Optimal. Leaf size=35 \[ \frac {x \left (a+d x^3\right )^{1+n} \, _2F_1\left (1,\frac {4}{3}+n;\frac {4}{3};-\frac {d x^3}{a}\right )}{a} \]
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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 = {252, 251}
\begin {gather*} 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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 251
Rule 252
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] Result contains higher order function than in optimal. Order 6 vs. order 5 in
optimal.
time = 0.13, size = 196, normalized size = 5.60 \begin {gather*} \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 F_1\left (1+n;-n,-n;2+n;-\frac {i \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{d} x\right )}{\sqrt {3} \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)} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.03, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.41, size = 11, normalized size = 0.31 \begin {gather*} {\rm integral}\left ({\left (d x^{3} + a\right )}^{n}, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 4.91, size = 34, normalized size = 0.97 \begin {gather*} \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 {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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