\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},-n;\ \frac {4}{3};\ -\frac {d\,x^3}{a}\right )}{{\left (\frac {d\,x^3}{a}+1\right )}^n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 10.54, size = 34, normalized size = 0.97 \[ \frac {a^{n} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, - n \\ \frac {4}{3} \end {matrix}\middle | {\frac {d x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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