Optimal. Leaf size=38 \[ \frac {x^2 \left (a+b x^3\right )^{5/3} \, _2F_1\left (1,\frac {7}{3};\frac {5}{3};-\frac {b x^3}{a}\right )}{2 a} \]
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Rubi [A]
time = 0.01, antiderivative size = 51, normalized size of antiderivative = 1.34, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {372, 371}
\begin {gather*} \frac {x^2 \left (a+b x^3\right )^{2/3} \, _2F_1\left (-\frac {2}{3},\frac {2}{3};\frac {5}{3};-\frac {b x^3}{a}\right )}{2 \left (\frac {b x^3}{a}+1\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 372
Rubi steps
\begin {align*} \int x \left (a+b x^3\right )^{2/3} \, dx &=\frac {\left (a+b x^3\right )^{2/3} \int x \left (1+\frac {b x^3}{a}\right )^{2/3} \, dx}{\left (1+\frac {b x^3}{a}\right )^{2/3}}\\ &=\frac {x^2 \left (a+b x^3\right )^{2/3} \, _2F_1\left (-\frac {2}{3},\frac {2}{3};\frac {5}{3};-\frac {b x^3}{a}\right )}{2 \left (1+\frac {b x^3}{a}\right )^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 5.94, size = 51, normalized size = 1.34 \begin {gather*} \frac {x^2 \left (a+b x^3\right )^{2/3} \, _2F_1\left (-\frac {2}{3},\frac {2}{3};\frac {5}{3};-\frac {b x^3}{a}\right )}{2 \left (1+\frac {b x^3}{a}\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int x \left (b \,x^{3}+a \right )^{\frac {2}{3}}\, dx\]
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 = 0.36, size = 13, normalized size = 0.34 \begin {gather*} {\rm integral}\left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} x, 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 = 0.41, size = 39, normalized size = 1.03 \begin {gather*} \frac {a^{\frac {2}{3}} x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {5}{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 [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int x\,{\left (b\,x^3+a\right )}^{2/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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